Contents
Chapter 1. What is differential topology? 1
Chapter 2. Embedding into Euclidean space 3
Chapter 3. Tubular neighbourhoods 7
3.A. Smoothening maps 11
3.B. Classification of 1-dimensional manifolds 12
Chapter 4. Sard’s theorem 15
Chapter 5. Transversality 19
5.A. Classifying vector bundles 23
Chapter 6. Degree of a map 29
Chapter 7. Pontryagin-Thom construction 32
7.A. Possible modifications of the Thom’s theorem 38
Chapter 8. Index of a vector field 40
8.A. Morse theory 43
Chapter 9. Function spaces 48
0
CHAPTER 1
What is differential topology?
The main subject is the study of topological (i.e. global) properties of smooth manifolds.
Typical questions involve:
• Is it possible to embed every smooth manifold in some Rk
, k 0?
• What properties of smooth maps are generic? A typical example is transversality,
e.g. transversality to 0 is demonstrated by the following two examples
pictures of x2
(x − 3) (not generic) and (x + 1)(x − 1)(x − 3) (generic)
The second function is generic because changing it slightly does not change the
situation. However a perturbation of the first function will either miss the x-axis
or will have two “generic” intersections.
• Classification of manifolds. Already Riemann in the late 19th
century described
all (closed) surfaces. They are classified by orientability and genus (“the number
of holes”).
picture of a sphere with a handle One can read this information off homology.
The next step is dimension 3. Poincar´e was pursuing this question. He asked
whether Betty numbers (the dimensions of the free parts of homology - i.e. all that
people at that time knew of homology) where enough to distinguish (compact and
oriented) non-diffeomorphic 3-manifolds. He showed that it is not. In fact even
the homology groups do not suffice (an example supplied by Poincar´e): if M is
a manifold with a perfect group ([π1(M), π1(M)] = π1(M)) then H1(M) = 0 =
H2(M) by Poincar´e duality and the manifold has the same homology as S3
(it is socalled
homology sphere) but is not even homotopy equivalent to it as π1(S3
) = 0.
His next question was the following. Is every compact simply connected 3-manifold
diffeomorphic to S3
? This is known as Poincar´e conjecture and was solved only
100 years later by Perelman.
Remark: one can always find a map Mm
→ Sm
of degree 1. If π1(M) = 0 and
H∗(M) ∼= H∗(Sm
) then this map is a homotopy equivalence.
We have the following diagram relating a 3-manifold M and S3
diffeo ⇒ homeo ⇒ htpy equiv ⇒ hlgy iso
Concerning the reverse implications: the last one is valid for simply connected
M, the middle one is what is the real Poincar’e conjecture. This is very difficult in
dimension 3 (and also 4), easier for dimension ≥ 5 but still very hard. The reverse
of the first implications holds in dimension 3 but is not true in higher dimensions.
1
1. WHAT IS DIFFERENTIAL TOPOLOGY? 2
For example on S7
there are 28 distinct differentiable structures (when one is
concerned about orientations, otherwise 15).
The conclusion should be now that the classification problem is too difficult.
Another idea of Poincar´e (still in the 19th
century) was to consider a much
weaker relation on compact manifolds: M is called bordant to N, denoted M N,
if the disjoint union M N ∼= ∂W is a boundary of some compact manifold W
with boundary. For example Sm
∅ as Sm
= ∂Dm+1
.
Classify manifolds up to bordism! We denote the set of bordism classes of
manifolds by N∗ and observe that it has a natural structure of a ring with respect
to and ×. In 50’s Thom showed that N∗ is isomorphic to homotopy groups
π∗(MO) of some space (or rather spectrum) MO. Moreover Thom was able to
compute this ring and
N∗
∼= Z/2[xn | n ≥ 2, n = 2t
− 1]
where deg(xn) = n i.e. xn is represented by a manifold of dimension n. At the
same time Pontryagin identified a similar bordism ring Ωfr
∗ of the so-called framed
manifolds with the stable homotopy groups of spheres, the main object of study
of algebraic topology. The way to prove both these identifications is the same
and is called the Pontryagin-Thom construction. This will be our first goal in the
lecture.
CHAPTER 2
Embedding into Euclidean space
Manifold will always be meant to be smooth, Hausdorff and second countable1
.
Definition 2.1. A topological space X is called paracompact if every open covering
of X has a locally finite refinement.
Here V is a refinement of U if ∀V ∈ V ∃U ∈ U : V ⊆ U. A covering U is called locally
finite if every point x ∈ X has a neighbourhood V which intersects only a finite number
of elements of U, i.e. {U ∈ U | U ∩ V = ∅} is finite.
Theorem 2.2. For a connected Hausdorff locally Euclidean space X the following conditions
are equivalent
a) X is second countable
b) X = ∞
i=0 Ki where each Ki+1 is a compact neighbourhood of Ki
c) X is paracompact
Proof. “a)⇒b)”: We need the following
Sublemma. If X is second countable then every open covering has a countable sub-
covering.
Proof of the sublemma. Let U be a covering and {Vi}∞
i=0 a basis for the topology.
For each Vi choose (if possible) U ∈ U such that Vi ⊆ U and call it Ui. Then the claim is
that {Ui}∞
i=0 is a subcovering. nice picture
Let us continue with the proof of the theorem now. Cover X by open sets Ui with
compact closure and we can assume that this collection is countable. We construct Ki
inductively starting with K0 = ¯U0. In the inductive step cover Ki by Uj1 , . . . , Ujn and form
Ki+1 = ¯Ui+1 ∪ n
k=1 Ujk
.
“b)⇒c)”: Let U be any covering. For each i let Ui
j1
, . . . , Ui
jni
cover Ki − int Ki−1. Then
{Ui
jk
∩ (int Ki+1 − Ki−1) | i = 0, . . . ; k = 1, . . . , ni}
does the job. nice picture
The reverse implications are similar.
Definition 2.3. A (smooth) partition of unity subordinate to a covering U of M is a
collection of smooth functions λi : M → R, i ∈ I, such that the following conditions are
satisfied
• supp λi is contained in some set U ∈ U from the covering
1having a countable basis for its topology
3
2. EMBEDDING INTO EUCLIDEAN SPACE 4
• the collection {supp λi | i ∈ I} is locally finite
• λi = 1 which makes sense thanks to the local finiteness property
Remark. If λi is a partition of unity subordinate to V and V is a refinement of U then
λi is also a partition of unity subordinate to U.
Remark. Choosing for each λi a definite U ∈ U for which supp λi ⊆ U we can add
those λi’s corresponding to a single U to get λU with supp λU ⊆ U. In this way we get a
partition of unity which is indexed by the covering U itself.
Now we can start proving the following theorem.
Theorem 2.4. Partitions of unity exist subordinate to any open covering.
If I am not mistaken then ρ(x) = e− 1
x+1 e− 1
1−x should be a function which is positive on
(−1, 1) and zero elsewhere. Similarly we get a function ρ(x1) · · · ρ(xm) : Rm
→ R, positive
on (−1, 1)m
and zero elsewhere. pictures
Lemma 2.5. Let K ⊆ U ⊆ Rm
with K compact and U open. Then there exists a smooth
function λ : Rm
→ R+ = [0, ∞) such that λ is positive on K and supp λ ⊆ U.
Proof. Cover K by a finite number of open cubes Vi. Modifying ρ slightly we find λi
which is positive on Vi and zero elsewhere. Then put λ = λi.
Proof of Theorem 2.4. Enough to prove for U locally finite and with U ∈ U ⇒ ¯U
compact and contained in a coordinate chart. This is so as for any U we can find a
refinement of this form. Also we can assume that U is countable as any covering possesses
a countable subcovering. Therefore let U = {Ui}∞
i=0. We construct λi’s inductively. We
start with λ0 any function which is positive on U0 − ∞
i=1 Ui and with supp λ0 ⊆ U0. We
denote V0 := λ−1
0 (0, ∞). In the inductive step we choose a function λj which is positive on
Uj − j−1
i=0 Vi − ∞
i=j+1 Ui and with supp λj ⊆ Uj. Then we denote Vj := λ−1
j (0, ∞) and the
induction step is finished. Clearly the local finiteness is satisfied as U is supposed to be
such. By construction ∞
i=0 λi > 0 on M. To finish the proof we replace λj by
λj
∞
i=0 λi
.
Theorem 2.6. Let M be a compact manifold. Then there exists an embedding of M
into Rk
for some k 0.
Proof. Let ϕi : Ui → Rm
, i = 1, . . . , k, be a covering of M by coordinate charts
(i.e. k
i=1 Ui = M. Let λi, i = 1, . . . , k be a partition of unity subordinate to U = {Ui}k
i=1.
The claim is that the map
j = (λ1, λ1 · ϕ1, . . . , λk, λk · ϕk) : M → R(m+1)k
(with λi · ϕi the obvious extension - by zero - of the map with the same name from Ui to
M). Clearly j is injective: if x and y are two points with j(x) = j(y) then certainly there
is i such that 0 = λi(x) = λi(y). Then in λi(x) · ϕi(x) = λi(y) · ϕi(y) we can divide by this
common value to conclude that ϕi(x) = ϕi(y) and then refer to the injectivity of ϕi. The
2. EMBEDDING INTO EUCLIDEAN SPACE 5
proof of injectivity of the tangential map follows the same idea: let (x, v) ∈ TxM be such
that j∗(x, v) = 0. Then all (λi)∗(x, v) = 0 and therefore
0 = (λi · ϕi)∗(x, v) = λi(x) · (ϕi)∗(x, v) + (λi)∗(x, v) · ϕi(x) = λi(x) · (ϕi)∗(x, v)
Again one of the λi(x) is nonzero and therefore (ϕi)∗(x, v) must be zero. But as ϕi is an
embedding this implies v = 0.
Aside. An embedding j : M → N is a diffeomorphism of M with a submanifold
j(M) ⊆ N. Equivalently (HW) j is an injective immersion and a homeomorphism onto its
image. Therefore for M compact the notions of an embedding and an injective immersion
coincide.
It is possible to decrease k from the theorem to 2m + 1. We need (an easy version of)
Sard’s theorem.
Definition 2.7. A subset S ⊆ Rm
has measure 0 if for each ε > 0, S can be covered
by a sequence Ci of cubes with
∞
i=0
Vol(Ci) < ε
Remark. This notion is closed under countable unions.
Remark. If S has measure 0 then S does not contain any nonempty open subset.
Lemma 2.8. Let f : U → Rm
, U ⊆ Rm
, be a smooth map and S ⊆ U a subset of
measure 0. Then f(S) has measure 0.
Proof. Cut U into countably many compact cubes Ci On each Ci the derivatives of
f are bounded, i.e.
|fu(x)| ≤ Ki ∀x ∈ Ci, u ∈ Sm−1
nice picture Now we compute
|f(x + v) − f(x)| =
1
0
fv(x + tv)dt ≤
1
0
|fv(x + tv)|dt ≤
1
0
Ki|v|dt = Ki|v|
In other words the lengths increases at most Ki-times, the image of a cube of side a lies
in a cube of side a
√
mKi. If S ∩ Ci is covered by cubes of total volume ε then f(S ∩ Ci)
is covered by cubes of total volume ε · (
√
mKi)m
which can be made arbitrarily small and
thus f(S ∩ Ci) has measure 0. As f(S) = i f(S ∩ Ci) the proof is finished.
Corollary 2.9. The property of having measure 0 is diffeomorphism invariant. In
other words it does not depend on the chart.
Therefore the following definition makes sense.
Definition 2.10. A subset S ⊆ M of a smooth manifold M has measure 0 if it is so
in each chart.
2. EMBEDDING INTO EUCLIDEAN SPACE 6
Theorem 2.11 (Sard’s theorem, easy version). Let f : Mm
→ Nn
be a smooth map
with m < n. Then im f has measure 0 in N.2
Proof. This is a local problem and so we can assume
f : Rm
→ Rn
extend f to a map
Rn
= Rm
× Rn−m pr1
−−−→ Rm f
−−→ Rn
and observe that im f = f(Rm
× 0) and Rm
× 0 has measure 0 in Rn
.
Theorem 2.12. Any compact smooth manifold Mm
can be embedded into R2m+1
.
Remark. M being compact is not important (but M being second countable is).
Remark. In fact M can be embedded into R2m
but the proof given here could not
work picture of a trefoil knot where the projection as in the proof does not exist.
Proof. Let M be embedded in some Rk
. This is possible by Theorem 2.6. The strategy
is to find a direction (or rather a line ) in which to project to reduce the dimension k by
1.
M   j
// Rk p
−−→ Rk
/
The problem is to make sure that p◦j is still an embedding. This is possible if k > 2m+1:
• p ◦ j injective: consider the following map
M × M − ∆
g
−−→ RPk−1
(x, y) −→ [j(x) − j(y)]
The injectivity of p ◦ j is equivalent to ∈ im g.
• p ◦ j immersion: consider the following map
STM
h
−−→ RPk−1
(x, v) −→ [j∗(x, v)]
Then equivalently ∈ im h.
All together, p ◦ j is an embedding iff ∈ im g ∪ im h. If k > 2m + 1 then im g ∪ im h has
measure 0 by Theorem 2.11 and thus its complement is nonempty.
Remark. The same proof shows that Mm
can be immersed into R2m
.
2Here it is important that M is second countable. Taking M = N but with discrete topology and
f = id is an example where the theorem would fail otherwise.
CHAPTER 3
Tubular neighbourhoods
Our situation in this chapter is that we have a submanifold M ⊆ N and we want to
describe a neighbourhood of M in N in a nice way.
Digression (about Riemannian metrics).
Theorem 3.1. Every manifold possesses a Riemannian metric.
Proof. The simplest proof would be: embed M into R2m+1
and induce a metric
from there. But we only proved the existence of an embedding for compact manifolds. We
exhibit a different proof: let Ui be a covering of M by charts and λi a subordinate partition
of unity. On Ui we can choose a Riemannian metric gi using the chart. Then the required
Riemannian metric on M is i λigi. This is possible because the set of Riemannian metrics
in T∗
x M ⊗ T∗
x M is convex.
Every manifold M has a metric (is a metric space). A possible proof refers to a general
statement that every second countable completely regular topological space is metrizable.
We give a geometric proof of this fact for M connected. Let g be a Riemannian metric on
M and define a length of a piecewise smooth curve γ : [0, 1] → M by the formula
(γ) =
1
0
g(˙γ(t), ˙γ(t))dt
where ˙γ : [0, 1] → TM denotes the derivative of γ (evaluate the tangential mapping
γ∗ : [0, 1] × R → TM on unit vectors (x, 1)).
Now we can define a metric on M by a formula
dg(x, y) = inf{ (γ) | γ : [0, 1] → M piecewise smooth, γ(0) = x, γ(1) = y}
To see that the topology induced by dg is correct go local. Then in Rm
the metric dg is
equivalent to the standard metric which is de for the standard Riemannian metric on Rm
.
The equivalence comes from a relation Ke ≤ g ≤ Le (for some K, L > 0) holding near
some fixed point. The same relation then holds for the induced metrics: Kde ≤ dg ≤ Lde
and so they induce the same topology.
Note. A geodesic γ is a solution to a certain second order differential equation
˙γ ˙γ = 0
where is the Levi-Civita connection associated with the Riemannian metric.
7
3. TUBULAR NEIGHBOURHOODS 8
For (x, v) ∈ TM denote by
t → ϕ(t, x, v)
the geodesic starting at x with velocity v, i.e. ˙γ(0) = (x, v). The curve ϕ(t, x, v) is defined
fot t in some neighbourhood of 0. Also ϕ(t, x, v) only depends on tv, not on t and v
separately
ϕ(t, x, v) = ϕ(1, x, tv) = expx(tv)
under a notation expx v = ϕ(1, x, v). We get a smooth map
V
⊆
ϕ
// M
R × TM
with V an open neighbourhood of 0 × TM. If C ⊆ M is a compact subset then STM|C is
also compact and there exists ε > 0 such that ϕ(t, x, v) = expx tv is defined for all x ∈ C,
v ∈ TxM with |v| = 1 and |t| < ε. In other words expx v is defined for all x ∈ C and
v ∈ TxM with |v| < ε.
Alltogether exp is defined on a neighbourhood of the zero section M ⊆ TM
U
⊆
exp
// M
TM
Consider the map
(π, exp) : U −→ M × M
(x, v) −→ (x, expx v)
We claim that (π, exp) is a diffeomorphism near the zero section. Clearly when restrcted
to the zero section M ⊆ TM the map (π, exp) coincides with the diagonal embedding
∆ : M → M × M. We check what its differential at the zero section is. First we observe
that at a zero section the tangent bundle of any vector bundle canonically splits
Lemma 3.2. Let p : E → M be a vector bundle. Then there is a short exact sequence
of vector bundles over E
0 → p∗
E → TE → p∗
TM → 0
which splits canonically at the zero section.
Proof. First we define the maps in the sequence. We think of the pullback p∗
TM as
a subset of E ×TM and write its elements as pairs. Similarly for p∗
E but here we must be
careful: the first coordinate serves as the point in the base whereas the second coordinate
as the vector. The first map is
(u, v) → d
dt t=0
(u + tv)
This map is clearly injective and its image is called the vertical subbundle of TE. The
second map in the sequence is induced by projection p∗ : TE → TM to the base. Clearly
3. TUBULAR NEIGHBOURHOODS 9
this sequence is exact. Restricting to the zero section M ⊆ E we obtain a short exact
sequence
0 → E → TE|M → TM → 0
which splits by the map j∗ : TM → TE|M induced by the inclusion j : M ⊆ E. Its image
is called the horizontal subbundle.
Now we compute (π, exp)∗ at the zero section. On the vertical subbundle
(π, exp)∗((x, 0), (0, v)) = ((x, x), (0, v))
(this is d
dt t=0
(x, expx tv)) and on the horizontal subbundle
(π, exp)∗((x, 0), (v, 0)) = ((x, x), (v, v))
(which is ∆∗(x, v)). Such vectors clearly span T(x,x)(M × M) and we can conclude that
(π, exp) is an embedding of a neighbourhood of the zero section M ⊆ TM onto a neighbourhood
of the diagonal ∆(M) ⊆ M × M by the following lemma.
Lemma 3.3. Let M ⊆ N be a compact submanifold, f : N → P a smooth map. If f is
an injective immersion at M (f|M injective and (f∗)|M injective rather than (f|M )∗ injective
- not sufficient) then there is a neighbourhood U of N such that f|U is an embedding.
Proof. Only need that there is a neighbourhood of M on which f is injective as M has
a final system of compact neighbourhoods: if f|V is injective choose an open neighbourhood
U of M for which ¯U is a compact subset of V and then f|¯U is a homeomorphism onto its
image and thus so is f|U . As we may also assume that f is an immersion on U, f|U is an
embedding.
Therefore assume that there is no neighbourhood of M on which f is injective. Remembering
that N is a metric space and exhibiting this for an 1/n-neighbourhood of M
we obtain sequences xi and yi of points in N such that f(xi) = f(yi). We can also assume
that both xi and yi converge, say to x and y respectively, both lying in M. As f|M is
injective we must have x = y and as f is an immersion at x necessarily xi = yi for i big
enough.
In fact with the same amount of effort one can prove a stronger result implying the same
conclusion, namely that (π, exp) maps a neighbourhood of M in TM diffeomorphically onto
a neighbourhood of ∆(M) in M × M, even for non-compact manifolds:
Lemma 3.4. Let M ⊆ N be a smooth submanifold, f : N → P a smooth map. If f
is a local diffeomorphism at M and f|M a homeomorphism onto its image then there is a
neighbourhood U of N such that f|U is an embedding.
Proof. Again we only need to show that f is injective on a neighbourhood of M.
Cover M by open sets Ui ⊆ N such that f is a diffeomorphism on a neighbourhood Vi of
each ¯Ui. Since f is also a homeomorphism we can (after restriction) assume that Ui satifies
f(Ui) ∩ f(M) = f(Ui ∩ M) and further that f(Ui) is locally finite. We denote U = Ui
and consider the following subset of P:
W = {y ∈ f(U) | if x, x ∈ U both belong to f−1
(y) then x = x }
3. TUBULAR NEIGHBOURHOODS 10
By construction f(M) ⊆ V and it is enough to show that W is a neighbourhood of f(M) as
then the (well-defined!) inverse of f is smooth by the implicit function theorem. Therefore
let y ∈ W. By local finiteness there is only a finite number of f(Ui)’s whose closure contains
y, let us denote the corresponding indices by i1, . . . , ik. In particular the unique (inside U)
preimage of y lies in the intersection ¯Ui1 ∩ · · · ∩ ¯Uik
. We claim that the open set
[f(U) ∩ f(Vi1 ∩ · · · ∩ Vik
)] −
i∈{i1...,ik}
f(Ui)
is contained in W. This is because for z from this set the only way f(x) = z could happen
for x ∈ U is that x ∈ Vi1 ∩ · · · ∩ Vik
and as f is injective on this set there is only one
possible choice of such x.
In general we consider now an embedding j : M   // N. A normal bundle ν(j) of j is
the vector bundle TM⊥
, the orthogonal complement of TM inside j∗
TN. This definition
depends on the choice of metric but it is possible to give one which does not: ν(j) =
j∗
TN/TM.
Remark. Let 0 → E → F → F/E → 0 be a short exact sequence of vector bundles
and choose a Riemannian metric on F. Then we get a diagram
0 // E // F // F/E // 0
E⊥
OO
∼=
<<
Therefore every short exact sequence of vector bundles (in general over a paracompact
topological space) split.
Example 3.5. For the diagonal embedding ∆ : M → M × M the normal bundle is
ν(∆) = (TM ⊕ TM)/TM ∼= TM
Here TM sits in TM ⊕TM diagonally. The orthogonal complement of TM is (if we impose
orthogonality of the two summands TM)
{(v, −v) ∈ TM ⊕ TM | v ∈ TM} ∼= TM
Example 3.6. Let j : M  
// Rk
be an embedding of M into Rk
and consider ν(j).
For any other embedding j : M   // Rk
and the corresponding ν(j ) we get a relation
ν(j) ⊕ (Rk
× M) ∼= ν(j) ⊕ TM ⊕ ν(j ) ∼= (Rk
× M) ⊕ ν(j )
Such vector bundles ν(j), ν(j ) are calles stably isomorphic (i.e. they are isomorphic after
adding trivial vector bundles - of possibly different dimensions). We also associate to ν(j)
a formal dimension −m, intuitively replacing the defining relation ν(j) ⊕ TM ∼= Rk
× M
by ν(j) ⊕ TM = 0 and therefore thinking of ν(j) as −TM.
3.A. SMOOTHENING MAPS 11
Construction 3.7. Thinking of ν(j) as TM⊥
, exp provides a map
exp : ˚DεTM⊥
−→ N
an embedding of a neighbourhood ˚DεTM⊥
of the zero section of TM⊥
as a neighbourhood
of M. The proof is exactly the same as for ∆ : M → M × M. Note that this construction
depends on the Riemannian metric on N.
Definition 3.8. A tubular neighbourhood of j : M   // N is a pair (E, ι) where p :
E → M is a smooth vector bundle and ι : E → N is an embedding of E as a neighbourhood
of M in such a way that ι|M = j (here M means the zero section).
Note. Here E must be isomorphic to the normal bundle of j:
E // TE|M
ι∗
// TN|M
p
// TM⊥
TM
OO
j∗
::
with p the orthogonal projection. As TE|M is a direct sum E ⊕ TM and pι∗|TM = pj∗ = 0
the composition across the top row must be an isomorphism.
Theorem 3.9. Every compact submanifold M ⊆ N has a tubular neighbourhood.
Proof. Use TM⊥ ∼= ˚DεTM⊥ exp
−−−→ N.
In particular a tubular neighbourhood provides a retraction
r : ι(E)
ι−1
−−−→ E
p
−−→ M
a nice picture of the retraction
Now we show an interesting application of tubular neighbourhoods (and in particular
of the above retraction).
3.A. Smoothening maps
Let M and N be compact manifolds and ι : ν(N) → Rk
a tubular neighbourhood of
some embedding N   // Rk
. Our aim is to find for each continuous map f0 : M → N a
smooth approximation, i.e. a smooth map f1 : M → N with a homotopy f : I × M → N
for which f(0, −) = f0 and f(1, −) = f1. In fact we will find a small homotopy f so that
f1 will be close to f0 but we will make this precise only later.
Theorem 3.10. Approximations exist for every f0 : M → N. If f0 is already smooth
in a neighbourhood of a closed subset K ⊂ M then the homotopy f can be chosen to be
constant on a (possibly smaller) neighbourhood of K (and in particular f1 ≡ f0 near A).
Proof. Find ε > 0 such that every Bε(x), x ∈ N lies in the tubular neighbourhood of
N. Cover M by an open covering U with the following properties:
• U ∈ U ⇒ ∀x, y ∈ U : |f0(x) − f0(y)| < ε
• U ∈ U ⇒ either U ⊆ M − K or f0|U is smooth
3.B. CLASSIFICATION OF 1-DIMENSIONAL MANIFOLDS 12
Let λU be a subordinate partition of unity and let gU be smooth maps U → Rk
satisfying
• ∀x ∈ U : |gU (x) − f0(x)| < ε
• if U ∩ K = ∅ then gU = f0
Such a collection exists - it is either dictated by the second condition or, when U ⊆ M −K,
one can choose x0 ∈ U and put gU (x) = f0(x0) constant. Define g : I × M → Rk
by the
following formula
g(t, x) = (1 − t)f0(x) + t
U∈U
λU (x)gU (x)
Easily im g lies in the tubular neighbourhood and one can compose with the retraction r
to obtain the required f(t, x) = r(g(t, x)).
3.B. Classification of 1-dimensional manifolds
The classification we present here works equally well for manifolds with boundary. First
we define all the needed notions (and slightly more).
In the same way manifolds are modeled on the Euclidean space Rm
manifolds with
boundary are modeled on the halfspace
Hm
= R− × Rm−1
= {(x1, . . . , xm) ∈ Rm
| x1 ≤ 0}
To do differential topology on manifolds with boundary it is important to define what a
smooth map is.
Definition 3.11. A map ϕ : Hm
→ Hn
is called smooth if all the partial derivatives
(possibly one-sided) are continuous. This is the standard definition but for convenience
our working (and equivalent) definition will be: ϕ is smooth if it can be locally extended to
a smooth map Rm
→ Rm
. Again a map is called a diffeomorphism if it is smooth together
with its inverse.
Definition 3.12. A smooth manifold with boundary is a Hausdorff second countable
topological space M with a chosen equivalence class of atlases - a collection of charts
satisfying
• each ϕ : U → Hm
is a homeomorphism of an open subset U ⊆ M with an open
subset of Hm
• the transition maps ψϕ−1
: Hm
→ Hm
(partially defined) are diffeomorphisms
The set ∂M of all points which in some (and then in any) coordinate chart lie in ∂Hm
=
{0} × Rm−1
is called the boundary of M.
Definition 3.13. A tangent bundle TM of a manifold with boundary M is defined in
the same way with THm
= Rm
× Hm
, i.e. we allow all vectors even those pointing out of
the manifold (therefore the kinetic definition of TM is not appropriate).
Definition 3.14. A subset M ⊆ N of a manifold with boundary N is called a neat
submanifold if there is a covering of M by charts ϕ : U → Hn
such that
ϕ(M ∩ U) = Hm
∩ ϕ(U)
3.B. CLASSIFICATION OF 1-DIMENSIONAL MANIFOLDS 13
Here Hm
⊆ Hn
consists precisely of all (x1, . . . , xn) with only first m coordinates nonzero1
.
Theorem 3.15. Let ϕ : M → N be a smooth map, ∂N = ∅, and suppose that both ϕ
and ϕ|∂M are submersions near ϕ−1
(y) where y ∈ N is some fixed point. Then ϕ−1
(y) is a
neat submanifold.
Proof. This is a local problem so we can assume that ϕ : Hm
→ Rn
and y = 0. The
statement is classical on the interior of M, thus we assume that x ∈ ∂Hm
is such that
ϕ(x) = 0. As ϕ|∂Hm is a submersion at x we can choose among x2, . . . , xm an n-tuple of
coordinates such that the derivatives of ϕ with respect to them give an isomorphism. We
assume that these are xm−n+1, . . . , xm. Writing now
Hm ∼= Hm−n
× Rn
we define a smooth map
ψ : Hm (pr1,ϕ)
−−−−−→ Hm−n
× Rn ∼= Hm
where pr1 : Hm
→ Hm−n
denotes the projection onto the first (m − n) variables. Easily
dψ(x) is an isomorphism and so by the inverse function theorem ψ is a diffeomorphism
near x (but note that the inverse function theorem does not imply that the inverse ψ−1
takes Hm
into Hm
, this follows from our definition of ψ). In the chart ψ the subset ϕ−1
(0)
becomes
ψ(ϕ−1
(y)) = {(x1, . . . , xm−n, 0, . . . , 0) ∈ Hm
}
A serious HW: every compact neat submanifold has a tubular neighbourhood.
From now on we will distinguish between compact manifolds without boundary - they
will be called closed - and compact manifolds with boundary which we will refer to simply
as compact manifolds.
Now we are ready to classify 1-dimensional manifolds.
Theorem 3.16. Every connected 1-dimensional manifold is diffeomorphic to exactly
one of the following list
• [0, 1]
• [0, ∞)
• R
• S1
(following Goodwillie we do not call ∅ connected as π0(∅) = ∗).
Proof. Let M be a connected 1-dimensional manifold and choose a Riemannian metric
on it. Let v ∈ TxM be a unit tangent vector and consider a geodesic determined by it:
ϕ : J −→ M
t −→ expx tv
1Loosely speaking, a neat submanifold should be perpendicular to the boundary. In particular ∂M is
not a neat submanifold of M although it is a submanifold (a notion which we will not define).
3.B. CLASSIFICATION OF 1-DIMENSIONAL MANIFOLDS 14
Here J is an interval in R but as M could have some boundary, the same could be true
of J. As ϕ is a submersion (∀t ∈ J : | ˙ϕ(t)| = 1) the image of ϕ is open and for the same
reason it is also closed2
. As M is connected ϕ is surjective. Suppose first that it is also
injective. Then ϕ is a diffeomorphism and this gives us the first three possibilities. Assume
now that for some a < b we have ϕ(a) = ϕ(b). As ˙ϕ(a) and ˙ϕ(b) are two unit vectors in
the same 1-dimensional space Tϕ(a)M necessarily ˙ϕ(a) = ± ˙ϕ(b).
• ˙ϕ(a) = − ˙ϕ(b): in this case ϕ(a + t) = ϕ(b − t) which for t = (b − a)/2 gives a
contradiction ˙ϕ((a + b)/2) = − ˙ϕ((a + b)/2).
• ˙ϕ(a) = ˙ϕ(b): in this case ϕ(a + t) = ϕ(b + t) and ϕ is periodic with period (b − a).
Then necessarily J = R. Let T be the smallest period of ϕ (which easily exists).
Then ϕ induces a map
˜ϕ : S1 ∼= R/TZ −→ M
which is now injective and therefore a diffeomorphism.
Remark. Surjective submersions are quotients in the category of smooth manifolds.
This is because surjective submersions have local sections.
M //
surjective
submersion

P
N
←−−− a factorization exists
iff it exists in Set
>>
In particular such quotients are diffeomorphic iff they have the same kernel. In the previous
proof this could be used as an explanation why S1 ∼= M in the last case - both are quotients
of R.
2One can partition M into images of geodesics, each of which is open.
CHAPTER 4
Sard’s theorem
Definition 4.1. Let f : M → N be a smooth map. A point x ∈ M is called a critical
point of f if f∗ : TxM → Tf(x)N has rank strictly lower than n = dim N. The set of critical
points is called a critical set of f and denoted Σf . A point y ∈ N is called a critical value
if it is an image of a critical point, y ∈ f(Σf ). Otherwise it is called a regular value.
Theorem 4.2 (Sard). The set of critical values has measure 0 in N.
Corollary 4.3 (Brown). The set of regular values of any smooth map f : M → N is
dense in N.
Before giving the proof of Sard’s theorem we show an application.
Theorem 4.4 (Brouwer). There is no retraction Dn
→ ∂Dn
= Sn−1
.
Proof. First we show that there is no smooth retraction. Suppose r : Dn
→ Sn−1
is
such a retraction. Take a regular value y ∈ Sn−1
(which exists by Sard’s theorem). Then
r−1
(y) is a compact 1-dimensional neat submanifold of Dn
and contains y as a boundary
point. There must exist a second boundary point. But as r(z) = z for all z ∈ Sn−1
the
only z ∈ Sn−1
in r−1
(y) is y itself, a contradiction. picture
To reduce to the smooth case we use smooth approximations. Let r : Dn
→ Sn−1
be
a continuous retraction. We define a new retraction r : Dn
→ Sn−1
by pasting together r
on the disc of radius 1/2 and the radial projection on the annulus. picture We apply the
relative version of the smoothening and find a smooth r : Dn
→ Sn−1
with r = r near
Sn−1
.
In the course of the proof of Sard’s theorem we need a version of Taylor formula which
is also of interest independently.
Lemma 4.5. Let f : U → Rn
be a smooth function defined on some convex subset U of
Rm
. Then for any two points x, y ∈ U the following formula holds
f(y) = f(x) + · · · + (Tk
x f)(x)(y − x)⊗k
+ · · · + (Tl
xf)(y)(y − x)⊗l
(all Tk
x f are applied to x except the last one, which is applied to y) where
Tk
f : U × U −→ hom (Rm
)⊗k
, Rn
(x, y) −→ (Tk
x f)(y)
is also smooth and (Tk
x f)(x) = 1
k!
f(k)
.
15
4. SARD’S THEOREM 16
Proof. We construct Tk
f inductively starting with k = 1:
f(y) − f(x) = f(x + t(y − x))
1
0
=
1
0
f (x + t(y − x)) dt (y − x)
and we denote (T1
x f)(y) =
1
0
f (x + t(y − x)) dt. As a function of x and y this is a smooth
map
T1
f : U × U → hom(Rm
, Rn
)
Iterating this construction we get Tk
x f = T1
x · · · T1
x f with
Tk
f : U × U −→ hom((Rm
)⊗k
, Rn
)
smooth and a Taylor formula1
f(y) = f(x) + · · · + (Tk
x f)(x)(y − x)⊗k
+ · · · + (Tl
xf)(y)(y − x)⊗l
Comparing the derivatives of the two sides we get Tk
x (f)(x) = 1
k!
f(k)
.
Proof of Sard’s theorem. Firstly we can assume that m ≥ n, the remaining cases
are taken care of by Theorem 2.11. Also the problem is local so we can assume
W
⊆
f
// Rn
Rm
We partition the critical set Σf according to the order of vanishing derivatives
Σ1 = {x ∈ Σf | df(x) = 0}
Σ2 = x ∈ Σf
df(x) = 0 but there is a nonvanishing
derivative of order at most m/n at x
Σ3 = {x ∈ Σf | all derivatives of order at most m/n vanish at x}
We need to show that each f(Σi) has measure 0. This will be proved in 3 steps.
III. f(Σ3) has measure 0: let C ⊆ W be a compact cube with side a. It is enough to
show that f(C ∩ Σ3) has measure 0. Let x ∈ C ∩ Σ3 and let y be another point in C.
According to Lemma 4.5 we can write
f(y) = f(x) + (Tl
xf)(y)(y − x)⊗l
where l is the smallest integer greater than m/n. Thinking of Tl
f as a function
C × C × (Rm
)l
−→ Rn
we get a bound (Tl
xf)(y)(v1 ⊗ · · · ⊗ vl) < K for all x, y ∈ C and vi ∈ Sm−1
. Therefore
|f(y) − f(x)| = (Tl
xf)(y)(y − x)⊗l
< K|y − x|l
(4.1)
1What this formula roughly says is that after subtracting from f its Taylor polynomial at x of order
(l − 1) the remainder remains smooth even after “dividing” by (y − x)⊗l
.
4. SARD’S THEOREM 17
Now cover C by km
cubes of side a/k, k some positive integer. Then C ∩ Σ3 is covered
by at most km
cubes of side a/k each containing a point from Σ3. By our bound (4.1) the
image f(C ∩ Σ3) is covered by cubes of total volume at most
cst ·km
· (a/k)l n
= cst ·k−n(l−m/n)
(at most km
cubes in Rn
of side at most (a/k)l
). As k → ∞ this tends to 0.
This finishes the proof for m = 1 as then n = 1 and Σf = Σ3 which is covered by step
III. We proceed further by induction on m, i.e. we prove steps II and I assuming the whole
theorem proved for all smaller m.
II. We show that f(Σ2) has measure 0. Let x ∈ Σ2. There is a multiindex I and
numbers i, j such that
∂xI
fj(x) = 0
∂xi
∂xI
fj(x) = 0
We will partition Σ2 according to different I, i, j and denote these ΣI,i,j
2 . It is enough to
show that the various parts ΣI,i,j
2 have image of measure 0. Near the point x ∈ ΣI,i,j
2 the
function
∂xI
fj : W → R
is a submersion and therefore X = (∂xI
fj)−1
(0) is near x a submanifold of dimension m−1.
Clearly ΣI,i,j
2 ⊆ Σf|X
(it is important here that we assume that all the first derivatives are
zero!) and consequently
f(ΣI,i,j
2 ) ⊆ f(Σf|X
)
which has measure 0 by induction hypothesis.
I. We show that f(Σ1) has measure 0. Let x ∈ Σ1 and assume for simplicity that
∂x1 f1(x) = 0 (otherwise change coordinates). Define a map
g = (f1, x2, . . . , xm) : W → Rm
with differential 



∂x1 f1 ∂x2 f1 · · · ∂xm f1
0
...
0
E




Therefore g is a diffeomorphism on a neighbourhood U of x
U
f
//
g ∼=

Rn
gU
¯f=fg−1=(x1, ¯f2,..., ¯fn)
==
Clearly f(U ∩ Σf ) = ¯f(Σ¯f ) (the critical values are diffeomorphism invariant) and so it is
enough to show that ¯f(Σ¯f ) has measure 0. Denote the last n − 1 coordinates of ¯f by
hx1 (x2, . . . , xm) := ( ¯f2, . . . , ¯fn)(x1, . . . , xm)
4. SARD’S THEOREM 18
Then we can write Σ¯f =
x1∈R
{x1} × Σhx1
and similarly
¯f(Σ¯f ) =
x1∈R
{x1} × hx1 (Σhx1
)
Fubini theorem gives the following formula with µ denoting the measure
µ( ¯f(Σ¯f )) = µ(hx1 (Σhx1
)) dx1 = 0 dx1 = 0
The second equality follows from the induction hypothesis as each hx1 is a smooth map
Rm−1
→ Rn−1
.
Remark. The version of Fubini theorem that we need here is rather easy to prove,
first we state it: let C be a compact subset of Rn ∼= R × Rn−1
. Denote by Rn
t the subset
{t} × Rn−1
⊆ Rn
and similarly Ct = C ∩ Rn
t . This version says that if each Ct has measure
0 then so does C.
To prove this let us cover, for each t, Ct by a countable family Ki of cubes of total
volume ε. By compactness there is a small interval It around t such that the products
It × Ki cover C ∩ (It × Rn−1
). The image of C under projection to the first coordinate is a
compact subset and thus lies in a closed interval, say of length a. This interval is covered
by the intervals It and one can pick from them a finite subset It1 , . . . , Itk
of intervals of
total length at most 2a. Then the total volume of cubes we used to cover C ∩ (Iti
× Rn−1
)
is at most 2aε and by varying ε can be dropped arbitrarily low.
CHAPTER 5
Transversality
Definition 5.1. Let there be given smooth maps as in the diagram
M
f
// N M
f
oo
The maps f and f are said to be transverse if for each x ∈ M, x ∈ M with y = f(x) =
f (x ) the following holds
f∗(TxM) + f∗(Tx M ) = TyN
We denote this fact by f f .
picture
We will need a special case when f is an embedding j : A   // N of a submanifold.
A _
j

M
f
// N
Then f is said to be transverse to A, denoted f A, if it is transverse to the embedding
j, i.e. if for each x ∈ M with f(x) ∈ A
f∗(TxM) + Tf(x)A = Tf(x)N
The following reformulation will be useful: for x ∈ f−1
(A) consider the composition
TxM −→ Tf(x)N −→ (TN/TA)f(x) = ν(j)f(x)
with ν(j) the normal bundle of the embedding j. Then f A iff this map is surjective for
all x ∈ f−1
(A).
Example. A submersion is transverse to every submanifold.
Example. Let A = {y} ⊆ N. Then f A iff f is a submersion near (or at) f−1
(y) iff
y is a regular value of f.
In the proceeding the following notation will be very useful. A codimension of an
embedding j : A   // N is the difference
codim j = dim N − dim A = the dimension of the fibre of ν(j)
of the dimensions.
Example. Let f : M → N be any smooth map and j : A  
// N. If dim M < codim j
then f A is equivalent to im f ⊆ N − A.
19
5. TRANSVERSALITY 20
Proposition 5.2. Let f : M → N be smooth and j : A  
// N a submanifold (which
is neat in the case ∂N = ∅) such that f A and f|∂M A. Then f−1
(A) is a neat
submanifold of the same codimension as A. Moreover the map g in the following diagram
is a fibrewise isomorphism
f−1
A
f
//
 _
j

A _
j

ν(j )
g
//

ν(j)

M
f
// N f−1
A
f
// A
Remark. The slogan here is: “transverse pullbacks exist”. More generally when f f
then M ×A M is a submanifold of M × M and we get a pullback
M ×N M //

M
f

M
f
// N
Serious HW: prove this (by reducing to the proposition).
Proof. This is again a local problem and so we can assume
Hm f=(f1,f2)
−−−−−−→ Rn−d
× Rd
with A = Rn−d
× {0}. The condition f A translates to the component f2 being a
submersion near f−1
(A) and similarly for f|∂M A. Therefore locally f−1
(A) = f−1
2 (0) is
a neat submanifold by Theorem 3.15.
Remark. A neat submanifold is a submanifold (which we have not defined) A ⊆ N
such that ∂A = A ∩ ∂N and which is transverse to ∂N.
Sard’s theorem says that most of the points are transverse to a given f : M → N. We
will now generalize this to submanifolds. In fact we will approach this problem from the
other side: we will prove that “most” of the maps f : M → N are transverse to a given A.
a picture of how this is natural
Lemma 5.3. Consider the following diagram with f A and any smooth map g : P →
M.
f−1
(A)
f
//
 _
j

A _
j

P g
// M
f
// N
Then g f−1
(A) if and only if fg A.
5. TRANSVERSALITY 21
Proof. Clearly for x ∈ P the conditions fg(x) ∈ A and g(x) ∈ f−1
(A) are equivalent.
For such x consider the diagram
ν(j )g(x)
∼= // ν(j)fg(x)
TxP
g∗
OO
(fg)∗
88
In this diagram g∗ is surjective iff (fg)∗ is surjective. This completes the proof.
A serious HW: generalize this to arbitrary j (not necessarily an embedding).
Remark (A very important remark!). Most of the statements we proved for compact
manifolds work (with the same proof) for general manifolds near any given compact subset.
Two (important!) examples are
• Let C ⊆ M a compact subset of a smooth manifold M. Then there exists an
embedding of some neighbourhood U of C in some Rk
, k 0.
For a proof, instead of covering M by a finite number of charts and producing a
map, cover C and write down the same formula. It will be an embedding of some
neighbourhood U.
• Let C ⊆ M   j
// N. Then some neighbourhood U of C has a tubular neighbourhood,
i.e. there is an embedding
ι : ν(j)|U → N
extending the embedding j.
Here exp is still defined and an embedding near C, i.e. on some ˚Dεν(j)|U . picture
We are now ready to prove our main theorem.
Theorem 5.4. Let M be a compact smooth manifold and A ⊆ N a manifold, both A and
N boundaryless. Then for any f0 : M → N there exists a smooth homotopy f : I ×M → N
such that f(0, −) = f0 and f(1, −) A. Moreover if f0 is transverse to A near some closed
subset K ⊆ M then f can be chosen constant near K.
Proof. Embed a neighbourhood of f0(M) ⊆ N in some Rk
with a tubular neighbourhood
ι : ν(N)|U
∼=
−−→ V ⊆ Rk
. First assume that K = ∅ and define
F : Rk
× M −→ Rk
(v, x) −→ f0(x) + v
For a small v we land in the tubular neighbourhood V and can define
G : ˚Dε× M
F
−−→ V
r
−→ U ⊆ N
This is clearly a submersion and in particular G A. Assuming this we consider G−1
(A)
and a regular value v for its projection to Rk
.
G−1
(A) ⊆ Rk
× M −→ Rk
5. TRANSVERSALITY 22
We claim that for this regular value
G(v, −) : M → N
is transverse to A. This is proved by examining the following diagram.
G−1
(A) //
 _

A _

{v} × ˚Dε
  //

˚Dε× M G
//
pr

N
{v}   // ˚Dε
Moreover formula (t, x) → G(tv, x) then defines the required homotopy.
Now we describe what has to be changed when K = ∅. As was observed it is enough to
alter F so that it is independent of v near K but still transverse to A. To do this choose
a function λ : M → [0, 1] for which λ−1
(0) is a neighbourhood of K where f0 is transverse
to A. Then F : (v, x) → f0(x) + λ(x)v is the required alteration: at poits (v, x) with
λ(x) > 0 it is still a submersion while, if λ(x) = 0 then necessarily dλ(x) = 0 too and
F∗(TvRk
× TxM) = (f0)∗(TxM).
Remark. Again we have a “near a compact subset” version for a compact subset
C ⊆ M of a noncompact manifold M: there is a homotopy g : I × V → N defined on a
neighbourhood V of C with g(0, −) = f0|V and with g(1, −) A (and also relative version
with K: then f is constant near K ∩ V ). Now we describe a variant of this version.
Choose a function ρ : M → [0, 1] such that ρ ≡ 1 on a neighbourhood U of C and with
support in V . Now we can define a global homotopy
f : I × M −→ N
(t, x) −→ g(ρ(x)t, x)
This homotopy has the following properties
• f(0, −) = f0
• (f(1, −)|U ) A
• f is constant near K and outside of a neighbourhood V of C (which in fact can
be chosen arbitrarily)
Using this version we can prove
Theorem 5.5. Let E → M be a smooth bundle over a compact manifold M. Let
E ⊆ E be a subbundle (locally a subset of the form F × U ⊆ F × U). For every section
s0 : M → E there exists a homotopy of sections s : I × M → E with s(0, −) = s0 and
s(1, −) E . Again if s0 is transverse to E near some closed subset K then s can be
chosen constant near K.
5.A. CLASSIFYING VECTOR BUNDLES 23
Proof. Cover M by open subsets U1, . . . , Uk over which we have trivializations
E|Ui

∼= // F × Ui
~~
Ui
identifying E |Ui
with F × Ui. Now observe that a section (f, id) of a trivial bundle
F × U → U is transverse to F × U iff f is transverse to F . In this way we translate the
situation over each Ui to one where Theorem 5.4 (or rather the remark following it) could
be applied.
Let λi be a partition of unity subordinate to U1, . . . , Uk. According to Theorem 5.4
there are homotopies of sections si
: I × M → E with the following properties
• s1
(0, −) = s0 and sj+1
(0, −) = sj
(1, −) for j ≥ 1
• each sj
(1, −) E near supp λj
• sj
is constant near K ∪ i<j supp λi
The concatenation of s1
, . . . , sk
is the required homotopy.
Serious HW: show that the relation of being smoothly homotopic (relative K, i.e. by a
homotopy constant near K) is transitive.
An application to vector bundles is the following.
5.A. Classifying vector bundles
We study the set of (linear) monomorphisms Rn
→ Rk
by studying its complement1
.
Denote by homr(Rn
, Rk
) the subset of hom(Rn
, Rk
) of maps of rank r.
Lemma 5.6. homr(Rn
, Rk
) is a smooth submanifold of hom(Rn
, Rk
) of codimension
(n − r)(k − r).
Proof. Let ϕ0 ∈ hom(Rn
, Rk
) and assume that
ϕ0 =
A0 B0
C0 D0
with the block A0 of size r × r invertible. Then ϕ0 ∈ homr(Rn
, Rk
) iff D0 = C0A−1
0 B0.
Therefore on a neighbourhood of such ϕ0 every ϕ =
A B
C D
has A invertible and we get
a chart
GL(r) × hom(Rn−r
, Rr
) × hom(Rr
, Rk−r
) −→ homr(Rn
, Rk
)
(A, B, C) −→
A B
C CA−1
B
1More precisely we study connectivity of the set of monomorphisms by studying the codimension of
its complement.
5.A. CLASSIFYING VECTOR BUNDLES 24
with codimension equal to the size of the D-block. As after after a linear change of
coordinates on Rn
and Rk
we can assume that the topleft r × r block of any element of
homr(Rn
, Rk
) is invertible this finishes the proof.
We can now reprove the existence of an inverse of a vector bundle, BUT we get a
relative version which will be crucial in our classification.
Construction 5.7. Let E → M be a vector bundle. Then hom(E, Rk
) is a vector
bundle with fibre hom(Ex, Rk
) over x. In fact it is canonically isomorphic to E∗
⊕ · · · ⊕
E∗
(k-times). We will also need its subbundle hominj(E, Rk
) consisting of all injective
homomorphisms.
Proposition 5.8. Let E → M be a smooth vector bundle over a compact manifold M
with the dimension of the fibre n. Then every smooth section
s : ∂M → hominj(E, Rk
)|∂M
extends to a section over M provided k ≥ m + n.
Proof. Extend s arbitrarily to a section
˜s : M → hom(E, Rk
)
This is possible using partition of unity and the fact that the fibre is convex (contractible).
Now we can homotope ˜s to a section t which is transverse to all homr(E, Rk
), 0 ≤ r < n.
This is done using Theorem 5.5 and an easy fact that these are subbundles. But as m <
(n − r)(k − r) for all such r the transversality is equivalent to t not meeting homr(E, Rk
).
Therefore
t : M → hominj(E, Rk
)
Remark. Such a section is equivalent to an inclusion of vector bundles
E   // Rk
× M
Any complementary subbundle (e.g. E⊥
) gives an inverse of E. We will be interested in
yet another reformulation of a section of hominj(E, Rk
).
The fibre of hominj(E, Rk
) is hominj(Rn
, Rk
) = Vk,n, the Stiefel manifold of n-frames
in Rk
(n-tuples of linearly independent vectors). Now Vk,n has a natural transitive action
of GL(n) with the orbit space Gk,n, the Grassmann manifold of all n-dimensional linear
subspaces in Rk
. The projection map is identified with
Vk,n −→ Gk,n
(e1, . . . , en) −→ span(e1, . . . , en)
There is a description of Gk,n as a homogenous space O(k)/(O(n) × O(k − n)). This
identification is constructed from the obvious transitive action of O(k) on Gk,n. By acting
on the standard subspace Rn
⊆ Rk
we obtain a map
O(k) → Gk,n
5.A. CLASSIFYING VECTOR BUNDLES 25
sending a matrix to the span of its first n columns (which is clearly smooth). It is easy to
identify the stabilizer of Rn
as the group of matrices of the form
∗ 0
0 ∗
This is what is understood by O(n) × O(k − n). The above map O(k) → Gk,n therefore
factorizes through an immersion
O(k)/(O(n) × O(k − n)) −→ Gk,n
which is then easily seen to be injective and thus a diffeomorphism. What will be important
for us is that Gk,n is a closed manifold.
There is a canonical n-dimensional vector bundle γk,n over Gk,n sometimes also called
tautological2
γk,n
vector
bundle

{(P, x) | P ∈ Gk,n, x ∈ P}
 _

j
$$
Gk,n Gk,n × Rkoo // Rk
Lemma 5.9. There is a bijection
{fibrewise isomorphisms E
f
−−→ γk,n}
∼=
−−→ {sections of hominj(E, Rk
)}
sending f to the composition jf : E → Rk
thought of as a section of hominj(E, Rk
).
Proof. Given a section t the corresponding f must be
Ex u  // (im t(x), t(x)(u))
E
f
//

γk,n ⊆ Gk,n × Rk

M
f
// Gk,n
and it is enough to show that im t : M → Gk,n is smooth. Locally (when E = Rn
×U → U
is trivial) the section t is given by a smooth map U → hominj(Rn
, Rk
) = Vk,n and im agrees
with the canonical projection Vk,n → Gk,n.
Remark. More generally for a principal G-bundle P → B and a G-space X there is a
bijection between sections of the associated bundle P ×G X → B and G-equivariant maps
P → X. This specializes to the previous lemma by taking X = Vk,n after one replaces the
vector bundles by the respective principal GL(n)-bundles.
2An alternative definition goes as follows. The projection Vk,n → Gk,n is in fact a principal GL(n)bundle
and γk,n is the associated vector bundle γk,n = Vk,n ×GL(n) Rn
.
5.A. CLASSIFYING VECTOR BUNDLES 26
If in the diagram
E
f
//

γk,n

M
f
// Gk,n
the homomorphism f is a fibrewise isomorphism we call f : M → Gk,n a classifying map
for E. This is because in such a situation E ∼= f∗
γk,n is determined by f.
Theorem 5.10. Let M be a closed(?) smooth manifold. There is a bijection
{homotopy classes of maps M
f
−−→ Gk,n}
∼=
{isomorphism classes of vector bundles E → M}
sending f to the pullback bundle f∗
γk,n provided k > m + n.
Proof. We will first show that this mapping is well-defined (i.e. f0 ∼ f1 implies
f∗
0 γk,n
∼= f∗
1 γk,n), a fact usualy proved topologically (in much greater generality). First
observe that it is enough to show that for every vector bundle p : E → I × M, E|{0}×M
∼=
E|{1}×M . For then a homotopy f : I ×M → Gk,n yields a pullback bundle f∗
γk,n → I ×M
with f∗
γk,n|{t}×M = f∗
t γk,n.
f∗
γk,n|{t}×M
//

f∗
γk,n
//

γk,n

{t} × M   //
ft
33I × M
f
// Gk,n
Our idea will be to compare the two restrictions by the flow of some vector field. We start
with I × M where we can compare3
the two ends via a canonical vector field
Y (t, x) =
d
ds s=t
(s, x)
Let X ∈ Γ(TE) be a vector field on E such that
• X|I×M = Y (agrees with Y on the zero section)
• p∗X = Y (projects to Y )
It is easy to find such X locally, on Rn
× J × U with J ⊆ I, U ⊆ Rn
, we take
X(v, t, x) =
d
ds s=t
(v, s, x)
3The comparison is reflected in the fact that the flow of Y (which is FlX
s (t, x) = (s + t, x)) produces a
diffeomorphism FlY
1 = id : {0} × M → {1} × M.
5.A. CLASSIFYING VECTOR BUNDLES 27
As both conditions are affine we can construct X globally using a partition of unity and
get a flow
FlX
: W
⊆
// E
R × E
(note that W is not open as E has boundary). Using our second condition on X
p FlX
(t, v) = FlY
(t, p(v)) = (t + p1(v), p2(v))
Therefore restricting to W ∩ (R × E{0}×M ) we get
ϕ : W
⊆
// E
[0, 1] × E|{0}×M
now with W open. Also W contains [0, 1] × ({0} × M), therefore a neighbourhood of the
form
[0, 1] × ˚DεE|{0}×M
In particular restricting to this disc bundle we finally obtain
ψ = ϕ(1, −) : ˚DεE|{0}×M → E{1}×M
This is a diffeomorphism onto its image (it is defined by a flow, its inverse is given by
following the flow for time −1). Note that ψ is not necessarily linear but we will linearize
it (by taking a derivative) using its 2 properties
• ψ({0} × M) = {1} × M (preserves the zero section)
• v ∈ ˚DεE(0,x) ⇒ ψ(v) ∈ E(1,x) (fibrewise over identity)
Taking the derivative at the zero section we get
T(E|{0}×M )|{0}×M
∼=
ψ∗

E|{0}×M ⊕ TM
α 0
0 id

T(E|{1}×M )|{1}×M
∼= E|{1}×M ⊕ TM
with α : E|{0}×M → E|{1}×M the required isomorphism.
Now we continue with proving the bijectivity of the mapping. We saw in Proposition 5.8
and Lemma 5.9 that it is surjective and thus it is enough to show that any two classifying
maps f0, f1 : M → Gk,n for E are homotopic. Consider
(I × E)|∂(I×M)
//
**

γk,n

I × E
77

∂(I × M)
**
// Gk,n
I × M
77
5.A. CLASSIFYING VECTOR BUNDLES 28
Here the map ∂(I×M) → Gk,n is fi on {i}×M. As they are both classifying maps for E the
map (fibrewise isomorphism) on the top exists. According to Lemma 5.9 this determines
a section of hominj(I × E, Rk
) over ∂(I × M) which can be extended by Proposition 5.8 to
a section over I × M which in return provides an extension f : I × M → Gk,n.
CHAPTER 6
Degree of a map
Let f : M → N be a smooth map between closed oriented manifolds of the same
dimension n. The degree deg(f, y) is defined in the following way:
• pick a regular value y ∈ N of f
• observe that f−1
(y) is a discrete compact subset of N, hence finite
f−1
(y) = {x1, . . . , xk}
• at each xi the derivative of f is an isomorphism f∗xi
: Txi
M → TyN and define
the local degree of f at xi by
degxi
f =
1 f∗xi
preserves orientation
−1 f∗xi
reverses orientation
• define deg(f, y) = degxi
f
Observe that changing the orientation of M to the opposite one changes the sign of
deg(f, y), similarly for the orientation of N. Also observe that if M and N are nonoriented,
the degree makes sense mod 2.
To proceed further we need to know how boundary of a manifold inherits an orientation.
Let W be an oriented manifold with boudnary. The orientation of ∂W is characterized
by the following: at x ∈ ∂W choose any outward pointing e ∈ TxW. Then a basis
(e1, . . . , en) in Tx(∂W) is positive iff (e, e1, . . . , en) is positive in TxW. Therefore a boundary
of Hn+1
= R− × Rn
with its standard orientation is Rn
with its standard orientation.
Example 6.1. ∂(I × M) = ({1} × M) ∪ (−{0} × M) where the minus sign in front of
an oriented manifold denotes the same manifold with the opposite orientation.
Lemma 6.2. Let f : Wn+1
→ Nn
be a smooth map between compact oriented manifolds,
∂N = ∅. Let y ∈ N be a regular value for both f and f|∂W . Then deg(f|∂W , y) = 0.
Proof. f−1
(y) is a compact neat 1-dimensional submanifold, therefore a disjoint union
of circles (which must lie in the interior) and neatly embedded intervals. If γ : I → W is
one of them then ˙γ(0) is outward iff ˙γ(1) is inward. We have the following picture
I //
 _

{y}
 _

I × TyN = ν(γ) //

TyN

W // N I // {y}
29
6. DEGREE OF A MAP 30
We can think of ν(γ) as a neighbourhood of γ in W and, possibly after reversing γ, even
with the orientations agreing. Then
∂(ν(γ)) = ({1} × TyN) ∪ (−{0} × TyN)
As the derivative of f at γ in the directions “orthogonal” to γ agrees with the projection
I × TyN → TyN (the map of the normal bundles) it is the same (namely id) on both ends
but for opposite orientations of the source. In particular
degγ(0) f|∂W + degγ(1) f|∂W = 0
Summing over all such intervals γ we obtain the result.
Lemma 6.3. Let f0, f1 : M → N be two smooth homotopic maps (M and N closed of
the same dimension) such that y ∈ N is a regular value for both of them. Then deg(f0, y) =
deg(f1, y).
Proof. Start with any homotopy
˜f : I × M → N
we can find an approximation f : I ×M → N transverse to {y} with f ≡ ˜f near ∂(I ×M).
Thus both f and f|∂(I×M) have {y} as a regular value and so
deg(f|∂(I×M), y) = 0
As ∂(I × M) = ({1} × M) ∪ (−{0} × M) this means
0 = deg(f|∂(I×M), y) = deg(f1, y) − deg(f0, y)
Theorem 6.4. Let f : M → N be a smooth map between closed manifolds of the same
dimension with N connected. If y0, y1 are two regular values of f then
deg(f, y0) = deg(f, y1)
We denote this number by deg f. It only depends on the homotopy class of f.
Proof. The heart of the proof is (a strong form of) homogenity of M. Namely, there
exists a smooth homotopy
ϕ : I × M → N
with the following properties (where ϕt = ϕ(t, −)):
• each ϕt a diffeomorphism (such ϕ is then called a diffeotopy)
• ϕ0 = id and ϕ1(y1) = y0
Assuming for now that it exists construct a homotopy
F : I × M −→ N
(t, x) −→ ϕ(t, f(x))
6. DEGREE OF A MAP 31
Again F0 = f and F1 = ϕ1f and so y0 is a regular value for F|∂(I×M).
{y1} //

{y0}

M
f
// N
ϕ1
// N
Therefore deg(f, y0) = deg(F0, y0) = deg(F1, y0) = deg(ϕ1, y0) · deg(f, y1) but
deg(ϕ1, y0) = deg(ϕ0, y0) = 1
as ϕ0 = id. The homotopy invariance follows from the previous lemma.
Now we return to the existence of the diffeotopy ϕ. First we find such a diffeotopy
locally, but as we would like to extend it, it should be compactly supported (i.e. nonconstant
only on a compact subset). Let y ∈ Rn
and assume for simplicity that y ∈ Dn
. Choose a
smooth function ρ : Rn
→ [0, 1] with ρ ≡ 1 near Dn
and with compact supp ρ. This function
will extend the standard translation on Dn
into a compactly supported diffeotopy. Namely
take a vector field X : Rn
→ Rn
sending x to −ρ(x) · y. As X is compactly supported the
flow
FlX
: R × Rn
→ Rn
is globally defined. Now we can take ϕ = FlX
|[0,1]×Rn as X = −y on Dn
implies ϕ(1, y) = 0.
If this was in a chart we can extend ϕ by a constant homotopy to get a global diffeotopy.
Now we consider the following equivalence relation:
z z ⇔ ∃ a smooth diffeotopy from id to a map sending z to z
The local construction ensures that the orbits are open and therefore closed (as the complement
is the union of the other orbits). Thus there is only one orbit.
Remark. Let Mn
be a closed oriented connected smooth manifold. Then the set of
homotopy classes of maps M → Sn
is in a bijective correspondence with Z via the degree
map
deg : [M, Sn
]
∼=
−−→ Z
We will prove this result in a much greater generality in the next chapter.
CHAPTER 7
Pontryagin-Thom construction
We start with a geometric reformulation of the degree isomorphism
deg : [M, Sn
]
∼=
−−→ Z
as for different dimensions the simple counting of preimages could not work. Nevertheless
in this reformlulation we do assume m = n. The preimage of a regular point of a map
f : M → Sn
is a finite subset of M with signs - describing wether f preserves or reverses
orientation at the point in question.
[M, Sn
]
deg
//

Z
{finite subsets of M with signs}/ ∼
ε
44
Having in mind that the horizontal map should be an isomorphism and that we want in
our geometric reformulation to have the vertical map an isomorphism, so must be the map
denoted by ε which just add the signs of all the points. This requirement determines the
equivalence relation but we want a more geometric description.
It is interesting to describe the Pontryagin-Thom construction in this setting. It plays
an essential role in constructing the inverse of the vertical map. Starting with a finite
subset X of M with signs attached to them we choose disjoint coordinate discs around
each of them and then collapse the complement of their interiors to get the first map in
M
P-T constr.
−−−−−−−→
X
Sn class. of VB
−−−−−−−−→ Sn
The second arrow just maps each copy Sn
onto Sn
by a map of degree equal to the sign
associated to that point.
Let us now start with the general setting. Let M and N be closed smooth manifolds.
We saw the interply between maps and submanifolds: if f : M → N is transverse to a
submanifold A   // N then f−1
(A) ⊆ M is again a submanifold. For homotopies we get
Lemma 7.1. If f0 ∼ f1 and both are transverse to A then f−1
0 (A) is cobordant to f−1
1 (A)
(in M - to be explained after the proof).
32
7. PONTRYAGIN-THOM CONSTRUCTION 33
Proof. Let f : I × M → N be a homotopy from f0 to f1. If it is not smooth we can
smoothen it rel ∂(I × M) where it already is smooth1
. Also we can make it transverse
to A rel ∂(I × M). Then f−1
(A) is a neat submanifold of I × M and ∂(f−1
(A)) =
{0} × f−1
0 (A) ∪ {1} × f−1
1 (A) as for neat submanifolds in general ∂B = (∂W) ∩ B.
Definition 7.2. Two compact submanifolds B0, B1 ⊆ M, ∂B0 = ∂B1 = ∂M = ∅,
are called cobordant in M if there exists a neat submanifold B ⊆ I × M such that ∂B =
{0} × B0 ∪ {1} × B1 .
Question. Is it possible to find A ⊆ N of codimension d in such a way that the
mapping t below is a bijection?
t :
homotopy classes of
maps f : M → N
−→
cobordism classes of submanifolds
B ⊆ M of codimension d
Here start with [f], make it transverse to A and define t([f]) = [f−1
(A)].
To answer the question we need a construction
Definition 7.3. Let E → M be a vector bundle. Then the Thom space Th(E) of E
is
Th(E) = DE/SE
the quotient of the associated disc bundle by its boundary, the sphere bundle. Th(E) is
only a topological space (not a manifold) but easily the composition
E ∼= ˚DE   // DE → DE/SE = Th(E)
is a topological embedding and so Th(E) = E ∪ {∞} is a smooth manifold away from ∞.
Also if M is compact then Th(E) is the one-point compactification of E.
Remark. There is an alternative way of defining Th(E). Choosing a metric on E
(which is only important in the smooth case anyway) we can now replace the typical
fibre Rd
by the one-point compactification (Rd
)+
= Sd
⊆ Rd
× R where O(d) acts by
O(d) ∼= O(d) × 1 ⊆ O(d + 1). In this way we obtain a sphere bundle E+
. As the inverse
of the stereographic projection Rd   // Sd
is O(d)-equivariant it induces an embedding
E   // E+
Now E+
has two sections, the zero section and ∞ : M → E+
. We define Th(E) =
E+
/∞(M) = E ∪ {∞}. The main property of Th(E), namely that Th(E) − M ∗
can now be easily verified: as the sphere (Rd
)+
is symmetric in the R-direction (as an
O(d)-space). Therefore Th(E) − M ∼= E/M and a contraction of this comes from the
homotopy
h : I × E −→ E
(t, (x, v)) −→ (x, tv)
1What we need is a homotopy which is smooth near ∂(I × M). Such a homotopy can be constructed
from f by pasting the concatenating the constant homotopy on f0 with f and the constant homotopy on
f1. picture
7. PONTRYAGIN-THOM CONSTRUCTION 34
which easily passes to E/M.
Now we can answer the question: the solution is
N = Th(γk,d) and A = Gk,d ⊆ γk,d, as the zero section
Now we will explain why this should be a good choice. If we have a map which is transverse
to A and take pullback
f−1
(A)
f
//
 _
j

A _
j

ν(j ) //

ν(j)

M
f
// N f−1
(A)
f
// A
This means that the normal bundle of any submanifold B ⊆ M must be a pullback of ν(j)
and the best choice is ν(j) = γk,d, the universal bundle. To construct the inverse of t we
need the Pontryagin-Thom construction.
Let j : B   // M be a submanifold with a tubular neighbourhood
ι : ν(j)   // M
Also denote U = im ι. The Pontryagin-Thom construction is the map
j!
: M → Th(ν(j))
defined by the following
• j!
|U : U
ι−1
−−−→ ν(j) → ν(j)/(ν(j) − ˚Dν(j)) ∼= Dν(j)/Sν(j) = Th(ν(j))
• j!
|M−U is constant onto ∞.
picture This clearly defines a continuous map j!
: M → Th(ν(j)). In fact j!
is smooth on a
neighbourhood of B. As we prefer to think of the Thom space as Th(ν(j)) = ν(j) ∪ {∞}
we will now alter the definition of j!
to
j!
|U : U
ι−1
−−−→ ν(j)   // Th(ν(j))
(obtained by replacing ν(j) with ˚Dν(j) ∼= ν(j), the result remains continuous). An advantage
of this point of view is that any fibrewise isomorphism E → E of vector bundles
induces a continuous map Th(E) → Th(E ) which is even smooth away from ∞.
Aside. One can give a more symmetric description of the map j!
by embedding M into
Rk
. Choosing a tubular neighbourhood of B ⊆ Rk
inside some tubular neighbourhood of
M ⊆ Rk
the Pontryagin-Thom construction for the embedding B ⊆ Rk
factors
Rk // //
$$
Th(ν(M))
j!

Th(ν(B))
The Poincar´e duality can be expressed as Th(ν(M)) being dual to M. This construction
then describes the dual map to j (in homology this map goes to cohomology by the Poincar´e
7. PONTRYAGIN-THOM CONSTRUCTION 35
duality then continues with the induced map j∗
in cohomology and finally returns to
homology by Poincar´e duality again).
Theorem 7.4 (Thom). The map
t :
homotopy classes of maps
f : M → Th(γk,d)
−→
cobordism classes of submanifolds
B ⊆ M of codimension d
is bijective provided k > n.
To define t let us call a continuous map f : M → Th(γk,d) transverse to Gk,d if it is
smooth near f−1
(Gk,d) and transverse to Gk,d in this neighbourhood.
Lemma 7.5. Every f : M → Th(γk,d) is homotopic to a map which is transverse to
Gk,d (plus a relative version).
Proof. Make f smooth near f−1
(Dγk,d) and make sure that no points outside of
f−1
(˚Dγk,d) get mapped into Gk,d by the perturbed map. Then make f transverse to Gk,d
near f−1
(Dγk,d) and again not mapping points outside of f−1
(˚Dγk,d) into Gk,d.
With this lemma in mind we define t from the theorem by choosing a representative f
of the homotopy class with f Gk,d and then taking f−1
(Gk,d). The relative version of
the lemma then guarantees that this prescription is independent of the choice of f: if two
such are homotopic, we can assume this homotopy to be transverse to Gk,d and thus giving
the required cobordism.
Proof. First we prove surjectivity. Let j : B   // M be a submanifold of codimension
d and considet the Pontryagin-Thom construction
g : M
j!
−−→ Th(ν(j))
Th(f )
−−−−−→ Th(γk,d)
where the second map comes from classifying ν(j):
ν(j)
f
//

γk,d

B
f
// Gk,d
The composition g is transverse to Gk,d with g−1
(Gk,d) = B.
The proof of injectivity is similar in the spirit but more complicated. Let B0 =
f−1
0 (Gk,d) ∼ f−1
1 (Gk,d) = B1 be cobordant in M. We proceed in few steps:
• First we reduce to the case B0 = B1. Let j : B  
// I × M be a cobordism,
∂B = {0} × B0 ∪ {1} × B1
and let ι : ν(j)  
// I × M be its tubular neighbourhood. Considering the
Pontryagin-Thom construction for j gives
I × M
j!
−−→ Th(ν(j)) −→ Th(γk,d)
7. PONTRYAGIN-THOM CONSTRUCTION 36
and thus a homotopy g0 ∼ g1 where Bi = g−1
i (Gk,d) Therefore it is enough to show
gi ∼ fi.
• We assume B = f−1
0 (Gk,d) = f−1
1 (Gk,d) with both fi Gk,d. There are pullback
diagrams
B
fi
//
 _
j

Gk,d
 _

ν(j)
fi
//

γk,d

M
fi
// N B
fi
// Gk,d
This implies from the classification theorem that f0 ∼ f1. Moreover this homotopy
is covered by a homomorphism of vector bundles
I × ν(j)
f
//

γk,d

I × B
f
// Gk,d
Our next goal is to extend this homotopy to M.
• The next step is to extend the above homotopy to a tubular neighbourhood of
I ×B. The main tool will be a linearization of maps between vector bundles. First
we explain what happens in the case of a single vector space. Let f : Rd
→ Rd
be
a diffeomorphism preserving 0. We can write
f(x) = f(0) + (T1
0 f)(x) · x
with T1
0 f : Rd
→ hom(Rd
, Rd
) smooth. The “zoom in” homotopy
h : (t, x) → 1/t · f(tx) = (T1
0 f)(tx) · x
It follows from the first description that each ht is a diffeomorphism whereas
from the second description it is obvious that h is smooth and h0 = dh(0). We
summarize this in
Theorem 7.6. Every diffeomorphism Rd
→ Rd
preserving 0 is homotopic
through such maps to its differential at 0.
Serious HW: Show how the last theorem implies Diff(Rd
, 0) GL(d) (this
requires understanding the topology on smooth maps).
Now we will proceed with a fibrewise version: let f : Rm
× Rd
−→ Rm
× Rd
be a smooth map with the following two properties:
f(Rm
× 0) ⊆ Rm
× 0 and f Rm
× 0
We do the linearization fibrewise:
(t, (x, v)) → (f1(x, tv), 1/t · f2(x, tv))
which is again smooth. This prescription makes sense between any vector bundles
and we get
7. PONTRYAGIN-THOM CONSTRUCTION 37
Theorem 7.7. Every smooth map f : E → E between total spaces of vector
bundles of the same dimension preserving and transverse to the zero section is
homotopic through such maps to the vector bundle homomorphism f .
M
f
//
 _
j

M _
j

E ∼= ν(j )
f
//

ν(j) ∼= E

E
f
// E M
f
// M
Now we return to the proof of injectivity. Denoting by U some tubular neighbourhood
of B where both fi are smooth the vector bundle homomorphism f
from the previous step is a homotopy between linearizations fi of fi|U . All together
on U we have a homotopy, which we denote by f, as a concatenation of
f0 ∼ f0 ∼ f1 ∼ f1
provided by the linearizations and the homotopy f .
• The last step in the proof is to extend the homotopy f to M. Here we will make
use of the contractibility of Th(γk,d) − Gk,d. Namely we are given a map
{0} × M ∪ I × U ∪ {1} × M
f0∪f∪f1
−−−−−−→ Th(ν(γk,d))
which we want to extend to I × M. Replacing U by the subset corresponding to
Dν(j) we can cut out the interior ˚Dν(j) to obtain
N = I × (M − ˚Dν(j)))
a manifold with corners (to be defined later). Our problem can then be formulated
as extending the map f : ∂N → Th(γk,d) − Gk,d to N. As Th(γk,d) − Gk,d
is contractible f ∼ ∞ is homotopic to the constant map with value ∞. This
constant map can be extended to N easily so the proof is finished by the next
theorem.
Definition 7.8 (Manifolds with corners). Manifolds with ccrneres are modelled on
(R−)m
, the rest is “the same” with a couple of remarks. Unlike for manifolds with boundary,
manifolds with corners are closed under products. That is why the manifold N from the
proof of Thom’s theorem is a manifold with corners. There are also certain disadvantages:
e.g. tangent vectors are not represented even by one-sided curves. More seriously, the
boundary of a manifold with corners is not a manifold
Theorem 7.9. Let N be a compact manifold with corners. Then ∂N   // N has a
homotopy lifting property.
Proof. Let X be an inward pointing vector field on N with no zeroes on ∂N. Locally
this is easy X = (−1, . . . , −1) on (R−)m
, to get X globally glue the local ones by the
partition of unity. The flow of X is defined on
FlX
: R+ × N → N
7.A. POSSIBLE MODIFICATIONS OF THE THOM’S THEOREM 38
Restrict to ∂N to get ϕ : R+ × ∂N → N. Observe that ϕ is injective: if ϕ(t, x) = ϕ(t , x )
with t ≤ t then FlX
(t − t, x) = x which is impossible as X is pointing inwards. Also im ϕ
is an open neighbourhood of ∂N. This is easily seen in a chart. picture
The neighbourhood of ∂N homeomorphic to R+ × ∂N is called a collar (if N only had
a boundary then even diffeomorphic). Start with f : ({0} × N) ∪ (I × ∂N) −→ X. Using
a retraction I × R+ → ({0} × R+) ∪ (I × {0}) we can extend f to a map
f : ({0} × N) ∪ (I × (R+ × ∂N)
the collar of ∂N
)
Now choose λ : R+ → I such that λ(0) = 1 and supp λ is compact and put
f (t, x) = f (ρ(x) · t, x)
where ρ : N → I is a function defined by
ρ|R+×∂N = λ ◦ pr1
ρ|N−(R+×∂N) ≡ λ ◦ 0
7.A. Possible modifications of the Thom’s theorem
• Get rid of the “in M” setting to classify manifolds of dimension d up to an abstract
cobordism. For that we need a manifold where all d-dimensional manifolds embed
and similarly for their cobordisms. S2d+2
will do but we need further tools, namely
the relative embedding theorem.
• Replace Th(γk,d) by the Thom space of a universal vector bundle for vector bundles
with a structure. Examples of these are:
– Th(˜γk,d) to get cobordism classes of submanifolds B ⊆ M with an orientation
of the normal bundle. It M was oriented this is equivalent to an orientation
of B. Here ˜γk,d is a pullback of the universal as in the diagram
˜γk,d
//

γk,d

˜Gk,d
// Gk,d
with ˜Gk,d → Gk,d the double covering obtained by considering the d-dimensional
subspaces of Rk
together with an orientation. For ˜Gk,d we have
[M, ˜Gk,d] ∼=
isomorphism classes of oriented
vector bundles over M
7.A. POSSIBLE MODIFICATIONS OF THE THOM’S THEOREM 39
– Sd
= Th(Rd
→ ∗) to get cobordism classes of submanifolds with (a cobordism
class of) a trivialization of the normal bundle. This is because the pullback
E //

Rd

M // ∗
provides a trivialization of the vector bundle E ∼= M × Rd
. In particular
[Mn
, Sn
] is in bijection with finite subsets of M with orientations of the tangent
spaces at the points. Is M is oriented this is equivalent to a sign. Two
such subsets of a connected manifold M are cobordant iff the total signs
agree (if M is oriented) or if the number of points agree mod 2 (if M is
non-orientable). This identifies
[Mn
, Sn
] ∼=
Z M oriented
Z/2 M non-orientable
Serious HW: fill in details.
– complex, symplectic structures
CHAPTER 8
Index of a vector field
The idea behind the index is roughly the same as with degrees: if X ∈ Γ(TM) we
count zeros of X with appropriate signs.
More generally we define an intersection number. Let M, N be smooth manifolds with
M closed and j : A   // N a closed submanifold with m + a = n, i.e. dim M = codim j.
Assume that f : M → N is transverse to A. Then f−1
(A) is a discrete subset of M, hence
finite. If we also assume that M, N and A are oriented then for x ∈ f−1
(A) we can say
whether the composition
TxM
f∗
//
∼=
11Tf(x)N // // Tf(x)N/Tf(x)A = ν(j)f(x) (8.1)
preserves or reverses orientation and put #x(f, A) = ±1 accordingly. This is because ν(j)
inherits orientation from N and A. Namely, if (e1, . . . , ea, ea+1, . . . , en) is positive in Tf(x)N
and (e1, . . . , ea) is positive in Tf(x)A then ([ea+1], . . . , [en]) is positive in ν(j)f(x). We define
the intersection number #(f, A) of f with A to be
x∈f−1(A)
#x(f, A)
Remark. If A = {y} with its canonical orientation then #(f, A) = deg(f, y).
Theorem 8.1. If f0, f1 : M → N are two homotopic maps, both transverse to A then
#(f0, A) = #(f1, A).
Proof. Make the homotopy transverse to A and take f−1
(A), a union of intervals
and circles. For the intervals compare the local intersection numbers at the two boundary
points.
Therefore the intersection number provides a map
[M, N]
#(−,A)
−−−−−→ Z
given by first approximating a map by one transverse to A and then taking the intersection
number.
Definition 8.2. Let X ∈ Γ(TM) be a vector field and assume that X M (M ⊆ TM
as the zero section). The index Ind X of X is defined to be #(X, M) (under the assumption
that M is oriented).
40
8. INDEX OF A VECTOR FIELD 41
Corollary 8.3. If X0, X1 are any two vector fields with X0 M, X1 M then
#(X0, M) = #(X1, M).
Proof. The required homotopy X0 ∼ X1 is given by (1 − t)X0 + tX1.
We denote the local index at x ∈ X−1
(0) by Indx X. To compute it we choose a
coordinate chart around x and write X in this chart
X : U → U × Rm
Composing with the projection
f : U
X
−−→ U × Rm pr
−−→ Rm
the local index is precisely the degree of this composition Indx X = degx(f) as the differential
df(x) equals the composition (8.1) defining the local intersection number.
Observe that by changing the orientation of U we correspondingly change the orientation
of Rm
and thus the index Indx X does not depend on the orientation.
As sections transverse to a subbundle exist by the bundle version of the transversality
theorem the index Ind X depends only on M (only if M is closed!) and we denote it from
now on by χ(M) as we will prove shortly that it equals the Euler characteristics of M. First
we describe χ(M) as a (unique) obstruction to the existence of a nonvanishing (nowhere
zero) vector field: obviously if there exists X ∈ Γ(TM) nowhere zero then χ(M) = 0 as
there are no intersections to count.
Theorem 8.4. Let M be a connected closed1
manifold. If χ(M) = 0 then there exists
a nonvanishing vector field on M.
Proof. We will give the proof in several steps.
Step I. “reduce to a local problem”: Let X ∈ Γ(TM) be any section transverse to M
and let X−1
(0) = {x1, . . . , xk}. We choose a coordinate disc D and move the points xi
inside D one by one. There is a diffeomorphism ϕ1 : M
∼=
−−→ M sending x1 into D. In the
next step we use a compactly supported2
diffeomorphism
ϕ2 : M − {ϕ1(x1)}
∼=
−−→ M − {ϕ1(x1)}
sending x2 into D. Having a compact support, ϕ2 extends to a diffeomorphism
ϕ2 : M
∼=
−−→ M
preserving ϕ1(x1). The composition
ϕ = ϕk ◦ · · · ◦ ϕ1
1This assumption cannot be removed. All noncompact manifolds possess a nonvanishing vector field,
as well as all manifolds with boundary.
2supp ψ = {x | ψ(x) = x}
8. INDEX OF A VECTOR FIELD 42
is a diffeomorphism of M mapping X−1
(0) into D. Therefore
Y = ϕ∗ ◦ X ◦ ϕ−1
: M // TM
M
X //
ϕ
OO
TM
ϕ∗
OO
satisfies Y −1
(0) ⊆ D and we reduced to a local problem.
Step II. “reduce index of Y to a degree”: In a chart we consider
f : U → Rm
Y = (id, f) : U → U × Rm
Recall that Indxi
X = degxi
f where xi ∈ Y −1
(0). For simplicity we put xi = 0. Choose a
small disc Di around xi not containing any other points of Y −1
(0). We claim that for the
map
g : ∂Di −→ Sm−1
y −→
f(y)
|f(y)|
Ind0 Y = deg g. This is because on Di we have a homotopy
ft(y) = 1/t · f(ty)
from df(0) to f and therefore the local indices are the same as well as the degrees of the
corresponding maps g. Now df(0) : Di → Rm
is a restriction of a linear isomorphism
and therefore homotopic through such to an orthogonal map (a “movie” version of GramSchidt
orthogonalization process). This homotopy applied to both f and g reduces via
homotopy invariance to the case when f itself is orthogonal and thus g = f|∂Di
. In this
case deg0 f = det f and as f preserves the outward pointing vectors deg g = det f too3
.
Step III. “putting everything together”: We have in a coordinate chart U a disc D
containing all the zeros {x1, . . . , xk} of a vector field Y and around each xi a disc Di. We
may assume that they are disjoint and contained in D. Putting W = D − k
i=1 Di the
above map g can be extended to W
g : W → Sm−1
y →
f(y)
|f(y)|
As g is smooth we know that deg g|∂W = 0. By step II. we have
k
i=1
deg g|∂Di
=
k
i=1
Indxi
Y = χ(M) = 0
3A basis (e2, . . . , em) in T(∂Di) is positive iff (e1, e2, . . . , em) is positive in TDi where e1 is outward
pointing. Applying the differential of g we obtain (f(e2), . . . , f(em)). As f(e1) is still outward pointing
this new basis has the same orientation in TSm−1
as (f(e1), f(e2), . . . , f(em)) has in TRm
. In particular
it is oriented iff f preserves orientation.
8.A. MORSE THEORY 43
and this implies deg g|∂D = 0. But we know (either from algebraic topology or preferably
from the homework) that
[∂D, Sm−1
]
deg
∼=
// Z
and hence g|∂D ∼ ∗. Consequently g extends to a map
˜g : D → Sm−1
As ˜g|∂D = g|∂D ∼ f|∂D : ∂D −→ Rm
− {0} there is also an extension
˜f : D → Rm
− {0}
of f|∂D (by the homotopy extension property of ∂D → D) defining a new vector field ˜Y
on D with no zeros. We can extend ˜Y to the whole M by the original Y as they agree on
∂D. Then ˜Y is nowhere vanishing.
8.A. Morse theory
Now we will identify χ(M) with the Euler characteristics. We will consider special
vector fields arising from functions that will provide a cell decomposition.
A function f : M → R is called Morse if for all critical points x ∈ M (points where
df(x)) the second derivative at x
TxM ⊗ TxM
d2f(x)
−−−−−→ R
is a nondegenerate summetric bilinear form. Such critical points are called nondegenerate.
The index of f at a nondegenerate critical point x is the index of d2
f(x), i.e. the number
of negative eigenvalues.
Aside. To define the second derivative we take the analytic approach. We express
d2
f(x) in a chart. If ϕ : Rm
→ Rm
is a local diffeomorphism sending y to x then
d2
(f ◦ ϕ)(y) = · · · = (d2
f(x)) · (dϕ(y) ⊗ dϕ(y)) + df(x) · d2
ϕ(y)
In the case that df(x) = 0 the second term vanishes and the second derivative is independent
of the chart chosen to define it.
Geometrically let X, Y ∈ TxM and extend then to local vector fields. Then define
d2
f(x)(X, Y ) = X(Y (f))(x)
but one has to show independence of the extensions.
In general for maps f : M → Rn
the second derivative is defined on
TxM ⊗ TxM → coker df(x)
and to make d2
f independent even of the charts on the target manifold (which is not what
we were after with Morse functions) we need to further restrict to
(ker df(x)) ⊗ (ker df(x)) → coker df(x)
8.A. MORSE THEORY 44
An equivalent condition for f to be Morse is that the differential df : M → T∗
M is
transverse to the zero section M: locally
df : U −→ U × Rm
x −→ (x, ∂x1 f, . . . , ∂xm f)
and the nondegeneracy of x translates to d(pr2 ◦ df)(x) being an isomorphism, which is
just the transversality. We will prove later that such functions exist and even such that
f(xi) = f(xj) for different critical points xi = xj.
a picture of a torus with the height function
From the Taylor expansion near a nondegenerate critical point 0
f(y) = f(0) + 1/2 · d2
f(0)(y ⊗ y) + the remainder
In fact we can ignore the remainder completely by choosing a good coordinate chart around
x, the so-called Morse chart.
Theorem 8.5 (Morse’s Lemma). Let x ∈ M be a nondegenerate critical point of index
i then there is a chart ϕ around x in which
(f ◦ ϕ−1
)(y1, . . . , ym) = f(x) − y2
1 − · · · − y2
i + y2
i+1 + · · · + y2
m
Remark. This means that f is 2-determined at x: for any function agreeing with f
up to derivatives of order 2 at x there is a diffeomorphism relating the two functions
U
∃g ∼=

f
%%
R
˜U
˜f
::
This is because ˜f must be Morse too and the Morse charts for f and ˜f combine to produce
the diffeomorphism g. Another point of view is that r-determined functions are polynomial
(but not vice versa), for they agree with their Taylor polynomial and hence they only differ
by a coordinate change.
Proof. Take any chart ψ : M → Rm
around x with ψ(x) = 0 and write
(f ◦ ψ−1
)(y) = f(x) + T2
0 (f ◦ ψ−1
)(y)(y ⊗ y)
The map T2
0 (f ◦ ψ−1
) : U → homsym(Rm
⊗ Rm
, R) is smooth and at 0
T2
0 (f ◦ ψ−1
)(0) = 1/2 · d2
(f ◦ ψ−1
)(0)
is nondegenerate, thus nondegenerate for all y near 0. For such y one can find a linear
transformation Qy such that
T2
0 (f ◦ ψ−1
)(y) ◦ (Qy ⊗ Qy)
is A =
−Ei 0
0 Em−i
. Setting y = Qyz or z = Q−1
y y
T2
0 (f ◦ ψ−1
)(y)(y ⊗ y) = A(z ⊗ z)
8.A. MORSE THEORY 45
Then η : y → Q−1
y y transforms f into the wanted form
Rm
η

M
ψff
xx
f
%%
Rm
z→f(x)+A(z⊗z)
// R
Assuming for a bit that Q is smooth we compute
dη(0)(v) = Q−1
0 v =⇒ dη(0) = Q−1
0 invertible
Hence we are left with the smoothness of Q. Let us fix a definite choice of Q0 in the form
of a product of matrices corresponding to elementary column operations. The entries of
each of these matrices is a rational function of the entries of d2
(f ◦ ψ−1
)(0). Therefore in
a neighbourhood of 0 one can use the same functions to get Qy.
Let us call a Riemannian metric g on M compatible with f if there exists a Morse chart
around each critical point (a chart from the Morse’s lemma) in which g is the standard
metric.
Lemma 8.6. Let f be a Morse function on M. Then there exists a Riemannian metric
compatible with f.
Proof. Glue local Riemannian metrics as usual and use compatible local metrics near
every critical point.
A gradient vector field grad f of f is a vector field dual (with respect to the metric) to
df, i.e.
df(X) = grad f, X ∀X ∈ TM
As grad f corresponds to df under the isomorphism TM ∼= T∗
M and df M so is
grad f M. Thus we can use the gradient vector field for computing χ(M). Before doing
so we first describe how M is built in steps, a process controlled by the Morse function f.
On the complement of the critical set in addition to grad f we also consider
X =
grad f
| grad f|
Its main property is that f increases linearly along its flowlines
d
dt t=t0
f(FlX
t (x)) = Xf(FlX
t0
(x)) = df(X(FlX
t0
(x))) = grad f, X(FlX
t0
(x)) = 1
Therefore f(FlX
t (x)) = f(x) + t. This proves the following theorem
Theorem 8.7. If a < b are such that there is no critical value in [a, b] then
f−1
[a, b] ∼= f−1
(a) × [a, b]
8.A. MORSE THEORY 46
in a strong sense. Namely the following diagram commutes
(x, t)
_

f−1
(a) × [a, b]
pr
))
∼=

[a, b]
FlX
t−a(x) f−1
[a, b]
f
55
picture
Proof. The inverse is given by y → (FlX
a−f(y)(y), f(y)).
The outstanding question is: what happens to the level surface f−1
(c) when passing
though a critical level? picture of a cell and the deformation
Theorem 8.8. Let a < b be two regular values of f such that there is precisely one
critical point in f−1
[a, b], say of index i. Then there is a i-cell
Di   ϕ
// f−1
[a, b]
with ϕ(∂Di
) ⊆ f−1
(a) such that f−1
[a, b] deformation retracts onto f−1
(a) ∪ϕ Di
.
Proof. The (idea of the) proof was summarized in a picture which I am unable to
draw in TeX.
Now we will finish the determination of χ(M). For our Morse function f with all critical
points in distinct levels we choose a0, . . . , ak ∈ R such that for the critical points x1, . . . , xk:
f(xj) ∈ (aj−1, aj). Let ij be the index of xj and abbreviate
Mj = f−1
(−∞, aj)
so that M0 = ∅ and Mk = M. picture We showed in the last theorem that
M = Mk Mk−1 ∪ Dik
...
Mj Mj−1 ∪ Dij
...
M1 Di1
(and i1 = 0, ik = m).
Remark. This implies that M is homotopy equivalent to a CW-complex with cells
of dimensions i1, . . . , ik. For the above homotopy equivalences can be expressed in saying
that
Sij−1
fj
//
 _

Mj−1

Dij // Mj
8.A. MORSE THEORY 47
is a homotopy pushout. By induction Mj−1 is homotopy equivalent to a CW-complex Xj−1
with cells of dimensions i1, . . . , ij−1. The homotopy invariance of the homotopy pushout
together with a homotopy of fj to a cellular map ˜fj gives Mj Xj−1 ∪˜fj
Dij
= Xj. Now
Xj is a CW-complex as the Dij
is glued to the (ij − 1)-skeleton.
In the proceeding we will only need a weaker version. We denote the usual (homological)
Euler characteristics by
χh(M) =
m
j=0
(−1)j
dim Hj(M)
Proposition 8.9. With the notation from above χh(M) = k
j=1(−1)j
.
Proof. Denoting the inclusion Sij−1
→ Mj−1 by fj let us consider the mapping cylinder
Mfj
of fj. Obviously Mfj
Mj−1 and contains a copy of Sij−1
and we form a long
exact reduced homology sequence of the pair (Mfj
, Sij−1
):
· · · // ˜Hp(Sij−1
) // ˜Hp(Mfj
) // ˜Hp(Mfj
/Sij−1
) // · · ·
As Mfj
Mj−1 and Mfj
/Sij−1
Mj for the Euler characteristics we get
(χh(Mj−1) − 1) = (−1)i1−1
+ (χh(Mj) − 1)
or in other words χh(Mj) = χh(Mj−1) + (−1)ij
. By induction and M = Mk, χh(M0) = 0
we obtain the result.
To identify χ(M) with χh(M) we only need
Proposition 8.10. The index of the vector field grad f at xj equals (−1)ij
.
Proof. Locally in the Morse chart (with xj = 0)
f(y, z) = f(xj) − |y|2
+ |z|2
(y, z) ∈ Rij
× Rm−ij
Then (grad f)(y, z) = (−2y, 2z) and Ind0 grad f = (−1)ij
as the sign of the determinant of
grad f.
CHAPTER 9
Function spaces
To proceed further we need to topologize the set C∞
(M, N) of smooth maps between
manifolds M, N. This topology should reflect all the differentiable qualities of maps: then
maps transverse to a closed submanifold A ⊆ N will form an open subset as transversality
is an open condition on the first derivative. It would not be open if only the values of
maps were considered. Similarly Morse functions are defined by conditions on the first and
second derivative.
To get hold of smooth maps and in particular to their derivatives, we need to describe
these geometrically. Namely we define Jr
(M, N) to be the set (of r-jets of maps M → N)
of equivalence classes of pairs (x, f) where x ∈ M and f : U → N a smooth map defined
on a neighbourhood U of x. The equivalence relation is: (x, f) ∼ (x , f ) iff x = x and
f agrees with f at x = x up to derivatives of order r (expressed in but independent of
charts around x and f(x)). We denote [(x, f)] = jr
xf. The point x is called the source and
f(x) the target of the jet jr
xf. In this way we obtain the map
Jr
(M, N)
(σ,τ)
−−−−→ M × N
and denote Jr
x(M, N) = σ−1
(x), the jets with source x. Locally for open sets U ⊆ Rm
,
V ⊆ Rn
Jr
(U, V ) ∼= U × V ×
r
k=1
homsym (Rm
)⊗k
, Rn
The factor U corresponds to the source, V to the target and homsym (Rm
)⊗k
, Rn
to the
k-th derivative. The canonical identification is therefore
jr
xf → (x, f(x), df(x), . . . , dr
f(x))
(and the inverse map could be realized by polynomials as there is exactly one polynomial
or degree at most r with prescribed value and derivatives up to order r at a given point
x). Varying these over charts on M and N we obtain an atlas on Jr
(M, N) and in fact a
bundle structure1
on
Jr
(M, N)
(σ,τ)
−−−−→ M × N
We will need later the composition of jets: we define jr
yf ◦ jr
xg := jr
x(f ◦ g) provided that
y = g(x). One needs to check that this is well-defined and therefore provides a canonical
map
Jr
(N, P) ×N Jr
(M, N) → Jr
(M, P)
1Moreover the chain Jr
(M, N) → · · · → J1
(M, N) → J0
(M, N) = M × N consists of affine bundles.
48
9. FUNCTION SPACES 49
which is moreover smooth2
. Another important tool is the jet prolongation
jr
: Cr
(M, N) −→ C0
(M, Jr
(M, N))
f −→ (x → jr
xf) =: jr
f
In fact jr
f is a section of the bundle σ : Jr
(M, N) → M and captures all the derivatives
of f. It is good for inducing topology on Cr
(M, N) once we say what the right topology
on C0
(M, N) is. In fact it turns out to be equally easy to topologize the set of continuous
sections, the situation we would like to consider anyway.
For this part let X be a topological space and p : Y → X a space over X, one may
think of it as a bundle. We would like to describe a convenient topology on the set Γ(Y )
of continuous sections of p : Y → X. If X is compact Hausdorff and Y metric (in fact one
only needs metric on each fibre) then Γ(Y ) could be equipped with a metric
d(f, g) = max
x∈X
d(f(x), g(x))
In the case of the product projection X × Z → X we use the notation C(X, Z) instead of
Γ(X × Z) (which does not tell what the map p is).
Lemma 9.1. The topology induced by the above metric on Γ(Y ) is generated by the open
sets
O(W) = {f ∈ Γ(Y ) | the image of f lies in W}
with W ⊆ Y varying over all open subsets3
.
Proof. Let g ∈ Γ(Y ) and ε > 0. Defining W as {y | d(y, g(p(y))) < ε} we easily verify
O(W) = Bε(g). On the other hand for g ∈ O(W) we take ε = infy /∈W d(y, g(p(y))) and
verify that ε > 0: this is because ε ≥ d(Y − W, g(X)) > 0. Clearly Bε(g) ⊆ O(W).
This topology behaves nicely (with respect to composition, evaluation, etc.). It extends
to noncompact spaces in two ways
• the strong topology ΓS(Y ) is generated again by sets O(W) as W ⊆ Y varies over
all open subsets. Alternatively for X paracompact and Y metric we can exhaust
a neighbourhood basis of a fixed g via sets of the form
N(g, ε) = {f ∈ Γ(Y ) | d(f(x), g(x)) < ε(x)}
with ε ranging over all positive functions X → (0, ∞). To verify this we are
required to find for every open neighbourhood W of the image of g a positive
function ε such that
{y ∈ Y | d(y, g(p(y))) < ε(p(y))} ⊆ W
2Locally one takes two polynomials of degree r and then substitutes one into the other, truncating the
part of degree bigger then r. The coefficients of the result are smooth functions of the coefficients of the
original polynomials.
3Typically neighbourhoods of the image of a fixed section g : X → Y . In this case the set of all such
O(W) provide a neighbourhood basis of g.
9. FUNCTION SPACES 50
Choose for each x ∈ X some εx > 0 such that B2εx (g(x)) ⊆ W. Then for
z ∈ Ux := g−1
Bεx (g(x)) we have Bεx (g(z)) ⊆ W. Let λx be a partition of unity
subordinate to (Ux). Then the required ε is ε = x∈X λxεx.
• the weak topology ΓW (Y ) is generated by res−1
K O(W) where K is a compact subset
of X and W an open subset of Y |K = p−1
(K). Here
resK : Γ(Y ) → Γ(Y |K)
is the restriction map. The weak topology is usually referred to as the compactopen
topology.
Homework: show that a sequence (fi) in CS(M, N) converges iff fi are eventually
constant outside of some compact subset of M where it converges uniformly.
Remark. Another point of view on the weak topology is as the limit topology
C(X, Y ) = lim
K⊆X
C(K, Y )
which is necessary (if one wants to have a categorically well-behaved internal hom functor)
at least on the category of compactly generated Hausdorff spaces:
X compactly generated ⇐⇒ X = colim
K⊆X
K
What we are describing here is an extension of C(−, Y ) from the category of compact
Hausdorff spaces to its closure CGHaus under colimits in the category of all Hausdorff
spaces. Again CW (−, −) turns CGHaus into a cartesian closed category, a very useful
property.
For the purpouses of differential topology the strong topology is better, we will therefore
restrict ourselves just to that and drop the index “S”. We can now use the map
jr
: Cr
(M, N) −→ Γ(Jr
(M, N))
to induce topology from the right hand side, i.e. endow Cr
(M, N) with the subspace topology.
The topology on C∞
(M, N) is defined as a limit (intersection)
C∞
(M, N) = lim
r
Cr
(M, N)
In concrete terms, the neighbourhood basis of g ∈ C∞
(M, N) consists of (the intersections
with C∞
(M, N) of) the neighbourhoods of g in Cr
(M, N), r ≥ 0. Alternatively one can
define
J∞
(M, N) = lim
r
Jr
(M, N)
together with j∞
: C∞
(M, N) → Γ(J∞
(M, N)) and induce the topology from there (which
is the best way). The most useful property of C∞
(M, N) happens to be that it satisfies
the Baire property.
Definition 9.2. A subset of a topological space X is called residual if it contains a
countable intersection of open dense subsets of X. A topological space X is called Baire
if every residual subset is dense.
9. FUNCTION SPACES 51
Recall that every complete metric space is Baire. We will now generalize this significantly.
Let Y → X be a space over X with Y equipped with a metric. A subset Q ⊆ Γ(Y )
is called uniformly closed if it contains the limit of any uniformly convergent sequence in
Q.
Proposition 9.3. If X is paracompact and Y complete metric then any uniformly
closed Q ⊆ Γ(Y ) is Baire (in the strong topology).
Remark. By taking X = ∗ and Q = C(∗, Y ) = Y we deduce that every complete
metric space is Baire.
Proof. Let U ⊆ Q be a nonempty open subset and (Ui) a sequence of open dense
subsets of Q. We must show that U ∩ ∞
i=0 Ui = ∅. To achieve this we inductively
construct a sequence of sections fi ∈ Q together with positive functions εi : M → (0, ∞)
in such a way that ε0 ≤ 1, εi ≤ 1/2 · εi−1 and
Q ∩ N(fi, 2εi) ⊆ Ui ∩ N(fi−1, εi−1)
This is possible because Ui being open dense must intersect N(fi−1, εi−1) in a nonempty
open subset. Choosing any fi from this intersection εi is guaranteed by the fact that
N(fi, ε) form a neighbourhood basis of fi (as X is paracompact). By our choice the
distance d(fi(x), fj(x)), i < j, is uniformly bounded by εi ≤ 2−i
. Hence (fi) converges
uniformly (as Y is complete) and the limit f = lim fi ∈ Q. Finally, for j > i, fj ∈ N(fi, εi)
and thus
f ∈ Q ∩ N(fi, εi) ⊆ Q ∩ N(fi, 2εi) ⊆ Ui
We are trying to complete the proof of the space Cr
(M, N) being Baire. First we find
a complete metric on every smooth manifold, in particular on Jr
(M, N).
Lemma 9.4. Let M be a manifold. Then there exists a proper function f : M → R+.
Proof. By definition f is proper if f−1
(K) is compact for all K compact. Recall the
decomposition of M into a union
M =
∞
i=0
Ki
of compact subsets Ki with Ki lying in the interior of Ki+1. Take Li = ˚Ki − Ki−2 and
consider its corresponding partition of unity λi. Then f = ∞
i=1 λi ·i is the required proper
function - outside of Ki, f(x) > i, thus f−1
[0, i] ⊆ Ki.
Theorem 9.5. Every smooth manifold admits a complete metric.
Proof. Let d : M × M → R+ be any metric on M. Define
d (x, y) = d(x, y) + |f(x) − f(y)|
where f is any proper function on M. Easily d is a metric and for the corresponing balls
around x, Bε(x) ⊆ Bε(x). On the other hand Bε(x) = {y | d (x, y) < ε} is open (with
9. FUNCTION SPACES 52
respect to the usual topology) and so the topologies induced by d and d coincide. It
remains to show that the metric d is complete. Hence let (xi) be a sequence which is
Cauchy with respect to d . Then f(xi) is necessarily bounded, say by L, and therefore
{xi} ⊆ f−1
[0, L], a compact set. Consequently (xi) has a limit point and converges.
Remark. This metric satisfies an additional property: the closed balls in this metric
are all compact. This is because BL(x) ⊆ f−1
[0, f(x) + L].
The last piece of the puzzle is the following proposition
Proposition 9.6. The image of
jr
: Cr
(M, N) → Γ(Jr
(M, N))
is uniformly closed. In particular it is a Baire space.
Proof. Let (jr
fi) be a sequence uniformly converging to g : M → Jr
(M, N). Then
we claim that g = jr
(τ ◦ g). This is a local problem where this means that when all
the derivatives dk
fi, k = 0, . . . , r, converge uniformly to gk
then gk
= dk
g0
, a well-known
fact.
Observe that the proof also works for r = ∞. We will state this as a theorem
Theorem 9.7. The space C∞
(M, N) is Baire.
Proof. As noted before the statement it only remains to show that J∞
(M, N) admits
a complete metric. By definition as a limit there is an inclusion
J∞
(M, N) ⊆
∞
r=0
Jr
(M, N)
as a susbet of compatible jets. This subset is clearly closed and thus the proof is finished
by refering to the standard fact that a countable product of complete metric spaces has a
complete metric. If the complete metrics on the factors are di with di ≤ 2−i
(which can be
achieved by replacing di by max{2−i
, di}) then the metric on the product is e.g.
d((xi), (yi)) =
∞
i=0
di(xi, yi)
Now we discuss certain naturality properties. From the diagram
C∞
(M, N)   //
f∗

Γ(J∞
(M, N))
f∗

C∞
(M, P)   // Γ(J∞
(M, P))
the map f∗ on the left is continuous. It is not the case that g∗
is continuous (unless g is
proper).
9. FUNCTION SPACES 53
Theorem 9.8. The canonical map
jr
: C∞
(M, N) → C∞
(M, Jr
(M, N))
is an embedding of a subspace.
Proof. There is a retraction τ∗ : C∞
(M, Jr
(M, N)) → C∞
(M, N), induced by the
target map. What we are left with is the continuity of
C∞
(M, N)
jr
−−→ C∞
(M, Jr
(M, N))
js
−−→ Γ(Js
(M, Jr
(M, N)))
This composition can be also written as
C∞
(M, N)
jr+s
−−−→ Γ(Jr+s
(M, N))
α∗
−−→ Γ(Js
(M, Jr
(M, N)))
where α∗ is induced by the canonical map α : Jr+s
(M, N) → Js
(M, Jr
(M, N)) sending
jr+s
x f to js
x(jr
f) once one checks this to be independent of the representing map f (within
the maps with the same (r + s)-jet at x) and also smooth. Both facts are easily verified
using polynomials.
We can now make certain statements involving “close” maps and “small” homotopies
precise. Let f : M → N be a smooth map and A ⊆ N a submanifold. We say that f is
transverse to A on K ⊆ A if f is transverse to A at each point x ∈ f−1
(K).
Proposition 9.9. Let M, N be smooth manifolds and A ⊆ N a submanifold. Let
K ⊆ A be a compact subset (or more generally K closed in N). Then the set
X = {f : M → N | f A on K}
is open in C∞
(M, N).
Proof. Clearly transversality is a condition on a 1-jet of f (i.e. condition on the values
and the first derivatives). Therefore we consider J1
(M, N) and its subset of jets fulfilling
our condition
W = {j1
xf | f(x) /∈ K or f∗(TxM) + Tf(x)A = Tf(x)N}
Then the set X from the statement can be also described as X = (j1
)−1
(O(W)) with
j1
: C∞
(M, N) → Γ(J1
(M, N))
It suffices to show that W is open as then X will be one of the generating open sets in
C∞
(M, N). As openness is a local property we can work in one of the charts on J1
(M, N),
i.e. reduce to the case
W ⊆ Rm
× Rn
× hom(Rm
, Rn
)
with A corresponding to a linear subspace Rk
⊆ Rn
and K ⊆ Rk
a closed subset. Then
W is defined by the requirement: either the component in Rn
lies in the complement of
K or the component in hom(Rm
, Rn
) when composed with the projection Rn
→ Rn
/Rk
is
surjective, a union of two open subsets.
9. FUNCTION SPACES 54
Corollary 9.10. Let M, N be smooth manifolds and A ⊆ Jr
(M, N) a submanifold.
Let K ⊆ A be a compact subset (or more generally K closed in Jr
(M, N)). Then the set
X = {f : M → N | jr
f A on K}
is open in C∞
(M, N).
Proof. One reduces to the previous proposition via the canonical continuous map
jr
: C∞
(M, N) → C∞
(M, Jr
(M, N)).
The situation of the previous corollary is called a jet transversality. A lot of situations
we were studying can be described by jet transversality.
Example 9.11. We saw that a smooth function f : M → R is Morse iff df : M → T∗
M
is transverse to the zero section. It is not hard to identify J1
(M, R) with R × T∗
M and
under this identification j1
f : M → J1
(M, R) becomes (f, df). Thus a function f is Morse
iff j1
f R × M with R × M ⊆ J1
(M, R) as the subset of all jets (at various points) of all
constant functions. In particular Morse functions form an open subset of C∞
(M, R).
Example 9.12. A map f : M → N between two smooth manifolds is an immersion iff
j1
f does not meet the subset of J1
(M, N) of the non-immersive jets. Locally
J1
(Rm
, Rn
) ∼= Rm
× Rn
× hom(Rm
, Rn
and f is an immersion iff j1
f misses all Rm
× Rn
× homr(Rm
, R)
for the rank r < m. We
computed earlier that each homr(Rm
, Rn
) is a submanifold of hom(Rm
, Rn
) with the lowest
codimension n − m + 1 for r = m − 1. Therefore when n ≥ 2m this condition is equivalent
to jet transversality (in this case with respect to a finite number of submanifolds).
The main theorem on jet transversality is the following.
Theorem 9.13 ((Thom’s) Jet Transversality Theorem). Let M, N be smooth manifolds,
∂N = ∅, and A ⊆ Jr
(M, N) a submanifold. Then the set
X = {f : M → N | jr
f A}
is residual in C∞
(M, N). It is moreover open provided that A is closed (as a subset).
Proof. We cover A by a countable number of compact subsets Ai with the following
properties.
• σ(Ai) lies in a coordinate chart Rm ∼= Ui on M.
• τ(Ai) lies in a coordinate chart Rn ∼= Vi on N.
We will show that each of the sets
Xi = {f : M → N | jr
f A on Ai}
is open dense. The openness part was proved in Corollary 9.10 so we are left with the
density.
We identify Ui with Rm
and Vi with Rn
using the charts (and thus reduce to a local
problem). Let f0 : M → N be a smooth map, we will find an arbitrarily close map which
is transverse to A on Ai. First we denote Wi = Ui ∩ f−1
0 (Vi) ⊆ Rm
the open subset where
9. FUNCTION SPACES 55
f0 can be written in the charts Ui and Vi. Let λ : Wi → R+ be a function with compact
support and which equals 1 on a neighbourhood Zi of σ(Ai) ∩ f−1
0 (τ(Ai)). Consider the
following family of smooth maps Wi → Rn
F : Jr
0 (Rm
, Rn
) × Wi −→ Rn
(jr
0p, x) −→ f0(x) + λ(x) · p(x)
Here p is the unique polynomial representative of the jet in question. Since λ is compactly
supported we can extend this family to a globally defined one, still denoted by F
F : Jr
0 (Rm
, Rn
) × M −→ N
For each α ∈ Jr
0 (Rm
, Rn
) take the r-jet of Fα to obtain a map
G : Jr
0 (Rm
, Rn
) × M −→ Jr
(M, N)
Our goal is to show that G A on a neighbourhood of Ai so that we can apply the
parametric transversality theorem to conclude that there is a sequence (αk) in Jr
0 (M, N)
converging to 0 so that each Gαk
A on Ai. This will finish the proof of density of Xi as
limk→∞ Fαk
= f0 in4
C∞
(M, N). To achieve this observe that on Zi we can express G in
the charts and, as λ ≡ 1 there,
G(jr
0p, x) = jr
x(f0 + p)
Thus G is in fact submersive on Jr
0 (Rm
, Rn
) × Zi. Now f0(σ(Ai) − Zi) is a compact subset
disjoint with τ(Ai). Therefore even for α ∈ Jr
0 (Rm
, Rn
) small the same will be true5
for
Fα. As for such α, G−1
α (Ai) is a subset of σ(Ai) disjoint with Zi and the transversality
holds on Zi the restricted map G is transverse to A on Ai. If we start instead of Ai with
its compact neighbourhood Bi ⊆ A (still with σ(Bi) ⊆ Ui and τ(Bi) ⊆ Vi) we get that the
(now even more) restricted map G is even transverse to A on Bi and in particular on the
interior of Bi - a manifold. Therefore we can invoke the parametric transversality theorem
to conclude the proof.
Therefore Morse functions form a residual subset of C∞
(M, R) and, for n ≥ 2m, so do
the immersions in C∞
(M, N).
Next we discuss a relative version of the jet transversality theorem. Let F ⊆ M be a
closed subset and s0 : F → Jr
(M, N) a section of the jet bundle over F. Then clearly
{s ∈ Γ(Jr
(M, N)) | s|F = s0}
is a uniformly closed subset (observe that this works even for r = ∞). Therefore the
corresponding subspace
C∞
(M, N)s0 = {f ∈ C∞
(M, N) | jr
f|F = s0}
is a Baire space. We then obtain the following generalization of Theorem 9.13:
4This is because the association α → Fα is continuous. This on the other hand follows from the fact
that the family F is constant outside of a compact set.
5As α → Fα is continuous this follows from the openness of the corresponding subset
O (M × (N − τ(Ai)) ∪ (M − (σ(Ai) − Zi)) × N) in C0
(M, N) = Γ(M × N → M).
9. FUNCTION SPACES 56
Theorem 9.14. Let M, N be smooth manifolds, ∂N = ∅, A ⊆ Jr
(M, N) a submanifold
and s0 : B → Jr
(M, N) a smooth section over some fixed closed (as a subset) submanifold
B ⊆ M for which s0 A. Then the set
X = {f ∈ C∞
(M, N)s0 | jr
f A}
is residual in C∞
(M, N)s0 . It is moreover open provided that A is closed (as a subset).
Proof. First note that when σ(A) ⊆ M − B this is proved by the same argument as
Theorem 9.13. Then write M − B as a countable union of compact subsets Ki and denote
Ai = A ∩ σ−1
(Ki) and correspondingly
Xi = {f ∈ C∞
(M, N)s0 | jr
f A on Ai}
Clearly X = Xi while each of the terms is residual (in fact open dense).
Corollary 9.15. Let M be a smooth manifold and B ⊆ M a closed (as a subset)
submanifold. Then every immersion f : B → Rn
extends to an immersion M → Rn
provided n ≥ 2m.
Proof. Suppose first that f extends to a smooth map ˜f : M → Rn
which is an
immersion near B. Then the last theorem applies to s0 = (j1 ˜f)|B to conclude that the set
X of immersions M → Rn
is residual in a nonempty space C∞
(M, Rn
)s0 (it contains ˜f).
In particular X itself is nonempty.
Thus it remains to find an extension ˜f. We assume here that B has codimension 0,
the only case we will need later6
. Then f extends locally at each point of ∂B and hence
to a neigbhbourhood of ∂B in M by the means of the partition of unity. The required
extension to M, automatically an immersion near B, is guaranteed by the contractibility
of Rn
.
To describe injective immersions (and later embeddings) in a way similar to immersions
we need a notion of a multijet. A multijet is a number of jets with different sources. More
formally denote by M(k)
the subset of the cartesian power Mk
of pairwise distinct k-tuples.
We define
J r
k (M, N) = (Jr
(M, N))k
|M(k)
An element of J r
k (M, N) is called a multijet. Again we have a multijet prolongation map7
j r
k : C∞
(M, N) → C∞
(M(k)
, J r
k (M, N))
Theorem 9.16 (Multijet Transversality Theorem). Let M, N be smooth manifolds,
∂N = ∅, and A ⊆ J r
k (M, N) a submanifold. Then the set
X = {f : M → N | j r
k f A}
is residual in C∞
(M, N).
6In the general case one assumes some nice position of B in M. This means that either B lies in
the interior of M or in the boundary or is a neat submanifold. In any case one extends f to a tubular
neighbourhood by a vector bundle argument which is left as a HW.
7A serious homework: explain why j r
k is NOT continuous.
9. FUNCTION SPACES 57
Proof. The proof follows that of Theorem 9.13 with the following modifications: we
require σ(Ai) to be contained in a product U1
i × · · · × Uk
i of disjoint charts on M. Then in
the density part we embed simlarly f0 into a family of maps
F : (Jr
0 (Rm
, Rn
))k
× M −→ N
and proceed analogously.8
Example 9.17 (injective maps). Let A ⊆ J 0
2 (M, N) ∼= M(2)
× N2
consists exactly of
those (x1, x2, y1, y2) with y1 = y2. Then f is injective iff j 0
2 f misses A. The codimension
of A is precisely n and thus for n > 2m, f is injective iff j 0
2 f A. In particular the set of
injective maps is residual in these dimensions. Combining with the result for immersions,
injective immersions are residual for n > 2m. To solve the problem for embeddings note
that f : M → N is an embedding as a closed submanifold of N iff f is a proper injective
immersion. We will prove that proper maps form an open subset. Then the existence of
embeddings of any manifold Mm
into R2m+1
follows from the previous.
Let N be equipped with a complete metric with compact balls (like the one constructed
in Theorem 9.5) and g : M → N be a proper map. Then
{f : M → N | d(f(x), g(x)) < 1}
is clearly an open neighbourhood of g and for any f in this set
f−1
(BL(y)) ⊆ g−1
(BL+1(y))
a compact subset.
Example 9.18 (special Morse functions). Let A ⊆ J 1
2 (M, R) consist precisely of the
pairs (j1
x1
f, j1
x2
f) with both x1 and x2 critical points of f and with f(x1) = f(x2). The
codimension of A is 2m + 1 and therefore a function f satisfies j 1
2 f A iff all the critical
points are in different levels and the set of such is residual. Together with the residuality
of the set of Morse functions, the same holds for these special Morse functions.
In fact special Morse functions are also stable: any function close to a special Morse
function f in C∞
(M, R) is equivalent to f (i.e. there are diffeomorphisms of M and R
relating the two functions).
Injective immersions are also stable. To get a stable version of immersions one needs
to restrict only to immersions with normal crossings picture.
The relative version of the multijet transversality theorem is hard even to state so we
content ourselves with the case of injective immersions. Let M be a compact manifold
with boundary and s0 : ∂M → N an injective smooth map.
Theorem 9.19. Assuming that n > 2m the set
X = {f ∈ C∞
(M, N)s0 | f is injective}
is residual in C∞
(M, N)s0 .
8A proof of openness should go as follows: the map
(jr
)k
: C∞
(M, N) → C∞
(Mk
, (Jr
(M, N))k
)
is continuous and Xi is a preimage of an open subset.
9. FUNCTION SPACES 58
Proof. Filling M − ∂M by a countable number of compact subsets Ki we again form
Xi = {f ∈ C∞
(M, N)s0 | f is injective on ∂M ∪ Ki}
Obvously X = Xi. By an easy modification of the proof of Theorem 9.16 we now prove
that each Xi is residual (or even open dense). We make sure that none of the charts used
on M meets both ∂M and Ki. We then have three possibilities on the pair U1
i , U2
i : if
both miss ∂M then we use the same argument, if both hit ∂M then f0 does not need to
be perturbed and in the mixed case we create a family only indexed by a single copy of
Jr
(Rm
, Rn
) which is still transverse to the diagonal.
Corollary 9.20. The forgetful map
the cobordism classes of closed
m-dimensional submanifolds of S2m+2 −→
the cobordism classes of closed
m-dimensional manifolds
is a bijection.
Proof. The surjectivity was proved already in Theorem 2.12. To prove injectivity let
W be a cobordism between two m-dimensional submanifolds M0, M1 of S2m+2
which we
think of as an embedding ι : ∂W   // ∂(I × S2m+2
). Removing a point from S2m+2
not
contained in M0 ∪ M1 we can replace S2m+2
by R2m+2
. We extend the embedding ι to a
neat embedding, also denoted by ι, of a collar V of ∂W in an obvious way. Denoting by
U a collar with U ⊆ V (e.g. the one corresponding to [0, 1) ⊆ R+) we are then asked to
extend an embedding
U − ∂W   // (0, 1) × R2m+2
to W − ∂W. The extension is provided by Corollary 9.15 and Theorem 9.19.
Corollary 9.21. There is a natural bijection
πm+d(Th(γk,d))
∼=
−−→
the cobordism classes of closed
m-dimensional manifolds
provided that d > m + 1 and k > m + d.