INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ČADEK
9. POINCARE DUALITY
Many interesting spaces used in geometry are closed oriented manifolds. Poincare duality expresses a remarkable symmetry between their homology and cohomology.
9.1. Manifolds. A manifold of dimension n is a Hausdorff space M in which each point has an open neighbourhood U homeomorphic to IRn. The dimension of M is characterized by the fact that for each x G M, the local homology group Hi(M, M — {x}; Z) is nonzero only for i = n since by excision and homotopy equivalence
Hi(M,M- {x};Z) = H,t(U,U - {x};Z) = H,t(Rn,Rn - {0};Z) = #j_1(S,n_1; Z). A compact manifold is called closed.
Example. Examples of closed manifolds are spheres, real and complex projective spaces, orthogonal groups 0(n) and SO(n), unitary groups U(n) and SU(n), real and complex Stiefel and Grassmann manifolds. The real Stiefel manifold Vn^ is the space of £>tuples of orthonormal vectors in IRn. The real Grassmann manifolds Grhk is the space of fc-dimensional vector subspaces of IRn.
9.2. Orientation of manifolds. Consider a manifold M of dimension n. A local orientation of M in a point x G M is a choice of a generator fix G Hn(M, M— {x}; Z) = Z.
To shorten our notation we will use Hi(M\A) for ll,[M. M A; Z) and _£P(M|/4) for H\M; M - A- Z) if A C M.
An orientation of M is a function assigning to each point x G M a local orientation fix £ Hn(M\x) such that each point has an open neighbourhood B with the property that all local orientations fiy foryEB are images of an element //# G Hn(M\B) under the map : Hn(M\B) -> Hn(M\x) where py : (M, M — {x}) -> (M,M - B) is the natural inclusion.
If an orientation exists on M, the manifold is called orientable. A manifold with a chosen orientation is called oriented.
Proposition. A connected manifold M is orientable if it is simply connected, i. e. every map S1 —^ M is homotopic to a constant map.
For the proof one has to know more about covering spaces and fundamental group. See [Hatcher], Proposition 3.25, pages 234 - 235.
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In the same way we can define an J?-orientation of a manifold for any commutative ring R. Every manifold is Z2-oriented.
9.3. Fundamental class. A fundamental class of a manifold M with coefficients in R is an element p G Hn(M; R) such that px*(p) is a generator of Hn(M\x; R) = R for each x G M where px : (M, 0) —> (M, M — {ic}) is the obvious inclusion. It is usual to denote the fundamental class of the manifold M by [M]. We will keep this notation.
If a fundamental class of M exists, it determines uniquely the orientation px = p^([M])of M.
Theorem. Let M be a closed manifold of dimension n. Then:
(a) If M is R-orientable, the natural map Hn(M;R) —> Hn(M\x;R) = R is an isomorphism for all x G M.
(b) If M is not R-orientable, the natural map Hn(M;R) —> Hn(M\x; R) = R is infective with the image {r G R; 2r = 0} for all x G M.
(c) Hi(M; R) = 0 for all i > n.
(a) implies immediately that very oriented closed manifold has just one fundamental class. It is a suitable generator of Hn(M; R).
The theorem will follow from a more technical statement:
Lemma. Let M be n-manifold and let A C M be compact. Then:
(a) Hi(M\A; R) = 0 for i > n and a G Hn(M\A; R) is zero iff its image px*{ct) G Hn(M\x; R) is zero for all x G M.
(b) // x H> px is an R-orientation of M, then there is      G Hn(M\A; R) whose image in Hn(M\x; R) is px for all x G A.
To prove the theorem put A = M. We get immediately (c) of the theorem. Further, the lemma implies that an oriented manifold M has a fundamental class [M] = pu and any other element in Hn(M; R) has to be its multiple in R. So we obtain (a) of the theorem. For the proof of (b) we refer to [Hatcher], pages 234 - 236.
Proof of Lemma. Since R does not play any substantial role in our considerations, we will omit it from our notation. We will omit also stars in notation of maps induced in homology. The proof will be divided into several steps.
(1) Suppose that the statements are true for compact subsets A, B and A fl B of M. We will prove them for A U B using the Mayer-Vietoris exact sequence:
0     Hn(M\A U5)^ Hn(M\A) © Hn(M\B) ^ Hn(M\A n B)
where = (pAa,pBa), ^(a,f3) = pAnBd - PAnEifi-
Hi(M\A U B) = 0 for i > n is immediate from the exact sequence. Suppose a G Hn(M\A U B) restricted to Hn(M\x) is zero for all x G A U B. Then pAct and are zeroes. Since $ is a monomorphism, a has to be also zero.
Take pA and such that their restrictions to Hn(M\x) are orientations. Then the restrictions to points x G A fl B are the same. Hence also the restrictions to A fl B coincide. It means ^(pA, Pb) = 0 and the Mayer-Vietoris exact sequence yields the
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existence of a in Hn(M\A U B) such that $(«) = (//^,//b). Therefore a reduces to a generator of Hn(M\x) for all rr G A U 5, and consequently, a = fiAuB-
(2) If M = Rn and A is a compact convex set in a disc D containing an origin 0, the lemma is true since the composition given by inclusions
Hi(Rn\D) —>• Hi(Rn\A) —>• Hi(Rn\0)
is an isomorhism.
(3) If M = Rn and A is finite simplicial complex in IRn, then A = IJ^=i ^ where Ai are convex compact sets. Using (1) and induction by m we can prove that the lemma holds in this case as well.
(4) Let M = Rn and A is an arbitrary compact subset. Let a G Hi(Rn\A) be represented by a relative cycle z G Zi(Rn, Rn — A). Let C C Rn — A be the union of images of the singular simplices in dz. Since C is compact, dist(C, A) > 0, and consequenly, there is a finite simplicial complex K D A such that C C Rn — K. (Draw a pisture.) So the chain z defines also an element G Hi(Rn\K) which reduces to a G Hi(Rn\A). If i > n, then by (3)      = 0 and consequently also a = 0.
Suppose that i = n and that a reduces to zero in each point x G A. K can be chosen in such a way that every its point lies in a simplex of K together with a point of A. Consequently, ax reduces to zero not only for all x G A but for all x G K. (Use the case (2) to prove it.) By (3) ax = 0, and therefore also a = 0.
The proof of existence of fiA G Hn(Rn\A) in the statement (b) is easy. Take /is G Hn(Rn\B) for a ball B D A and its reduction is fi^.
(5) Let M be a general manifold and A a compact subset in an open set U homeo-morphic to IRn. Now by excision
Hi(M\A) = Hi(U\A) = Hi(Rn\A)
and we can use (4).
(6) Let M be a manifold and A an arbitrary compact set. Then A can be covered by open sets Vi, V2, ■ ■ ■, Vm such that the closure of Vi lies in an open set Ui homeomorphic to Rn. Then by (5) the lemma holds for At = A fl Vi. By (1) and induction it holds also for      1 Ai = A. □
9.4. Cap product. Let X be a space. On the level of chains and cochains the cap product
D : Cn{X-R) <g> Ck{X- R) -> Cn_fc(X; R) is given for 0 < k < n by
a n    = v?(o-/[v0, t>i,..., vfc])o-/[vfc, Vfc+i, • • • ,fn]
where a is a singular n-simplex, 99 : Ck(X; R) —^ J? is a cochain and o"/[^o, fi, • • •, Vk] is the composition of the inclusion of Ak into the indicated face of An with a, and is given by zero in the remaining cases.
The proof of the following statement is similar as in the case of cup product and is left to the reader as an exercise.
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Lemma A. For a G Cn(X; R) and ip G Ck(X; R)
d(a n <p) = (-l)k(do- f] <p - a f] 5<p) This enables us to define
D : Hn(X; R) <g> #fc(X; R) -> Hn_k(X; R)
by
[a] n M = [o- n ^]
for all cycles a and cocycles ip. In the same way one can define
D : Hn(X, A; R) <g> #fc(X; R) -> Hn_k(X, A; R)
for any pair (X, A) and
n : Hn(X, A U 5; i?) <g> A; i?)     Hn_k(X, B; R)
for A, B open in X or subcomplexes of CW-complex X.
Exercise. Show the correctness of all the definitions above and prove the following lemma.
Lemma B (Naturality of cup product). Let f : (X, A) —> (Y,B). Then
Manf*(ß)) = Ma)nß
for all a G Hn(X, A; R) and ß G Hk(Y; R).
9.5. Poincare duality. Now we have all the tools needed to state the Poincare duality for closed manifolds.
Theorem (Poincare duality). If M is a closed R-orientable manifold of dimension n with fundamental class [M] G Hn(M; R), then the map D : Hk(M; R) Hn-k(M; R) defined by
D(ip) = [M] n ip
is an isomorphism.
Exercise. Use Poincare duality to show that the real projective spaces of even dimension are not orientable.
This theorem is a consequence of a more general version of Poincare duality. To state it we introduce the notion of direct limit and cohomology with compact support.
9.6. Direct limits. A direct set is a partially ordered set I such that for each pair l, k £ I there is A G / such that i < A and k < A.
Let GL be a system od Abelian groups (or Ä-modules) indexed by elements of a directed set I. Suppose that for each pair i < k of indices there is a homomorphism fLK : GL —^ GK such that fu = id and fK\fiK = fL\. Then such a system is called directed.
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Having a directed system of Abelian groups (or J?-modules) we will say that a G GL and b G GK are equivalent (a ~ b) if f\\(a) = fK\(b) for some A G /. The direct limit of the system {Gt}te/ is the Abelian group (J?-module) of classes of this equivalence
limGt = 0Gt/~.
iei
Moreover, we have natural homomorphism jL : GL —> limGt.
The direct limit is characterized by the following universal property: Having a system of homomorphism hL : GL —>• A such that hL = hKfiK whenever i < k, there is just one homomorphism
H : limGt -> A
such that hL = HjL.
It is not difficult to prove that direct limits preserve exact sequences. In a system of sets the ordering is usually given by inclusions.
Lemma. // a space X is the union of a directed set of subspaces XL with the property that each compact set in X is contained in some XL, the natural map
lim Hn(XL;R) -+Hn(X;R)
is an isomorphism.
The proof is not difficult, we refer to [Hatcher], Proposition 3.33, page 244.
9.7. Cohomology groups with compact support. Consider a space X with a directed system of compact subsets. For each pair (L,K), K C L, the inclusion (X,X- L) ^ (X,X - K) induces homomorphism Hk(X\K;R) -> Hk(X\L;R). We define the cohomology groups with compact support as
Hkc(X- R) = lini Hk(X\K; R).
If X is compact, then Hk(X; R) = Hk(X; R).
For cohomology with compact support we get the following lemma which does not hold for ordinary cohomology groups.
Lemma. // a space X is the union of a directed set of open subspaces XL with the property that each compact set in X is contained in some XL, the natural map
lim Hk(X,; R) -> Hk(X; R)
is an isomorphism.
Proof. The definition of natural homomorphism in the lemma is based on the following fact: Let U be an open subset in V. For any compact set K C U the inclusion (U, U — K)     (V, V — K) induces by excision an isomorphism
Hk(V\K;R) -+ Hk{U\K-R).
Its inverse can be composed with natural homomorphism Hk(V\K; R) —> Hk(V;R). By the universal property of direct sum there is just one homomorphism
Hk(U-R) -> Hk(V-R).
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So on inclusions of open sets Hk behaves as covariant functor and this makes the definition of the natural homomorphism in the lemma possible. The proof that it is an isomorphism (based on excision) is left to the reader. □
9.8. Generalized Poincare duality. Let M be an J?-orientable manifold of dimension n. Let K C M be compact. Let G Hn(M\K; R) be such a class that its reduction to Hn(M\x; R) gives a generator for each x G K. The existence of such a class is ensured by Lemma in 9.3. Define
DK : H\M\K) -> Hn_k(M; R) :    DK(<p) =      H </?.
If if C L are two compact subsets of M, we can easily prove using naturality of cap product that
for G Hk(M\K; R) and p : (M, M - L) ^ (M, M - K). It enables us to define the generalized duality map
DM : Hk(M; R) -> #n_fe(M; i?) :    D„(^) =      H (p
since each element 99 G H^(M;R) is contained in Hk(M\K; R) for some compact set /\ c .W.
Theorem (Duality for all orientable manifolds). If M is an R-orientable manifold of dimension n, then the duality map
DM:Hkc(M;R)^Hn_k(M;R)
is an isomorphism.
The proof is based on the following
Lemma. // a manifolds M be a union of two open subsets U and V, the following diagram of Mayer-Vietoris sequences
Hkc{U n V)-- Hkc{U) © Hk(V)-- Hk(M)-- Hk+1(U n V)
Dr
Dr
DV®DV
Hn-k(U nv)--        Hn-k(U) © Hn-k(V)--        Hn-k(M)--        Hn^x{U n V)
commutes up to signs.
The proof of this lemma is analogous as the proof of commutativity of the diagram in the proof of Theorem 8.4 on Thom isomorphism. So we omit it referring the reader to [Hatcher], Lemma 3.36, pages 246 - 247 or to [Bredon], Chapter VI, Lemma 8.2, pages 350 - 351.
Proof of Poincare Duality Theorem. We will use the following two statements
(A) If m = u U v where u and v are open subsets such that Djj, Dy and Djjny are isomorphisms, then Dm is also an isomorphism.
(B) If M = IJi^i Ui where Ui are open subsets such that U\ C U2 C U3 C ... and all Djj. are isomorphisms, then Dm is also an isomorphism.
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The former is an immediate consequence of the previous lemma and Five Lemma. To obtain the latter apply the direct limit to the short exact sequences
0 -> Hkc{Ui) ^ Hn_k(Ut) -> 0
and use the lemmas in 9.6 and 9.7. The proof of Duality Theorem will be carried out in four steps.
(1) For M = W1 we have
Hk(Wn) ^ Hk(An, dAn),   Hn(Wn\An) ^ Hn(An, dAn).
Take the generator fi G Hn(An,dAn) represented by the singular simplex given by identity. The only nontriavial case is k = n. In this case for a generator
up e Hn((AndAn)) = Eom(Hn((AndAn),R)
we get fi H ip = </?(//) = ±1. So the duality map is an isomorphism.
(2) Let M C Mn be open. Then M is a countable union of open convex sets Vi which are homeomorphic to IRn. Using the previous step and induction in statement (A) we show that the duality map is an isomorphism for every finite union of Vi. The application of statement (B) yields that the duality map Dm is an isomorphism as well.
(3) Let M be a manifold which is a countable union of open sets Ui which are homeomorphic to W1. Now we can proceed in the same way as in (2) using its result instead of the result in (1).
(4) For general M we have to use Zorn lemma. See [Hatcher], page 248. □
Corollary. The Euler characteristic of a closed manifold of odd dimension is zero.
Proof. For M orientable we get from Poincare duality and the universal coefficient theorem that
rankiJn_fc(M;Z) = rankFfc(M;Z) = rank Horn Hk(M; Z) = rankiJfc(M;Z)
Hence X(M) = ^"=0(-l)irankfli(M;Z) = 0 for n odd.
If M is not orientable, we get from the Poincare duality with Z2 coefficients that
n
^(-l)idimJHi(M;Z2) = 0.
i=0
Here the dimension is considered over Z2. Applying the universal coefficient theorem one can show that the expression on the left hand side equals to x(M). See [Hatcher], page 249. □
Remark. Consider an oriented closed smooth manifold M. The orientation of the manifold induces an orientation of the tangent bundle tm and we get the following relation between the Euler class of tm, the fundamental class of M and the Euler characteristic of M:
X(M)=e(rM)n[M].
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Particulary, for spheres of even dimension we get that the Euler class of their tangent bundle is twice a generator of Hn{Sn;'L). For the proof see [MS], Corollary 11.12.
9.9. Duality and cup product. One can easily show that for a G Cn(X;R), p> G Ck(X; R) and if> G Cn~fc(X; R) we have
ip(a fl tp) = (<p U ip)(a).
For a closed J?-orientable manifold M we define bilinear form
(*) Hk(M;R) x Hn-k(M;R) -+ R : (p,ifj) ^ (p U ^)[M].
A bilinear form A x 5 -)• is called regular if induced linear maps A —)• Hom(5, J?) and 5 —y B.om(A, R) are isomorphisms.
Theorem. Let M be a closed R-orientable manifold. If R is a field, then the bilinear
form (*) is regular.
If R = Z; £/zen £/ze bilinar form
Hk(M;Z)/Torsion Hk(M;Z) x Hn-k(M; Z)/Torsion Hn-k(M; Z) ^Z induced by (*) regular. Proof. Consider the homomorphism
Hn-k(M; R)       Hom(iJn_fc(M; R); R) ^ Hom(Hk(M; R),R).
Here fcftW) = ^(Z3) for Z5 € Hn_k(M; R) and ^ G Hn~k(M; R) and £>* is the dual map to duality. The homomorphism /i is an isomorphism by the universal coefficient theorem and D* is an isomorphism since so is D. Now it suffices to prove that the composition D*h is the homomorphism induced from the bilinear form (*). For ip G Hn-k{M- R) and p G Hk(M; R) we get
(D*h(4>)) (ip) = (hty)) D(tp) = (hty)) ([M] n <p) = i>([M] nv) = (vu ^)[M].
□
This theorem gives us a further tool for computing the cup product structure in cohomology of closed manifolds.
Corollary. Let M be a closed orientable manifold of dimension n. Then for every p> G Hk(M] Z) of infinite order which is not of the form p> = mp>\ for m > 1, there is ip G Hn-k(M; Z) such that p> U if> is a generator of Hn(M; Z) = Z.
Example. We will prove by induction that iJ*(CPn;Z) = 7L\uj\/(un+1) where u G iJ2(CPn; Z) is a generator. For n = 1 the statement is clear. Suppose that it holds for n — 1. From the long exact sequence for the pair (CPn, CPn_1) we get that
iT(CPn;Z) ^ iJ^CP^Z)
for i < 2n — 1. Now, using the consequence above for p> = uj we obtain that ujn is a generator of F2n(CPn; Z).
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9.10. Manifolds with boundary. A manifold with boundary of dimension n is a Hausdorff space M in which each point has an open neighbourhood homeomorphic either to IRn or to the half-space
K+ = {(x1,x2,...,xn) Gl"; xn>0}.
The boundary dM of the manifold M is formed by points which have all neighbourhoods of the second type. The boundary of a manifold of dimension n is a manifold of dimension n — 1. In a similar way as for a manifold we can define orientation of a manifold with boundary and its fundamental class [M] G Hn(M; dM; R).
Theorem. Suppose that M is a compact R-orientable n-dimensional manifold whose boundary dM is decomposed as a union of two compact (n — 1)-dimensional manifolds A and B with common boundary dA = dB = A H B. Then the cap product with the fundamental class [M] G Hn(M,dM; R) gives the isomorphism
DM : Hk(M, A; R) -> Hn-k(M, B; R).
For the proof and many other applications of Poincare duality we refer to [Hatcher], Theorem 3.43 and pages 250 - 254, and [Bredon], Chapter VI, Sections 9 and 10, pages 355 - 366.
9.11. Alexander duality. In this paragraph we introduce another version of duality.
Theorem (Alexander duality). If K is a proper compact subset of Sn which is a deformation retract of an open neighbourhood, then
Hi(Sn -K;Z) = Hn~i~1(K;Z).
Proof. For i ^ 0 and U a neighbourhood of K we have
Hi(Sn -K)^ H^-\Sn - K) by Poincare duality
= lim uHn-\Sn — K,U — K)   by definition
^ lim uH^iS12, U) by excision
= lim jjHn~t~1{U) connecting homomorphism
= Hn~i~1(K) K is a def. retract of some U
First three isomorphisms are natural and exist also for i = 0. So using these facts we have
H0(Sn -K) = Ker (H0(Sn - K) -> tf0(pt)) = Ker (H0(Sn - K) -> H0(Sn))
= Ker (lim Hn(Sn, U) -> Hn(Sn)^ = lim Ker (Hn(Sn, U) -> Hn(Sn)) = lim Hn-\U) = Hn-\K).
□
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Corollary. A closed nonorientable manifold of dimension n cannot be embedded as a subspace into IRn+1.
Proof. Suppose that M can be embedded into IRn+1. Then it can be embedded also in Sn+1. By Alexander duality
Fn_i (M; Z) = H1(Sn+1 — M;Z).
According to the universal coefficient theorem
H\Sn+1 - M; Z) = Hom(iJ1(1Sn+1 - M; Z), Z) © Ext(iJ0(1Sn+1 — M; Z))
is a free Abelian group. On the other hand
Z2 = #n(M; Z2) = Hn(M; Z) © Z2 © Tor^^M, Z), Z2).
According to (b) of Theorem 9.3 the tensor product has to be zero, and since iJn_i(M; Z) is free, the second summand has to be also zero, which is a contradiction. □
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