HOMEWORK 5
Exercises 2, 3 can be foud in Hatcher’s book p.133 ex.28. Those, who are interested
in homotopy equivalence of pairs might be also interested in ex.27.
Exercise 1. Let f : Sn
→ Sn
be a map of degree 2. Prove that f has a fixed point
(there exists x ∈ Sn
such that f(x) = x).
Hint: Use contradiction nad antipodal map.
Exercise 2. Compute the local homology groups of the following space: take the
edges of the tetrahedron (see them for example as [vi, vj], 0 ≤ i < j ≤ 3 and we have
vertices v0, v1, v2, v3). Now we add a vertex p into the barycentre, we connect the
point with all the other vertices and get edges [vi, p]. To finish, we add 2-simplices
[vi, vj, p] 0 ≤ i < j ≤ 3.
Compute the local homotopy groups of this space.
Exercise 3. We denote the space from the previous example X. We define ∂X = set
of points x of X such that Hn(X, X − {x0}) = 0. Compute the local homology groups
of ∂X.
Exercise 4. Let space X have the following (reduced) homology groups:
¯H1(X) = Z ⊕ Z2
¯H10(X) = Z2013
⊕ Z42
¯Hi(X) = 0, i ∈ {1, 10}
Create two spaces Y, Z such that they have the same homology groups as X and such
that Y is not homeomorphic to Z and prove that. Hint: Ignore ¯H10, and think of ways
how to glue D2
to S1
∨ S1
.
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