### Math 601 – Spring 2014 Homework #3 1. (Hatcher

```Math 601 – Spring 2014
Homework #3
1. (Hatcher, number 2.1.14) Determine whether there exists a short exact sequence
0 → Z4 → Z8 ⊕ Z2 → Z4 → 0.
More generally, determine which abelian groups A fit into a short exact sequence
0 → Zpm → A → Zpn → 0
with p prime. What about the case of short exact sequences
0 → Z → A → Zn → 0?
e n (X) ∼
e n+1 (SX) for all n, where SX is the suspension
2. (Hatcher, number 2.1.20) Show that H
=H
of X; SX = X × I/(x, 1) ∼ (y, 1) and (x, 0) ∼ (y, 0) for all x, y ∈ X.
e 1 (X/A) if X = [0, 1] and
3. (Hatcher, number 2.1.26) Show that H1 (X, A) is not isomorphic to H
1 1
A is the sequence 1, 2 , 3 , . . . together with its limit 0.
4. (Hatcher, number 2.2.2) Given a map f : S 2n −→ S 2n , show that there is some point x ∈ S 2n
with either f (x) = x or f (x) = −x. Deduce that every map RP2n −→ RP2n has a fixed
point. Construct maps RP2n−1 −→ RP2n−1 without fixed points from linear transformations
R2n −→ R2n without eigenvectors.
5. (Hatcher, number 2.2.8) A polynomial f (z) with complex coefficients, viewed as a map C −→
C, can always be extended to a continuous map of one-point compactifications fˆ : S 2 −→ S 2 .
Show that the degree of fˆ equals the degree of f as a polynomial. Show also that the local
degree of fˆ at a root of f is the multiplicity of the root.
6. Prove the Brouwer fixed point theorem: Any continuous map f : Dn −→ Dn admits a fixed
point.
7. By considering T 2 as R2 /Z2 , we see that SL(2, Z) acts on T 2 via homeomorphisms. For
φ ∈ SL(2, Z), let Yφ = (S 1 × D2 ) ∪φ (S 1 × D2 ). Determine H∗ (Yφ ) in terms of φ.
8. Given φ ∈ SL(2, Z), consider the mapping torus of T 2 with respect to φ:
Xφ = T 2 × I/(x, 1) ∼ (φ(x), 0). Compute H∗ (Xφ ) in terms of φ.
9. For any knot in R3 , compute H∗ (R3 \νK), where νK is a neighborhood of the knot.
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