Topology General Exam

August 21, 2019

Instructions: This is a four hour exam. Your solutions should be legible and clearly organized, written in complete sentences in good mathematical style. All work should be your own-no outside sources are permitted-using methods and results from the first year topology course topics. Each problem is worth the same number of points.

Problem 1

Let $f: S^{n} \rightarrow S^{n}$ be a smooth map with the property that $d f_{x}: T_{x} S^{n} \rightarrow T_{f(x)} S^{n}$ is injective for every $x \in S^{n}$.
a) Prove that $f$ is a diffeomorphism provided that $n \geq 2$.
b) Find a counterexample to part (a) in the case that $n=1$.

Problem 2

Let $M$ be a connected smooth manifold of dimension $m$ and let $N \subset M$ be a smooth submanifold of dimension $n=m-k$.
a) Show that if $k \geq 2$ then $M \backslash N$ is connected.
b) Prove that if $k \geq 3$ then the inclusion $M \backslash N \rightarrow M$ induces an isomorphism $\pi_{1}\left(M \backslash N, x_{0}\right) \rightarrow \pi_{1}\left(M, x_{0}\right)$ for $x_{0} \in M \backslash N$ any basepoint.

Problem 3

Let $M$ be a smooth compact manifold with nonempty boundary $N=\partial M$. Prove that there does not exist a retraction $M \rightarrow N$.

Problem 4

Let $M_{n} \cong \mathbb{R}^{n^{2}}$ denote the square $n \times n$ matrices with real entries.
a) Prove that $\mathrm{SL}_{n}(\mathbb{R})$, the matrices with determinant 1 , is a smooth submanifold of $M_{n}$.
b) Prove that $\mathrm{SL}_{n}(\mathbb{R})$ is not compact.

Problem 5

Let $K$ be a 2 -complex consisting of one 0 -cell, two 1 -cells labelled $\{a, b\}$, and a single 2 -cell attached along the loop $a b a^{-1} b$.
a) Find $H_{*}(K ; \mathbb{Z})$.
b) Find $H_{*}(K ; \mathbb{Z} / 2 \mathbb{Z})$.
c) Show that $K$ is homeomorphic to a smooth manifold.

Problem 6

Lef $f: S^{n} \rightarrow S^{n}$ be a continuous map.
a) Prove that if $f$ has no fixed points then $f$ is homotopic to the antipodal map.
b) Let $G$ be a group acting freely on $S^{n}$, where here $n$ is even. Prove that $G$ has order at most two. [Hint: construct a homomorphism to $\mathbb{Z} / 2 \mathbb{Z}$ using degree and argue that this map is injective.]

Problem 7

Let $X$ be the 1 -point union of two copies of the projective plane, $X=\mathbb{R} P^{2} \vee \mathbb{R} P^{2}$.
a) Find a presentation for $\pi_{1}(X)$ and describe $H_{1}(X ; \mathbb{Z})$ as a direct sum of cyclic groups.
b) Prove that every continuous map $f: X \rightarrow S^{1}$ is nullhomotopic.

Problem 8

Let $C$ be a circle that is smoothly embedded in $\mathbb{R}^{3}$. For example $C$ could be the trefoil knot:

but your methods should not be specific to this case. Find the homology groups $H_{*}\left(\mathbb{R}^{3} \backslash C ; \mathbb{Z}\right)$.