Topology General Exam

January 10, 2022

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 ( $X, A$ ) be a pair of topological spaces, and consider the union $X \cup C A$ of $X$ with the cone on $A$. Here we think of $C A=A \times[0,1] /\{(a, 1), a \in A\}$, and in $X \cup C A$ we identify $(a, 0) \in C A$ with $a \in A \subset X$.
a) Prove that $X$ is a retract of $X \cup C A$ if and only if $A$ is contractible in $X$, i.e., there is a homotopy from the inclusion $A \hookrightarrow X$ to a constant map.
b) Prove that $(X \cup C A) / X$ is homeomorphic to the suspension $S A$.

Problem 2

Let $A=\{(0,0, t) \mid-2 \leq t \leq 2\} \subseteq \mathbb{R}^{3}$ and $B=\left\{x \in \mathbb{R}^{3}| | x \mid=2\right\} \subseteq \mathbb{R}^{3}$.
a) Compute $\pi_{1}(A \cup B)$.
b) Compute the homology groups of $A \cup B$.

Problem 3

What are all the connected covering spaces of the 2-dimensional torus $T^{2}$, up to equivalence? Justify your answer.

Problem 4

Consider a commutative diagram of abelian groups

such that $j_{1} i_{1}=0=j_{2} i_{2}$. Prove that if $f_{1}$ and $f_{2}$ are isomorphisms and the sequence

$$A_{2} \xrightarrow{i_{2}} C \xrightarrow{j_{2}} B_{1}$$

is exact, then the maps

$$i_{1}+i_{2}: A_{1} \oplus A_{2} \rightarrow C \text { and }\left(j_{2}, j_{1}\right): C \rightarrow B_{1} \oplus B_{2}$$

are isomorphisms and that the sequence

$$A_{1} \xrightarrow{i_{1}} C \xrightarrow{j_{1}} B_{2}$$

is exact.

Problem 5

Let $M \subset \mathbb{R}^{3}$ be defined by the equations

$$\begin{array}{r} x^{2}+y^{2}-z^{2}=1 \\ z^{2}-x y=0 \end{array}$$

Prove that $M$ is a smooth manifold and find its dimension.

Problem 6

Let $M$ be a smooth compact manifold of dimension $n$, and $f: M \rightarrow \mathbb{R}^{n+1}-\{0\}$ a smooth map. Prove that there exists a line through the origin in $\mathbb{R}^{n+1}$ that intersects $f(M)$ in only finitely many points.

Problem 7

Prove that if $M$ is any smooth manifold, then the tangent bundle $T M$ is an orientable manifold.

Problem 8

Prove that any continuous map $f: S^{n} \rightarrow S^{n}$ with $\operatorname{deg}(f) \neq(-1)^{n+1}$ has a fixed point. As a suggestion, you might consider proving the contrapositive.