Topology General Exam

August 22, 2018

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.

Problem 1

Let $M$ be a smooth $n$-manifold and $f: M \rightarrow M$ a smooth map. Let

$$\Gamma_{f}=\{(x, f(x)) \mid x \in M\} \quad \text { and } \quad \Delta=\{(x, x) \mid x \in M\}$$

be the graph of $f$ and the diagonal submanifold, respectively, in $M \times M$.
Prove that if $x_{0} \in M$ is a point with $f\left(x_{0}\right)=x_{0}$, then the following are equivalent:
(i) The manifolds $\Gamma_{f}$ and $\Delta$ intersect transversely at ( $x_{0}, x_{0}$ ).
(ii) The linear map $d f_{x_{0}}: T_{x_{0}} M \rightarrow T_{x_{0}} M$ does not have 1 as an eigenvalue.

Problem 2

Let $S$ be an oriented surface of genus 2 without boundary.
(a) By describing $S$ as a polygon with certain pairs of edges identified and using the Seifert-Van Kampen theorem, or by another method, give a presentation for the group $\pi_{1}(S)$.
(b) Show that if a finite $G$ group acts freely on $S$, then $G$ must have order either 1 or 2.

Problem 3

Let $X$ be a smooth, compact $n$-manifold with boundary $\partial X$. Prove that there does not exist a retraction $X \rightarrow \partial X$, that is, show there exists no smooth map $f: X \rightarrow \partial X$ with $f(x)=x$ for all $x \in \partial X$.

Problem 4

(a) Let $M$ and $N$ be smooth connected closed ( $=$ compact without boundary) manifolds of the same dimension. Show that a submersion $f: M \rightarrow N$ will then be a finite sheeted covering map.
(b) Explain why if $M$ is a connected closed surface, and $f: M \rightarrow S^{2}$ is a submersion, then $f$ must, in fact, be a diffeomorphism.

Problem 5

Let $C_{*}$ and $D_{*}$ be chain complexes of abelian groups.
(a) Complete the definition: Two chain maps $f_{*}, g_{*}: C_{*} \rightarrow D_{*}$ are chain homotopic if $\ldots$.
(b) Show that if $f_{*}$ is chain homotopic to $g_{*}$, and $g_{*}$ is chain homotopic to $h_{*}$, then $f_{*}$ is chain homotopic to $h_{*}$.
(c) Prove that if $f_{*}, g_{*}: C_{*} \rightarrow D_{*}$ are chain homotopic chain maps, then

$$H\left(f_{*}\right)=H\left(g_{*}\right): H_{*}\left(C_{*}\right) \rightarrow H_{*}\left(D_{*}\right)$$

Problem 6

Suppose $X$ is a CW complex with $n$-skeleton $X_{n}$ for $n \geq 0$.
(a) Define the associated cellular chain complex: the groups $C_{n}^{C W}(X)$ and the differentials $d_{n}^{C W}: C_{n+1}^{C W}(X) \rightarrow C_{n}^{C W}(X)$. Then prove that $d_{n-1}^{C W} \circ d_{n}^{C W}=0$.
(b) Explain why $C_{n}^{C W}(X)$ is isomorphic to a free abelian group with one generator for each $n$-cell of $X$.

Problem 7

Prove that the subset of $\mathbb{R}^{3}$ determined by the following equations is a manifold:

$$\begin{aligned} 2 x^{2}+3 y+z & =6 \\ -x^{2}+2 y^{3}+z^{2} & =2 \end{aligned}$$

Also describe the tangent space to this manifold at the point $(1,1,1)$.

Problem 8

Let $S^{3} \stackrel{p_{1}}{\longleftrightarrow} S^{3} \vee S^{3} \xrightarrow{p_{2}} S^{3}$ be the two 'projection maps': the other sphere is collapsed to the basepoint. Then say that a map $f: S^{3} \rightarrow S^{3} \vee S^{3}$ has type ( $m, n$ ) if the degree of $p_{1} \circ f$ is $m$ and the degree of $p_{2} \circ f$ is $n$. Let $X_{f}=\left(S^{3} \vee S^{3}\right) \cup_{f} D^{4}$.

Compute the homology groups of $X_{f}$ if $f$ has type ( 8,6 ), describing the homology groups as direct sums of cyclic groups, as usual.