Topology General Exam

August 14, 2015

Name:

Instructions: This is a four hour exam and 'closed book'. There are seven problems: the first 6 are worth 15 points and no. 7 is worth 10 points, for a maximum total of 100 .

Problem 1

(a) Describe a connected double cover of $\mathbb{R} P^{2} \vee S^{1}$. (There is more than one correct answer.)
(b) What are the homology groups of your double cover?
(c) What is the fundamental group of your double cover?

Problem 2

Suppose that $M \xrightarrow{g} N$ is a smooth maps between smooth manifolds of dimensions $m$ and $n$ respectively. Let $z \in N$ be a regular value for $g$ and let $K=g^{-1}(z)$.
(a) Explain why $K$ will be orientable if $M$ is orientable.
(b) Now suppose that one also has a smooth map $L \xrightarrow{f} M$. Show that $z \in N$ will be a regular value for the composite $g \circ f$ if and only if $f$ is transverse to $K$.

Problem 3

(a) Complete the definition: Two chain maps $f_{*}, g_{*}: C_{*} \rightarrow D_{*}$ are chain homotopic if ...
(b) Prove that if $f_{*}, g_{*}: C_{*} \rightarrow D_{*}$ are chain homotopic chain maps, then

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

(c) Suppose that $h_{*}: D_{*} \rightarrow E_{*}$ is yet another chain map. Show that if $f_{*}$ is chain homotopic to $g_{*}$, then the composite $h_{*} \circ f_{*}$ is chain homotopic to $h_{*} \circ g_{*}$.

Problem 4

(a) Show that an $n$-dimensional Lie group $G$ is parallelizable, i.e., admits $n$ smooth vector fields that are linearly independent when evaluated at any point.
(b) Prove that $S^{2}$ does not admit a group structure making it into a Lie group.

Problem 5

(a) Describe a smooth atlas for $\mathbb{R} P^{3}$.
(b) Describe a C.W. complex structure for $\mathbb{R} P^{3}$.
(c) Describe the cellular chain complex associated to your answer to (b), and use this to compute $H_{*}\left(\mathbb{R} P^{3}\right)$.

Problem 6

(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 7

Let $C$ be the ’middle circle’ in the genus 2 surface $M$ as pictured:

Show that if $C^{\prime} \subset M$ is any other embedded circle transverse to $C$, then $C^{\prime}$ intersects $C$ in an even number of points.