Topology General Exam

August 26, 2017

Name:

Instructions: This is a four hour exam and 'closed book'. There are eight problems. Show your work using methods and results from the first year topology course topics. Results from one part of a problem can be assumed in later parts.

Problem 1

Let $f: \mathbb{R}^{2}-\{(0,0)\} \rightarrow \mathbb{R}^{3}$ be given by $f(x, y)=\left(x^{2}, y^{2}, x y\right)$.
(a) Prove that $f$ is an immersion, but not an embedding.
(b) Find all values of $c$ such that $f$ is transverse to the plane $\{(u, v, w) \mid v=c\} \subset \mathbb{R}^{3}$.

Problem 2

Two covering spaces over $X, p_{1}: X_{1} \rightarrow X$ and $p_{2}: X_{2} \rightarrow X$, are said to be isomorphic if there exists a homeomorphism $h: X_{1} \rightarrow X_{2}$ such that the diagram commutes:

Describe the isomorphism classes of covers of the space $\mathbb{R} P^{2} \times \mathbb{R} P^{2}$.

Problem 3

Suppose that $t \in \mathbb{R}$ is a regular value of a smooth map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$, and let $M=f^{-1}(t)$.
(a) Will $M$ necessarily have a nowhere vanishing normal vector field?
(b) Will $M$ necessarily have a nowhere vanishing tangent vector field?
(c) Will $M$ necessarily be orientable?

In each part, explain how to construct the vector field or orientation if it must exist, and give a counterexample, if it needn't.

Problem 4

(a) Define what it means for two chain maps between chain complexes to be chain homotopic. Then prove that if $f_{*}, g_{*}: C_{*} \rightarrow D_{*}$ are chain homotopic, then $f_{*}=g_{*}: H_{*}\left(C_{*}\right) \rightarrow H_{*}\left(D_{*}\right)$.
(b) Call a chain map a quasi-isomorphism if it induces an isomorphism on homology. Let

be a commutative diagram of chain complexes, such that each horizontal row is exact. Show that if $f_{*}$ and $g_{*}$ is a quasi-isomorphism then so is $h_{*}$.

Problem 5

Let $f: M \rightarrow N$ be a smooth map between manifolds of dimension $m$ and $n$ respectively. Let $D(f) \subset M \times M$ be the ’double point’ subspace:

$$D(f)=\{(x, y) \mid x \neq y \text { and } f(x)=f(y)\}$$

(a) Say that $f$ is self transverse if for all $(x, y) \in D(f), d_{x} f\left(T_{x} M\right)+d_{y} f\left(T_{y} M\right)=T_{f(x)} N$. Show that then $D(f)$ is a smooth manifold of $M \times M$, and find its dimension. [Hint: Show that if $f$ is self transverse, then the function $f \times f: M \times M-\Delta(M) \rightarrow N \times N$ is transverse to the diagonal $\Delta(N) \subset N \times N$.]
(b) Describe, with pictures and/or words, an example of a self transverse smooth function $f: S^{1} \rightarrow \mathbb{R}^{2}$ for which $D(f)$ is nonempty.
(c) Describe, with pictures and/or words, an example of a smooth function $f: S^{1} \rightarrow \mathbb{R}^{2}$ that is not self transverse.

Problem 6

(a) Show that there is no continuous map $g: S^{2} \rightarrow S^{2}$ such that, for all $x \in S^{2}, g(x) \neq x$ and $g(x) \neq-x$. [Hint: if such a $g$ exists, explain how it can be used to show that the antipodal map is homotopic to the identity map on $S^{2}$. Hmm ...]
(b) Show that every continuous map $f: \mathbb{R} P^{2} \rightarrow \mathbb{R} P^{2}$ has a fixed point. [If $f$ has no fixed point, use covering space theory to show that one can construct $g$ as in part (a).]

Problem 7

Suppose a finite group $G$ acts freely on the right of a Hausdorff space $X$. Let $X / G$ be the space of $G$-orbits - sets of the form $\{x g \mid g \in G\}$ - given the quotient topology.
(a) Show that the quotient map $X \rightarrow X / G$ is a covering space map. [Hint: you need to show that each point $x \in X$ has an open neighborhood $U$ such that all the translates of $U-$ the sets $U g$ for $g \in G$ - are disjoint.]
(b) How are the groups $G, \pi_{1}(X)$ and $\pi_{1}(X / G)$ related?
(c) Show that there exists a smooth 3 -manifold with a fundamental group that is both finite and non-abelian. [One approach: Recall that $S^{3}$ can be viewed as the quaternions of unit length.]

Problem 8

(a) $S^{2}$ has a CW complex structure with one 0 -cell and one 2 -cell, and then the associated ’product’ CW structure on $S^{2} \times S^{2}$ has one 0-cell, two 2-cells, and one 4-cell. Compute the homology groups of $S^{2} \times S^{2}$.
(b) If $M$ is a smooth connected $n$-dimensional manifold, let $\widetilde{M}$ denote $M$ with a small open $n$-ball removed. Show that $\widetilde{S^{2} \times S^{2}}$ is homotopy equivalent to $S^{2} \vee S^{2}$. [Hint: you can assume that the small open 4-ball is removed from the interior of the 4-cell.]
(c) The connected sum $M \# N$ of two $n$-manifolds admits a decomposition $M \# N=\widetilde{M} \cup \widetilde{N}$ with $\bar{M} \cap \bar{N}$ diffeomorphic to $S^{n-1}$. Compute the homology groups of ( $S^{2} \times S^{2}$ ) #( $S^{2} \times S^{2}$ ).