Topology General Exam

January 9, 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.

Problem 1

Let $f: M \rightarrow N$ be a smooth map between manifolds of dimension $m$ and $n$, respectively. Let $q \in N$ be a regular value for $f$, and let $X=f^{-1}(q) \subset M$.
a) Prove that if $M$ is orientable, then $X$ is orientable.
b) Let $g: K \rightarrow M$ be another smooth map, where $K$ is a smooth manifold. Prove that $q \in N$ is a regular value for $f \circ g$ if and only if $g$ is transverse to $X$.

Problem 2

a) Prove that any smooth map $f: S^{k} \rightarrow \mathbb{R}^{n}$ can be extended to a smooth map $F: D^{k+1} \rightarrow \mathbb{R}^{n}$, where $S^{k}=\partial D^{k+1}$ is the $k$-dimensional sphere and $D^{k+1}$ the unit ball of dimension $k+1$.
b) Let $M \subset \mathbb{R}^{n}$ be a smooth compact manifold of dimension $m$, and assume $k<n-m-1$. Prove that any smooth map $f: S^{k} \rightarrow \mathbb{R}^{n}-M$ can be extended to a smooth map $F$ : $D^{k+1} \rightarrow \mathbb{R}^{n}-M$.

Problem 3

Let $M$ and $N$ be the subsets of $\mathbb{R}^{3}$ defined by

$$M=\left\{x^{2}+y^{2}+z^{2}=1\right\} \quad N=\left\{x^{2}-y^{2}+z^{2}=c\right\}$$

for a real number $c$. Justify your responses to the following:
a) Determine all values of $c$ for which $M$ and $N$ are submanifolds of $\mathbb{R}^{3}$, and the intersection $M \cap N$ is transverse.
b) Determine all values of $c$ for which $M \cap N$ is a submanifold of $\mathbb{R}^{3}$.

Problem 4

Let $X$ and $Y$ be closed, compact, oriented manifolds of the same dimension. Let $f, g: X \rightarrow Y$ be two smooth maps. The graphs of $f$ and $g$ are the submanifolds of $X \times Y$ given by

$$\Gamma_{f}=\{(x, f(x)) \mid x \in X\} \quad \Gamma_{g}=\{(x, g(x)) \mid x \in X\}$$

oriented so that the diffeomorphisms $X \rightarrow \Gamma_{f}$ and $X \rightarrow \Gamma_{g}$ given by $x \mapsto(x, f(x))$ and $x \mapsto(x, g(x))$ are orientation-preserving.

The coincidence number of $f$ and $g$, written $C(f, g)$, is defined to be the intersection number $I\left(\Gamma_{f}, \Gamma_{g}\right) \in \mathbb{Z}$ (sometimes also written $\Gamma_{f} \cdot \Gamma_{g}$ ).
a) Prove that if $C(f, g) \neq 0$ then for any smooth maps $\tilde{f}, \tilde{g}: X \rightarrow Y$ such that $\tilde{f}$ and $\tilde{g}$ are homotopic to $f$ and $g$, respectively, there exists a point $x \in X$ such that $\tilde{f}(x)=\tilde{g}(x)$.
b) Let $f, g: S^{1} \rightarrow S^{1}$ be two smooth maps of degree $n$ and $m$, respectively. Prove that if $n \neq m$, then there is a point $x \in S^{1}$ with $f(x)=g(x)$.

Problem 5

Let $p: S^{n} \rightarrow \mathbb{R} P^{n}$ be the projection.
a) Let $f: \mathbb{R} P^{n} \rightarrow \mathbb{R} P^{n}$ be continuous. Show that then there exists a continuous map $\tilde{f}: S^{n} \rightarrow S^{n}$ such that $p \circ \tilde{f}=f \circ p: S^{n} \rightarrow \mathbb{R} P^{n}$.
b) Show that every continuous map $f: \mathbb{R} P^{2 k} \rightarrow \mathbb{R} P^{2 k}$ has a fixed point.

Problem 6

Suppose given a commutative diagram of abelian groups with exact rows:

Prove part of the Five Lemma: show that if $f_{2}$ and $f_{4}$ are monomorphisms, and $f_{1}$ is an epimorphism, then $f_{3}$ is a monomorphism. (Hint: start by showing that $x \in \operatorname{Ker}\left(f_{3}\right) \Rightarrow x \in \operatorname{Ker}\left(\alpha_{3}\right)$.)

Problem 7

Let $C \subset \mathbb{R}^{3}$ be the union of the $x$-axis and the $y$-axis. Compute $H_{*}\left(\mathbb{R}^{3}-C ; \mathbb{Z}\right)$. (Hint: note that $\mathbb{R}^{3}-C=\left(\mathbb{R}^{3}-x\right.$-axis $) \cap\left(\mathbb{R}^{3}-y\right.$-axis $)$.)

Problem 8

Let $a: S^{1} \rightarrow S^{1} \vee S^{1}$ and $b: S^{1} \rightarrow S^{1} \vee S^{1}$ respectively be the inclusion of the circle as the first and second wedge summand. Then $\pi_{1}\left(S^{1} \vee S^{1}\right)$ can be identified with the free group on $a$ and $b$. Let $f: S^{1} \rightarrow S^{1}$ represent the element $c \in \pi_{1}\left(S^{1} \vee S^{1}\right)$, and let $X_{f}=\left(S^{1} \vee S^{1}\right) \cup_{f} D^{2}$, the topological space obtained by identifying points on $S^{1}=\partial D^{2}$ with their images under $f$ in $S^{1} \vee S^{1}$. If $c=a(a b)^{4} a$, give a presentation of $\pi_{1}\left(X_{f}\right)$ and compute $H_{*}\left(X_{f} ; \mathbb{Z}\right)$, describing the homology groups as direct sums of cyclic groups, as usual.