Topology General Exam — August 2023

Instructions: This is a four hour exam. Your solutions should be legible and clearly organized, written in complete sentences in good mathematical style on your own paper. All work should be your own-no outside sources are permitted-using methods and results from the first year topology courses. Each problem is worth the same number of points.

Problem 1

Let $x_{i}, y_{i}, z, i=1, \ldots, n$, denote the coordinates on $\mathbb{R}^{2 n+1}$. Consider the 1 -form

$$w=\mathrm{d} z+\sum_{i=1}^{n} x_{i} \mathrm{~d} y_{i} .$$

Compute $w \wedge(\mathrm{~d} w)^{n}$ and show that it is a volume form. Here $(\mathrm{d} w)^{n}$ denotes the $n$-fold wedge product $\mathrm{d} w \wedge \ldots \wedge \mathrm{~d} w$.

Problem 2

Consider the subset $S$ of $\mathbb{R}^{4}$ defined by the equation $x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}=1$, and consider the function $f: \mathbb{R}^{4} \longrightarrow \mathbb{R}$ given by $f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=x_{1}$.

Show that $S$ is a submanifold of $\mathbb{R}^{4}$, and identify the critical points and critical values of the function $f$ restricted to $S,\left.f\right|_{S}: S \longrightarrow \mathbb{R}$.

Problem 3

Let $M$ be a compact manifold with boundary. Prove that there does not exist a smooth map $f: M \longrightarrow \partial M$ such that $f(x)=x$ for all $x \in \partial M$.

Problem 4

Let $M$ be a smooth manifold and $f: M \longrightarrow M$ a smooth map. Suppose that $p \in M$ is a fixed point of $f$, meaning $f(p)=p$. Prove that the following conditions are equivalent:
a) The graph $\{(x, f(x)) \mid x \in M\}$ of $f$ intersects the diagonal $\{(x, x) \mid x \in M\}$ transversely at the point $(p, p) \in M \times M$.
b) 1 is not an eigenvalue of the differential $d_{p} f: T_{p} M \longrightarrow T_{p} M$.

Hint. It may be helpful to consider the graph $\left\{\left(v, d_{p} f(v)\right) \mid v \in T_{p} M\right\}$ of the differential $d_{p} f$ and the conditions under which it is transverse to the diagonal $\left\{(v, v) \mid v \in T_{p} M\right\}$ in $T_{p} M \times T_{p} M$.

Problem 5

Suppose that the 3 -torus $T^{3}=S^{1} \times S^{1} \times S^{1}$ is written as a union $T^{3}=U_{1} \cup \cdots \cup U_{m}$ of open subets $U_{j}$ each homeomorphic to $\mathbb{R}^{3}$. Prove that for some $k \leq m$, the intersection $\left(U_{1} \cup \cdots \cup U_{k-1}\right) \cap U_{k}$ is either empty or disconnected.

Problem 6

Let $f: S^{n} \rightarrow S^{n}$ be a continuous map.
a) Prove that if $\operatorname{deg}(f) \neq(-1)^{n+1}$ then $f$ has a fixed point.
b) Suppose that $f(x)=f(-x)$ for all $x \in S^{n}$. Prove that $\operatorname{deg}(f)$ is even, and if in addition $n$ is even, then $\operatorname{deg}(f)=0$.

Problem 7

Let $X=\mathbb{R} P^{2} \vee \mathbb{R} P^{2}$.
a) Prove that $X$ does not admit a connected, regular, 3 -sheeted covering space.
b) Construct a connected (non-regular) 3-sheeted covering space of $X$.

Problem 8

Let $K \subset S^{3}$ be a smooth submanifold diffeomorphic to a circle, $K \cong S^{1}$ (so $K$ is a "knot" in the 3 -dimensional sphere). Use the axioms and properties of homology, including for example the long exact sequence of a pair and the Mayer-Vietoris sequence, to answer the following:
a) Find the relative homology groups $H_{j}\left(S^{3}, K\right)$ for all $j$.
b) Prove that $H_{1}\left(S^{3}-K\right) \cong \mathbb{Z}$.