Topology General Exam January 2023

Problem 1

Recall that if $X$ is a "nice" (meaning connected, locally path connected and semi-locally simply connected) topological space with basepoint $x_{0}$, then connected, based covering spaces of $X$ are in 1-1 correspondence with subgroups of $\pi_{1}\left(X, x_{0}\right)$. For such a space $X$, let $p: \widetilde{X} \rightarrow X$ be a universal covering map, and let $A \subset X$ be a "nice" subspace. Let $\widetilde{A} \subset \widetilde{X}$ be a path component of the preimage $p^{-1}(A)$. Prove that, with suitable choices of basepoints, the restriction $p: \widetilde{A} \rightarrow A$ is the covering space of $A$ corresponding to the kernel of the map $\pi_{1}(A) \rightarrow \pi_{1}(X)$ induced by inclusion.

Problem 2

a) For each $n>0$, let $S^{n}$ have the CW structure with a single 0 -cell and a single $n$-cell. Describe a CW structure on the product $S^{n} \times S^{m}$, and use it to compute the homology groups $H_{i}\left(S^{n} \times S^{m}\right)$.
b) Let $X$ be the space obtained from $S^{1} \times S^{2}$ by attaching a 2-cell along a map $\partial D^{2} \rightarrow S^{1} \times\{p t\}$ of degree $k$. Describe a CW structure on $X$ and use it to find the homology groups of $X$.

Problem 3

Let $f: S^{n} \rightarrow S^{n}$ be a continuous map that is homotopic to a constant map. Prove that there exists $x \in S^{n}$ with $f(x)=x$, and also a point $y \in S^{n}$ with $f(y)=-y$.

Problem 4

Let $M$ be a smooth, oriented manifold without boundary and $f: M \rightarrow \mathbb{R}$ a smooth map. For $t \in \mathbb{R}$ a regular value of $f$, prove that the submanifold $Y=f^{-1}(t)$ of $M$ is orientable.

Problem 5

Let $M$ be a smooth manifold covered by two open connected sets, $M=U_{1} \cup U_{2}$. Suppose $\omega$ is a 1 -form on $M$ and $f_{i}: U_{i} \rightarrow \mathbb{R}$ are smooth maps such that $\omega_{U_{i}}=d f_{i}, i=1,2$ Prove that if $U_{1} \cap U_{2}$ is connected then there is a smooth function $f: M \rightarrow \mathbb{R}$ such that $\omega=d f$. Find an example of a manifold $M=U_{1} \cup U_{2}$ where $U_{1} \cap U_{2}$ is not connected, and this conclusion does not hold.

Problem 6

Prove that the orthogonal group $O(n)$ (consisting of $n \times n$ real matrices whose rows are orthonormal) is a manifold. What is its dimension?

Problem 7

Let $A$ and $B$ each be copies of a Möbius band, and let $X=A \cup B$ be the space obtained by identifying the boundary circles of $A$ and $B$.
a) Find a presentation for the fundamental group of $X$.
b) Compute the homology groups of $X$ with $\mathbb{Z}$ coefficients and with coefficients in $\mathbb{Z} / 2 \mathbb{Z}$.

Problem 8

Consider the Möbius band $M$ with boundary curve $\gamma$. Denote the two circles in the wedge sum $S^{1} \vee S^{1}$ by $a, b$, and consider a continuous map $\partial M \longrightarrow S^{1} \vee S^{1}$ which sends $\gamma$ to the word $a b^{2} a b$. Does $f$ admit an extension to $M$ ?

Problem 9

A possible alternative to one of the above problems:

Let $f: A \longrightarrow B$ be a chain map. Denoting the differential in the chain complexes $A, B$ by $\mathrm{d}^{A}, \mathrm{~d}^{B}$ respectively, the mapping cone $C(f)$ is defined as the chain complex

$$\ldots \longrightarrow A_{n+1} \oplus B_{n} \longrightarrow A_{n} \oplus B_{n-1} \longrightarrow A_{n-1} \oplus B_{n-2} \longrightarrow \ldots$$

with the differential

$$\left(\begin{array}{cc} -d_{n+1}^{A} & 0 \\ f_{n+1} & d^{B} b_{n} \end{array}\right)$$

Prove that if $f, g$ are chain homotopic, then $C(f), C(g)$ are chain homotopy equivalent.