Topology - August 2021

TOPOLOGY QUALIFYING EXAM – AUGUST 2021

Instructions: Solve any FOUR out of the five problems. If you work on all five problems, your top four problem scores will count towards your exam score.

NOTE: No books, notes, or resources of any kind may be used during the exam.

All work must be shown, and all answers justified appropriately.

Problem 1

Suppose that $f$ is a continuous map from a sphere $S^n$ to the torus $T^2=S^1\times S^1$.

1. If $n\ge2$, prove that the map $f$ is nullhomotopic (i.e. homotopic to a constant map).

2. If $n=1$, does $f$ have to be nullhomotopic? Justify your answer.

Problem 2

1. Explain how to construct a CW complex $X$ with fundamental group $\pi_1(X)=\mathbb Z/2\mathbb Z*\mathbb Z/3\mathbb Z$.

2. Compute all the homology groups of the CW complex $X$ you constructed in part (a).

3. Is the answer to part (b) the same for all CW complexes $Y$ with fundamental group $\pi_1(Y)=\mathbb Z/2\mathbb Z*\mathbb Z/3\mathbb Z$?

Problem 3

Prove or find a counterexample for each of the following assertions:

1. Suppose that for connected CW-complexes $X,Y$, we have that $H_i(X)=H_i(Y)$ for all $i\ge0$. Then $X$ and $Y$ are homotopy equivalent.

2. Let $Z=\mathbb{RP}^2\vee\mathbb{RP}^2$ be the wedge of two projective planes. There is a covering space $W$ of $Z$ such that $\pi_1(W)=\mathbb Z$.

3. There is no retraction $r:D^2\to S^1$ from the disk to its boundary circle.

Problem 4

Prove that if $\mathbb R^m$ is homeomorphic to $\mathbb R^n$, then $m=n$.

Problem 5

Recall that the suspension $SX$ of a space $X$ is the quotient of $X\times I$ obtained by collapsing $X\times\{0\}$ to one point and $X\times\{1\}$ to another point. Prove that $\widetilde H_{n+1}(SX)\cong\widetilde H_n(X)$ for all $n\ge0$.