Topology - August 2024

Topology Qualifying Examination

M. Hedden, August 2024

Instructions: Solve four out of the five problems. Justify your claims either by direct arguments or by referring to theorems you know.

Problem 1

State and prove the Brouwer fixed point theorem.

Problem 2

Let $Z$ be a CW complex constructed from the 1-complex shown here:

by attaching three 2-cells via attaching maps that traverse the loops $a^5$, $b^3$, and $aba^{-1}b^{-1}$, respectively, and a single 7-cell (attached along any map)

1. Compute $\pi_1(Z)$.

2. Prove that any map $f:Z\to S^1$ is homotopic to a constant map.

Problem 3

1. Let $Y$ be any topological space. Using only the definition of singular homology, prove that $H_0(Y)$ is isomorphic to the free abelian group generated by the path components of $Y$.

2. Let $+$ denote the subspace of $\mathbb R^2$ consisting of the coordinate axes; that is,

$$+:=\{(x,y)\in\mathbb R^2\mid xy=0\}.$$

Prove there does NOT exist a homeomorphism $g:+\to +$ sending the origin $p=(0,0)$ to the point $q=(1,0)$. (Hint: Part (a) is there for a reason.)

Problem 4

Find all the connected covering spaces of $\mathbb{RP}^2\times\mathbb{RP}^2$. Justify your answer carefully. Which of these covering spaces are normal? Explain.

Problem 5

Let $X$ be the space consisting of a genus two surface $\Sigma$ and a 2-disk $D$, meeting along a circle as shown in the figure on the following page. Calculate the homology groups $H_i(X)$ for all $i$.