Topology - August 2025

Topology Qualifying Examination

August 2025

Instructions: Solve four out of the five problems below. Even if you attempt all the problems, indicate which problems you want graded. You must justify your claims either by direct arguments or by referring to theorems you know.

Important Note: You may NOT use the textbook or any notes during the exam.

Problem 1

Let $\mathbb{RP}^n$ be real projective $n$-space.

1. Using the CW structure with one cell in each dimension $0,1,\ldots,n$, compute $H_i(\mathbb{RP}^n;\mathbb Z)$ for all $i$. State precisely for which $n$ the top homology group is $\mathbb Z$ and for which it is $0$.

2. Compute the mod-2 cohomology groups $H^i(\mathbb{RP}^n;\mathbb Z/2\mathbb Z)$.

3. Describe the structure of the cohomology ring $H^*(\mathbb{RP}^n;\mathbb Z/2\mathbb Z)$. You don’t need to give a proof of your answer.

4. Let $m<n$. Prove that there is no retraction $r:\mathbb{RP}^n\to\mathbb{RP}^m$. (Hint: Use part (c).)

Problem 2

Let $X=S^1\vee S^1$ be the wedge of two circles with wedge point $x_0$. Let the loops generating the two circles be $a$ and $b$.

1. Compute $\pi_1(X,x_0)$ in in terms of $a,b$.

2. Classify (up to equivalence of coverings) all connected 2-sheeted covering spaces $p:Y\to X$. Give explicit labeled graph models for the total spaces $Y$, indicate which coverings are normal, and for each normal cover compute the deck transformation group.

3. Draw a picture of a covering space corresponding to the normal subgroup of $\pi_1(X,x_0)$ generated by $a,b$ and $(ab)^4$, and prove this covering space is indeed the correct one.

Problem 3

1. Let $D^2$ denote the 2-dimensional unit disc. Is there a continuous function $f:D^2\to D^2$ without fixed points?

2. Is there a connected, compact 4-dimensional manifold without boundary such that: $X$ is simply connected and $H_3(X,\mathbb Z)=\mathbb Z\oplus\mathbb Z$?

3. Suppose that $Y$ is an $n$-dimensional embedded submanifold of a simply connected $n+2$-dimensional manifold $X$. Is the complement $X\setminus Y$ simply connected?

Problem 4

Let $E\subset S^2$ be the equator of the 2-sphere and $f:E\to S^1$ be an $n$-sheeted covering map of $E$ onto $S^1$ (where $n>0$). Form the quotient space $X_n$ from the disjoint union $S^2\sqcup S^1$ by identifying each $x\in E\subset S^2$ with $f(x)\in S^1$.

1. Calculate the fundamental group of $X_n$.

2. The quotient map in part (a) gives a map $q:S^2\to X_n$. Calculate $H_2(X_n)$ and describe the map $q_*:H_2(S^2)\to H_2(X_n)$, induced by $q$.

3. Let $\widetilde X_n$ denote the universal cover of $X_n$. Show that for every $n$, there is a continuous map $S^2\to\widetilde X_n$ not homotopic to a constant map.

Problem 5

1. Let $\mathbb{RP}^2$ denote the real projective plane and let $M=S^1\times S^1\times S^1$. Prove that any continuous map

$$f:\mathbb{RP}^2\times\mathbb{RP}^2\to M$$

is null-homotopic.

2. Show that if $M$ is a compact contractible $n$-manifold for $n\ge1$, then $\partial M$ is a homology $(n-1)$-sphere. That is, we have $H_i(\partial M)=H_i(S^{n-1})$, for all $i$.