TOPOLOGY GENERAL EXAM FALL 2025

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 $X=S^{1} \times \mathbb{R} P^{2}$.
(a) Compute $\pi_{1}(X)$.
(b) Describe the universal cover of $X$ and explicitly describe each element in the group of deck transformations of the universal cover.

Problem 2

Let $x_{1}, x_{2}, \ldots, x_{10}$ be 10 distinct points on the 2 -torus $T^{2}$. Let $X$ be the quotient of $T^{2}$ obtained by identifying all 10 points.
(a) Compute $\pi_{1}(X)$.
(b) Compute $H_{n}(X)$ for all $n \geq 0$.

Problem 3

(a) Let $f: S^{1} \rightarrow \mathbb{R}$ be a continuous map. Show that there exists some $x \in S^{1}$ such that $f(x)=f(-x)$.
(b) Show that any continuous map $g: \mathbb{R} P^{2} \times \mathbb{R} P^{2} \rightarrow T^{2}$ is nullhomotopic.

Problem 4

Let ( $C, d$ ) and ( $C^{\prime}, d^{\prime}$ ) be chain complexes of abelian groups. The mapping cone of a chain map $f:(C, d) \rightarrow\left(C^{\prime}, d^{\prime}\right)$, denoted (cone $(f), d^{\prime \prime}$ ), is defined by

$$\operatorname{cone}(f)_{n}:=C_{n-1} \oplus C_{n}^{\prime}, \quad d^{\prime \prime}\left(c, c^{\prime}\right)=\left(-d(c), d^{\prime}\left(c^{\prime}\right)-f(c)\right) .$$

Prove that (cone $\left.(f), d^{\prime \prime}\right)$ is a chain complex.

Problem 5

Consider the family of smooth maps $f_{c}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ defined by

$$f_{c}(x, y)=\left(x, y, c x+y^{2}\right), \quad c \in \mathbb{R} .$$

Let $S \subset \mathbb{R}^{3}$ be the surface defined by

$$S=\left\{(x, y, z) \in \mathbb{R}^{3} \mid z=x^{2}+y^{2}\right\} .$$

(a) For which values of $c \in \mathbb{R}$ is the map $f_{c}$ transverse to the surface $S$ ?
(b) For what values of $c \in \mathbb{R}$ is the intersection of the image of $f_{c}$ with $S$ a manifold?

Problem 6

Consider the 3 -form

$$\theta=e^{w}(x d y \wedge d z+y d z \wedge d x+z d x \wedge d y)$$

on $\mathbb{R}^{4}$ where the coordinates are $x, y, z, w$. Let $H$ be the hemisphere $x^{2}+y^{2}+ z^{2}+w^{2}=1, w \geq 0$, with the orientation induced as being part of the boundary of the 4 -ball with the standard orientation. Compute

$$\int_{S} \mathrm{~d} \theta .$$

Problem 7

Let $p: M \rightarrow N$ be a smooth covering map between closed manifolds of some dimension $n$. Suppose $N$ is oriented. Show that $M$ is orientable. Pick an orientation on $M$ and compute the degree of $p$ in terms of the number of sheets of the covering.