Topology General Exam — January 11, 2024

Student ID: __________________________

Instructions: This is a four hour exam. Your solutions should be legible and clearly organized, written in complete sentences in good mathematical style. All work should be your own-no outside sources are permitted-using methods and results from the first year topology course topics. There are 8 problems; each problem is worth the same number of points.

Problem 1

Let $f, g: \mathbb{R}^{1} \longrightarrow \mathbb{R}^{2}$ be two smooth maps. Prove that the subset of $\mathbb{R}^{1} \times \mathbb{R}^{1} \times \mathbb{R}^{2}$ consisting of all points $(x, y, z) \in \mathbb{R}^{1} \times \mathbb{R}^{1} \times \mathbb{R}^{2}$ satisfying the equation $f(x)-g(x)=z$ is a submanifold. What is its dimension?

Problem 2

Consider two topological spaces $X, Y$ obtained from a disk by removing two smaller disks from its interior, and then identifying all three boundary curves to a single curve, respecting the orientations as shown in the figure. Compute the homology groups of both $X$ and $Y$ with $\mathbb{Z}$ coefficients.

Problem 3

Show that any smooth map $f: \mathbb{R} P^{n} \rightarrow \mathbb{R} P^{n}$ has a fixed point, when $n$ is even.

Problem 4

Prove that 1-forms $\theta_{1}, \ldots, \theta_{n}$ on a smooth $n$-manifold $M$ are linearly independent (thought of as sections of the cotangent bundle) if and only if $\theta_{1} \wedge \ldots \wedge \theta_{n}$ is non-vanishing.

Problem 5

Suppose the following diagram of abelian groups and homomorphisms commutes:

Assume that the rows and columns form exact sequences. Prove that there are isomorphisms $\operatorname{ker} \alpha \cong \operatorname{ker} \beta$ and coker $\alpha \cong \operatorname{coker} \beta$.

Problem 6

Let $M=\left\{x^{2}+y^{2}+z^{2}=1\right\} \subset \mathbb{R}^{3}$, and for a real number $c \neq 0$ let $N=\left\{x^{2}-y^{2}+z^{2}=c\right\}$. For which values of $c$ is the intersection between $M$ and $N$ transverse? For which values of $c$ is this intersection a submanifold of $\mathbb{R}^{3}$?

Problem 7

Let $\exp : \mathbb{C} \rightarrow \mathbb{C} \backslash\{0\}$ be the complex $\operatorname{exponential}$ map $\exp (z)=e^{z}$ (recall that $\exp$ is a universal covering map). If $X$ is a connected, locally path-connected space and $f: X \rightarrow \mathbb{C} \backslash\{0\}$ a continuous map, a logarithm of $f$ is a continuous map $g: X \rightarrow \mathbb{C}$ such that $\exp \circ g=f$. Prove that if such an $X$ has finite fundamental group, then any continuous map $f: X \rightarrow \mathbb{C} \backslash\{0\}$ has a logarithm.

Problem 8

The connected sum of the 2-torus $T$ and the projective plane $\mathbb{R} P^{2}$ is a smooth closed 2manifold $M$ whose fundamental group has a presentation $\pi_{1}(M)=\left\langle a, b, c \mid a b a^{-1} b^{-1} c^{2}=1\right\rangle$. Determine the number of connected, regular 3-fold covering spaces of $M$ up to equivalence.

Hint: First show that this number is the same as the number of surjections $\pi_{1}(M) \rightarrow \mathbb{Z} / 3 \mathbb{Z}$ up to equivalence, where two surjections $f, g$ are equivalent if and only if $g=\phi \circ f$ for some automorphism $\phi$ of $\mathbb{Z} / 3 \mathbb{Z}$.