Problem 1 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(5 points+5 points) Let $a, b$ denote the two standard generators of $\pi_1(X) \cong F_2$.

(a) Draw a picture of the directed graph of the covering space of $X$ that corresponds to the following subgroup
$$H = \langle a^2,\; b^2,\; ab^2 a,\; ba^2 b,\; (ab)^2 \rangle.$$
On each edge of the directed graph, you should draw an arrow for its direction and label “$a$” or “$b$” for its image under the covering map.

(b) Is the covering above a normal covering? Prove your claim.

Problem 2 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(10 points) Let $M \subset \mathbb{R}^n$ be a smooth submanifold of dimension $m < n - 2$. Prove that the complement $\mathbb{R}^n \setminus M$ is connected and simply-connected.

Problem 3 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(10 pts) Let $(M, g)$ be a complete Riemannian manifold. Is every Killing vector field $X$ on $M$ complete?

Problem 4 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(10pts) Let $M$ be an oriented Riemannian $4$-manifold. A $2$-form $\omega$ on $M$ is said to be self-dual if $*\omega = \omega$, and anti-self-dual if $*\omega = -\omega$.

(a) Show that every $2$-form $\omega$ on $M$ can be written uniquely as a sum of a self-dual form and an anti-self-dual form.

(b) On $M = \mathbb{R}^4$ with the Euclidean metric, determine the self-dual and anti-self-dual forms in standard coordinates.

Problem 5 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(15 pts) Classify all $2024$-dimensional complete Riemannian manifolds with constant sectional curvature $K = 1$.

Problem 6 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(15 points) Let $M$ be a $2$-dimensional manifold without boundary such that $\pi_1(M)$ is an infinite group. ($M$ is not necessarily compact.) Compute $\pi_n(M)$ for all $n \ge 2$. Prove your claim.

Problem 7 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(15 pts) Compute the Ricci curvature and scalar of $\big(\mathbb{R}^2,\; g = e^{-\pi(x^2+y^2)}(dx \otimes dx + dy \otimes dy)\big)$. Is $\big(\mathbb{R}^2,\; g = e^{-\pi(x^2+y^2)}(dx \otimes dx + dy \otimes dy)\big)$ a complete Riemannian manifold? Prove your claim.

Problem 8 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(10 points +5 points) Let $M, N$ be closed, connected, orientable manifolds of the same dimension $n$. Suppose that $f : M \to N$ induces a nonzero map
$$f_* : H_n(M; \mathbb{Z}) \to H_n(N; \mathbb{Z}).$$
Are the following statements true? If true, give a proof; if false, give a counter-example and prove that the example you give is indeed a counter-example.

(a) $\dim_{\mathbb{Q}} H_k(M; \mathbb{Q}) \ge \dim_{\mathbb{Q}} H_k(N; \mathbb{Q})$ for all $k$.

(b) $\operatorname{rank} H_k(M; \mathbb{Z}) \ge \operatorname{rank} H_k(N; \mathbb{Z})$ for all $k$. (Here, the rank of a group is defined to be the minimal number of generators. By this definition, $\operatorname{rank}(\mathbb{Z} \oplus \mathbb{Z}/2) = 2$.)

Problem 9 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(15 pts) Can $M = S^1 \times S^2$ admit a smooth Riemannian metric $g$ with $\operatorname{Ric}(g) = f \cdot g$ for some smooth function $f$ on $M$? Prove your claim.

Problem 10 (qiuzhen-qualify 2024-spring)
Section: Geometry & Topology

(15 points) Prove that $\mathbb{RP}^n \times S^2$ does not have an open cover consisting of $k$ contractible open subsets if $k \le n + 1$.