Problem 1 (qiuzhen-qualify 2025-fall)

Consider the topological space $X$ given by the gluing of the following polygonal pieces:

$146$, $52153$, $2346$.

Here the word $146$ means the labeling scheme for the 3-gon is $e_1 e_4 e_6^{-1}$, and so for the other words. Each $e_i$, $1 \le i \le 6$ is oriented counter-clock-wisely and the underline means the orientation of the edge is clock-wise.

Compute the homotopy groups $\pi_i(X)$ and singular homology groups $H_i(X; Z)$ for all $i \ge 1$.

Problem 2 (qiuzhen-qualify 2025-fall)

Let $M$ be a Riemannian manifold and $\mathcal{K}$ be a set of isometries of $M$. Suppose that $F$ is a nonempty subset of $M$ which is fixed by every element in $\mathcal{K}$. Show that each connected component of $F$ is a closed totally geodesic submanifold of $M$.

Problem 3 (qiuzhen-qualify 2025-fall)

Let $G$ be a compact Lie group of dimension $\ge 1$. Show that the Euler characteristics of $X$ vanishes.

Problem 4 (qiuzhen-qualify 2025-fall)

Let $H^n = \{(x_1, x_2, \ldots, x_n) \in R^n \mid x_n > 0\}$ be the upper half-space with the metric defined by $$ds^2 = \frac{1}{x_n^2}(dx_1^2 + \cdots + dx_n^2).$$ Let $\Sigma$ be a $k$-dimensional minimal submanifold of $H^n$, where $k < n$. Show that $\mathrm{Ric}_{\Sigma}(X) \le 0$, i.e., $\forall$ vector field $X$ on $\Sigma$, we have $\mathrm{Ric}(X, X) \le 0$.

Problem 5 (qiuzhen-qualify 2025-fall)

Suppose $X$ is a contractible, compact manifold of dimension $n \ge 1$. Prove that $$H_i(\partial X; Z) = H_i(S^{n-1}; Z)$$ for all $i$.

Problem 6 (qiuzhen-qualify 2025-fall)

Let $(M, g)\, $ be a Riemannian manifold and $\pi : \tilde{M} \to M$ be a covering map with the pullback metric $\pi^* g$ on $\tilde{M}$.

(a) Show that $M$ is complete if and only if $\tilde{M}$ is complete.

(b) Is the conclusion in (a) true if $\pi$ is only assumed to be a local isometry? If so, provide a proof. If not, provide a counterexample.

Problem 7 (qiuzhen-qualify 2025-fall)

Let $X$ be a connected finite-dimensional CW complex, $\tilde{X}$ its universal cover. Let $K(\pi_1(X), 1)$ be the Eilenberg-McLane space (i.e., the space whose fundamental group is $\pi_1(X)$ and whose higher homotopy groups vanish). Let $Y = \tilde{X} \times K(\pi_1(X), 1)$. Show that:

(a) $X$ and $Y$ have isomorphic homotopy groups;

(b) If $\pi_1(X)$ contains an element of order $2$, then $X$ and $Y$ are not homotopy equivalent.

Problem 8 (qiuzhen-qualify 2025-fall)

Let $M$ be a complete Riemannian manifold with nonnegative sectional curvature and $\gamma : [0, 1] \to M$ be a minimizing geodesic with $\gamma(0) = p$, $\gamma(1) = q$. Suppose that $X$ is a parallel vector field along $\gamma$ such that $X \perp \gamma'$. 

(a) Show that for sufficiently small $t > 0$, $$\mathrm{dist}(\exp_p(tX(0)), \exp_q(tX(1))) \le \mathrm{dist}(p, q).$$

(b) State the condition for which the equality in (a) holds (no proof needed).

Problem 9 (qiuzhen-qualify 2025-fall)

Let $X$ be the space of “a line contained in a plane in $C^3$”, namely a point $F \in X$ is described as a sequence $$\{0\} \subset l \subset P \subset C^3,$$ where $P$ is a complex plane and $l$ is a complex line.

(a) The $\mathrm{GL}(3, C)$-action on $C^3$ induces an action on $X$. Describe the stabilizer of the action and prove that $X$ is a smooth manifold.

(b) Compute the cohomology ring $H^*(X; R)$.

Problem 10 (qiuzhen-qualify 2025-fall)

Let $(M, g)$ be a complete compact oriented Riemannian manifold. We say a vector field $X$ on $M$ is a conformal Killing field if $\mathcal{L}_X g = f g$ for some smooth function $f$, where $\mathcal{L}_X$ denotes the Lie derivative with respect to $X$. If $\mathrm{Ric}(M) \le 0$, show that every conformal Killing field on $M$ is parallel. (Hint: Consider $\nabla_X X - \operatorname{div}(X)\, X$.)