Problem 1 (yau-contest 2022)
Section: Geometry and Topology

The topological space $X$ is obtained by gluing two tetrahedra as illustrated by the figure. There is a unique way to glue the faces of one tetrahedron to the other so that the arrows are matched. The resulting complex has 2 tetrahedra, 4 triangles, 2 edges and 1 vertex.

Show that $X$ can not have the homotopy type of a compact manifold without boundary.

Problem 2 (yau-contest 2022)
Section: Geometry and Topology

Suppose $(M, h)$ is a closed (i.e., compact without boundary) Riemannian manifold, and $h$ is a metric on $M$ with $\operatorname{sec}(h) \le -1$, where $\operatorname{sec}(h)$ is the sectional curvature. Suppose $\Sigma$ is a closed minimal surface with genus $g$ in $(M, h)$. Show that

$$\operatorname{Area}(\Sigma) \le 4\pi(g - 1).$$

Remark: A minimal surface is an immersed surface with constant mean curvature 0.

Problem 3 (yau-contest 2022)
Section: Geometry and Topology

For any topological space $X$, the $n$-th symmetric product of $X$ is the quotient of the Cartesian product $(X)^n$ by the action of the symmetric group $S_n$, which permutes the factors in $(X)^n$. This space is denoted by $SP^n(X)$, and the topology is the natural quotient topology induced from $(X)^n$.

Show that $SP^n(CP^1)$ is homeomorphic to $CP^n$. Here $CP^1$ and $CP^n$ are equipped with the manifold topology.

Problem 4 (yau-contest 2022)
Section: Geometry and Topology

Let $M$ be a complete noncompact Riemannian manifold. $M$ is said to have the geodesic loops to infinity property if for any $[\alpha] \in \pi_1(M)$ and any compact subset $K \subset M$, there is a geodesic loop $\beta \subset M \setminus K$, such that $\beta$ is homotopic to $\alpha$.

Show that if a complete noncompact Riemannian manifold $M$ does not have the geodesic loops to infinity property, then there is a line in the universal cover $\widetilde{M}$.

Remark: A line is a geodesic $\gamma : (-\infty, \infty) \to M$ such that $\operatorname{dist}(\gamma(s), \gamma(t)) = |s - t|$; a geodesic loop is a curve $\beta : [0,1] \to M$ that is a geodesic and $\beta(0) = \beta(1)$.

Problem 5 (yau-contest 2022)
Section: Geometry and Topology

A topological space $X$ is called an $H$-space if there exist $e \in X$ and $\mu : X \times X \to X$ such that $\mu(e,e) = e$ and the maps $x \mapsto \mu(e,x)$ and $x \mapsto \mu(x,e)$ are both homotopic to the identity map.

(a) Show that the fundamental group of an $H$-space is Abelian.

(b) Show that the sphere $S^{2022}$ is not an $H$-space.

Historic Remark: "$H$" was suggested by Jean-Pierre Serre in recognition of the contributions in Topology by Heinz Hopf.

Problem 6 (yau-contest 2022)
Section: Geometry and Topology

A hypersurface $\Sigma \subset \mathbf{R}^{n+1}$ is called a shrinker if it satisfies the equation

$$H(x) = \tfrac{1}{2} \langle x, \vec{n} \rangle.$$

Here $H$ is the mean curvature, which is $-\langle \operatorname{tr}_A \vec{n} \rangle$ where $A$ is the second fundamental form, $x$ is the position vector, and $\vec{n}$ is outer unit normal vector.

(a) Show that $S^n(\sqrt{2n})$, the sphere with radius $\sqrt{2n}$, is a shrinker.

(b) Show that any compact shrinker without boundary must intersect with $S^n(\sqrt{2n})$.