Problem 1 (yau-contest 2018)
Section: Geometry and Topology

Let $M$ and $N$ be smooth, connected, orientable $n$-manifolds for $n \ge 3$, and let $M\#N$ denote their connect sum.

(a) Compute the fundamental group of $M\#N$ in terms of that of $M$ and of $N$ (you may assume that the basepoint is on the boundary sphere along which we glue $M$ and $N$).

(b) Compute the homology groups of $M\#N$.

(c) For part (a), what changes if $n = 2$? Use this to describe the fundamental groups of orientable surfaces.

Problem 2 (yau-contest 2018)
Section: Geometry and Topology

Determine all of the possible degrees of maps $S^2 \to S^1 \times S^1$.

Problem 3 (yau-contest 2018)
Section: Geometry and Topology

Classify all vector bundles over the circle $S^1$ up to isomorphism.

Problem 4 (yau-contest 2018)
Section: Geometry and Topology

Suppose $C$ is a regular curve in the unit sphere $S^2$. For any point $W \in S^2$, there exists the only oriented great circle $S_W$ (determined by the right hand rule) in $S^2$ such that $W$ is the pole of $S_W$. Denote by $n(W)$ the number of points at which the oriented great circle $S_W$ and $C$ intersect. Prove the Crofton formula

$$\iint_{S^2} n(W)\,dW = 4L,$$

where $dW$ and $L$ is the area element of $S^2$ and the length of $C$, respectively.

Problem 5 (yau-contest 2018)
Section: Geometry and Topology

Let $M$ be an $n$-dimensional closed submanifold in the Euclidean space $\mathbb{R}^{n+p}$. Prove the following inequality

$$\int_M H^n\,dV \ge \operatorname{vol}(S^n),$$

where $H$ and $dV$ is the mean curvature (i.e., norm of the mean curvature vector) and the volume element of $M$, and $S^n$ is the standard unit sphere of dimension $n$.

Problem 6 (yau-contest 2018)
Section: Geometry and Topology

Let $M$ be an even dimensional compact and oriented Riemannian manifold with positive sectional curvature. Show that $M$ is simply connected.