Problem 1 (yau-contest 2014)
Section: Geometry and Topology

Compute the fundamental and homology groups of the wedge sum of a circle $S^1$ and a torus $T = S^1 \times S^1$.

Problem 2 (yau-contest 2014)
Section: Geometry and Topology

Given a properly discontinuous action $F : G \times M \to M$ on a smooth manifold $M$, show that $M/G$ is orientable if and only if $M$ is orientable and $F(g,\cdot)$ preserves the orientation of $M$. Use this statement to show that the Möbius band is not orientable and that $RP^n$ is orientable if and only if $n$ is odd.

Problem 3 (yau-contest 2014)
Section: Geometry and Topology

(a) Consider the space $Y$ obtained from $S^2 \times [0,1]$ by identifying $(x,0)$ with $(-x,0)$ and also identifying $(x,1)$ with $(-x,1)$, for all $x \in S^2$. Show that $Y$ is homeomorphic to the connected sum $RP^3\#RP^3$.

(b) Show that $S^2 \times S^1$ is a double cover of the connected sum $RP^3\#RP^3$.

Problem 4 (yau-contest 2014)
Section: Geometry and Topology

Prove that a bi-invariant metric on a Lie group $G$ has nonnegative sectional curvature.

Problem 5 (yau-contest 2014)
Section: Geometry and Topology

Let $M$ be the upper half-plane $R_+^2$ with the metric $$ds^2 = \dfrac{dx^2 + dy^2}{y^k}.$$ For which values of $k$ is $M$ complete?

Problem 6 (yau-contest 2014)
Section: Geometry and Topology

Given any nonorientable manifold $M$ show the existence of a smooth orientable manifold $M$ which is a double covering of $M$. Find $M$ when $M$ is $RP^2$ or the Möbius band.