# Extra Problems: Algebraic Topology

## Algebraic Topology

### Fundamental Group

• Compute $$\pi_1(X)$$ where $$X \coloneqq S^2/\sim$$, where $$x\sim -x$$ only for $$x$$ on the equator $$S^1 \hookrightarrow S^2$$.
• Hint: try cellular homology. Should yield $$[{\mathbf{Z}}, {\mathbf{Z}}/2{\mathbf{Z}}, {\mathbf{Z}}, 0, \cdots]$$.
• Show that if $$X = S^2 {\textstyle\coprod}_{\operatorname{id}} S^2$$ is a pushout along the equators, then $$H_n(X) = [{\mathbf{Z}}, 0, {\mathbf{Z}}^3, 0, \cdots]$$.

### Covering Spaces

• Describe all connected covering spaces of $${\mathbb{RP}}^2 \vee {\mathbb{RP}}^2$$.

### Homology

• Compute the homology of the Klein bottle using the Mayer-Vietoris sequence and a decomposition $$K = M {\textstyle\coprod}_f M$$
• Use the Kunneth formula to compute $$H^*(S^2\times S^2; {\mathbf{Z}})$$.
• Known to be $$[{\mathbf{Z}}, 0, {\mathbf{Z}}^2, 0, {\mathbf{Z}}, 0, 0, \cdots]$$.
• Compute $$H^*(S^2 \vee S^2 \vee S^4)$$
• Known to be $$[{\mathbf{Z}}, 0, {\mathbf{Z}}^2, 0, {\mathbf{Z}}, 0, 0, \cdots]$$.
• Show that $$\chi(\Sigma_g + \Sigma_h) = \chi(\Sigma_g) + \chi(\Sigma_h) - 2$$.