In this question, you may assume all spaces involved are triangulable.

(a) (i) State and prove the Mayer-Vietoris theorem. [You may assume the theorem that states that a short exact sequence of chain complexes gives rise to a long exact sequence of homology groups.]

(ii) Use Mayer-Vietoris to calculate the homology groups of an oriented surface of genus $g$.

(b) Let $S$ be an oriented surface of genus $g$, and let $D_{1}, \ldots, D_{n}$ be a collection of mutually disjoint closed subsets of $S$ with each $D_{i}$ homeomorphic to a two-dimensional disk. Let $D_{i}^{\circ}$ denote the interior of $D_{i}$, homeomorphic to an open two-dimensional disk, and let

$$T:=S \backslash\left(D_{1}^{\circ} \cup \cdots \cup D_{n}^{\circ}\right)$$

Show that

$$H_{i}(T)= \begin{cases}\mathbb{Z} & i=0 \\ \mathbb{Z}^{2 g+n-1} & i=1 \\ 0 & \text { otherwise }\end{cases}$$

(c) Let $T$ be the surface given in (b) when $S=S^{2}$ and $n=3$. Let $f: T \rightarrow S^{1} \times S^{1}$ be a map. Does there exist a map $g: S^{1} \times S^{1} \rightarrow T$ such that $f \circ g$ is homotopic to the identity map? Justify your answer.