I have a specific problem, but it generalizes to the problem on how to compute connecting homomorphisms of any shape or form. So the problem is the following:
Let $S^n$ be the unit sphere, obtined by glueing two copies of $\Delta^n$ along their boundaries. Denote the obvious two simplices by $\tau_{\pm} : \Delta_{\pm}^n \rightarrow S^n$ denoting the upper and lower hemisphere by inclusion. I want to show that $\tau_+ – \tau_-$ is a cycle, and that $[\tau_+ – \tau_-]$ generates $\tilde{H}_n(S^n)$.
Showing that $\tau_+ – \tau_-$ is a cycle is easy, since $\partial (\tau_+ – \tau_-) = 0$, by linearity and orientation of their boundaries. Now to show that they generate $\tilde{H}_n(S^n)$, one looks at the following part of the Mayer-Vietoris LES:
$$0 \longrightarrow H_n(S^n) \xrightarrow{\partial_*} \tilde{H}_{n-1}(\partial \Delta^n) \longrightarrow 0$$
By exactness, the connecting homomorphism is an isomorphism. Lets say it is known that $[\gamma_{n-1}] \in \tilde{H}_{n-1}(\partial \Delta^n)$ is the obvious generator, "going once around the boundary". If one can show that $\partial_*[\tau_+ – \tau_-] = [\gamma_{n-1}]$, we are done. But how do I show this?
I am tempted to just write that $\partial_*[\tau_+ – \tau_-] = [\partial \tau_+] = [\gamma_{n-1}]$, but I cannot justify the first equality.
Best Answer
By how the sequence is constructed, to compute the connecting homomorphism you are to take a representative of $[\tau_+ - \tau_-]$ in $C_n$, then take its preimage in $C_n(\Delta_{+}^n) \oplus C_n(\Delta_{-}^n),$ compute the boundary, and then take its preimage in $C_{n-1}(\partial \Delta^n).$ Take as the representative $\tau_+ - \tau_-$ itself. The preimage is $(\tau_+,-\tau_-).$ The boundary is $(\gamma_{n-1},-\gamma_{n-1}).$ The preimage of this under $C_{n-1}(\partial \Delta^n) \to C_{n-1}(\Delta_{+}^n) \oplus C_{n-1}(\Delta_{-}^n)$ is $\gamma_{n-1}$.