I am desperately trying to solve the following problem:
Let $X$ and $Y$ be topological spaces and $X * Y$ their join. Prove that there is a short exact sequence $$ 0 \to \tilde{H}_k(X * Y) \to \tilde{H}_{k-1}(X \times Y) \to \tilde{H}_{k-1}(X) \oplus \tilde{H}_{k-1}(Y) \to 0$$ Using Mayer-Vietoris it was quite easy to show, that there is a long exact sequence of that fashion. I only need to "cut" it in the right way, to get the short exact sequence. The two things I already figured out:
- The second map is $\phi := (p_1)_* \oplus (p_2)_*$, where $p_1$ and $p_2$ are the projections $X \times Y \to X$, $X \times Y \to Y$.
- The most elegant way to prove this is providing a split $$s : \tilde{H}_{k-1}(X) \oplus \tilde{H}_{k-1}(Y) \to \tilde{H}_{k-1}(X \times Y)$$ with $ {\rm id} = ( (p_1)_* \oplus (p_2)_*)\circ s$.
My approach: Choose $x_0 \in X, y_0 \in Y$ and define $\iota_X : X \to X \times Y$ by $x \mapsto (x,y_0)$ and $\iota_Y$ analogously. Let $s := (\iota_X)_* + (\iota_Y)_*$. If I am not mistaken we now get $$\phi \circ s \big( [\sigma], [\rho]\big) = \Big( [\sigma] + \big[\rm const _{x_0}\big] , [\rho] + \big[\rm const_{y_0}\big]\Big)$$ So only $\big[\rm const _{x_0}\big] =0$ remains to be proven. This is the point where I am stuck. I think I'm overlooking something, because this looks like it should be obvious to me, but it isn't …
Thanks in advance for your help!