Exact sequence for join and product

Statement

Let $X$ and $Y$ be topological spaces. Denote by $X*Y$ the join and by $X\times Y$ the product. The long exact sequence of reduced homology obtained using Mayer-Vietoris then splits into short exact sequences of the form:

$0\to {\tilde {H}}_{q+1}(X*Y)\to {\tilde {H}}_{q}(X\times Y)\to {\tilde {H}}_{q}(X)\oplus {\tilde {H}}_{q}(Y)\to 0$

Moreover, this short exact sequence splits so we get:

${\tilde {H}}_{q}(X\times Y)\cong {\tilde {H}}_{q+1}(X*Y)\oplus {\tilde {H}}_{q}(X)\oplus {\tilde {H}}_{q}(Y)$

Note that the above is not true for unreduced homology at $q=0$.

Applications

In the case when either $X$ or $Y$ is a sphere, the homology of $X\times Y$ can be computed in terms of the homology of $X$. This is because taking a join with a sphere, is equivalent to an iterated suspension, and the homology of an iterated suspension is simply obtained by displacing the homology of the original space.

In symbols:

${\tilde {H}}_{q}(X\times S^{m})={\tilde {H}}_{q-m}(X)\oplus {\tilde {H}}_{q}(X)+{\tilde {H}}_{q}(S^{m})$

The utility of this exact sequence in computing the homology of the product breaks down when we do not understand the homologies of the join. In this case, we use a more advanced tool such as the Kunneth formula.

Related results

• Kunneth formula
• Eilenberg-Zilber theorem

Proof

We can view $X*Y$ as the double mapping cylinder for the coordinate projections from $X\times Y$ to $X$ and to $Y$ and then apply the exact sequence for double mapping cylinder. This is a long exact sequence. To show that it separates into several short exact sequence, and that each one splits, it suffices to construct a section of the map from ${\tilde {H}}_{q}(X)\oplus {\tilde {H}}_{q}(Y)$ to ${\tilde {H}}_{q}(X\times Y)$. For $q\geq 1$, this section can be constructed geometrically, and for $q=0$ it can be constructed explicitly in terms of the description of reduced homology.