# Exact sequence for join and product

Template:Exact sequence for construction

Template:Exact sequence for construction

## Statement

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

Moreover, this short exact sequence *splits* so we get:

Note that the above is *not* true for unreduced homology at .

## Applications

In the case when either or is a sphere, the homology of can be computed in terms of the homology of . 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:

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

## Proof

We can view as the double mapping cylinder for the coordinate projections from to and to 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 to . For , this section can be constructed geometrically, and for it can be constructed explicitly in terms of the description of reduced homology.