Exact sequence for join and product
Moreover, this short exact sequence splits so we get:
Note that the above is not true for unreduced homology at .
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.
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.
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.