Exact sequence for join and product

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Template:Exact sequence for construction

Template:Exact sequence for construction

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

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 \ge 1, this section can be constructed geometrically, and for q = 0 it can be constructed explicitly in terms of the description of reduced homology.