U+2A33 Smash Product

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In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (X, x₀) and (Y, y₀) is the quotient of the product space X × Y under the identifications (x, y₀) ~ (x₀, y) for all x in X and y in Y. The smash product is itself a pointed space, with basepoint being the equivalence class of (x₀, y₀). The smash product is usually denoted X ∧ Y or X ⨳ Y. The smash product depends on the choice of basepoints (unless both X and Y are homogeneous).

One can think of X and Y as sitting inside X × Y as the subspaces X × {y₀} and {x₀} × Y. These subspaces intersect at a single point: (x₀, y₀), the basepoint of X × Y. So the union of these subspaces can be identified with the wedge sum $X\vee Y=(X⨿Y)/\sim$. In particular, {x₀} × Y in X × Y is identified with Y in $X\vee Y$, ditto for X × {y₀} and X. In $X\vee Y$, subspaces X and Y intersect in the single point $x_0\sim y_0$. The smash product is then the quotient

$X\wedge Y=(X\times Y)/(X\vee Y).$

The smash product shows up in homotopy theory, a branch of algebraic topology. In homotopy theory, one often works with a different category of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two CW complexes is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.