Smash product - Definition and Overview

In mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) X and Y is the quotient of the product space X &times Y under the identifications (x, y₀) ∼ (x₀, y) for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y. The smash product obviously depends on the choice of basepoints (unless, of course, 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} &times 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 ∨ Y. The smash product is then the quotient

<math>X \wedge Y = X \times Y / X \vee Y<math>.

The smash product is important in homotopy theory, a branch of algebraic topology.

Examples

• The smash product of two spheres S^m and Sⁿ is homeomorphic to the sphere S^m+n. In particular, for m = n = 1, the smash product of two circles is a quotient of the torus homeomorphic to the 2-sphere.

See also

• pointed space
• wedge sum
• homotopy category
• suspension (topology)

Example Usage of product