Smash product

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 × 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} × 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

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

The smash product is important in homotopy theory, a branch of algebraic topology. One reason why is that the smash product makes the homotopy category into a symmetric monoidal category, with the 0-sphere (two points) as unit. Symmetric here means that the smash product of spaces is commutative up to homotopy. Thus smash product behaves somewhat like the tensor product of modules. Furthermore, the suspension functor can be represented by smashing with a circle.

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.

The smash product can be used to define the reduced suspension:

$\Sigma X=S^{1}\wedge X$.

Adjoint relationship

Adjoint functors make the analogy between the tensor product and the smash product more precise. In the category of R-modules over a commutative ring R, the tensor functor (– ⊗_R A) is left adjoint to the internal Hom functor Hom(A,–) so that:

$\mathrm {Hom} (X\otimes A,Y)\cong \mathrm {Hom} (X,\mathrm {Hom} (A,Y)).$

In the category of pointed spaces, the smash product plays the role of the tensor product. In particular, if A is locally compact then we have an adjunction

$\mathrm {Hom} (X\wedge A,Y)\cong \mathrm {Hom} (X,\mathrm {Hom} (A,Y))$

where Hom(A,Y) is the space of based continuous maps together with the compact-open topology.

In particular, taking A to be the unit circle S¹, we see that the suspension functor Σ is left adjoint to the loop space functor Ω.