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Concept
Smash product
Summary
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 ∨ Y. The smash product is then the quotient
The smash product shows up in homotopy theory, a branch of algebraic topology. In homotopy theory, one often works with a different of spaces than the . 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.
The smash product of any pointed space X with a 0-sphere (a discrete space with two points) is homeomorphic to X.
The smash product of two circles is a quotient of the torus homeomorphic to the 2-sphere.
More generally, the smash product of two spheres Sm and Sn is homeomorphic to the sphere Sm+n.
The smash product of a space X with a circle is homeomorphic to the reduced suspension of X:
The k-fold iterated reduced suspension of X is homeomorphic to the smash product of X and a k-sphere
In domain theory, taking the product of two domains (so that the product is strict on its arguments).
For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g., that of compactly generated spaces), there are natural (basepoint preserving) homeomorphisms
However, for the naive category of pointed spaces, this fails, as shown by the counterexample and found by Dieter Puppe.
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