Smash product

The smash product called a topological structure. Given two dotted topological spaces X and Y of the base points x0 and y0 Considering first the product space X × Y with the identifier (x, y0) ~ (x0, y) for all x ∈ X and all y ∈ Y, the quotient of X × Y with this identification is the smash- product of X and Y, and is denoted by X ∧ Y. It usually depends on the chosen basis points.

If we identify the space X with X × { y0 } and { x0 } × Y with Y, then X and Y intersect in ( x0, y0) and their union gives the subspace X ∨ Y of X × Y. The Smash Product is the quotient

The smash product is especially important in homotopy theory, where it makes the homotopy category to a symmetric monoidal category, with the 0 - sphere (consisting of two points ) as neutral element. The smash product is commutative up to homotopy, ie X ∧ Y ∧ X and Y are not necessarily homeomorphic, but homotopic.

Examples

• The smash product of two spheres Sm and Sn is homeomorphic to the sphere Sm n The smash product of two circles is therefore a 2-sphere, which is the quotient of a torus.
• With the smash product can be called the reduced device to attach as a:

Functorial properties

In the category of topological spaces dotted the smash product has the following property, which is analogous to the tensor product of modules. For A locally compact the Adjunktionsformel applies

Where Stet (A, Y) the space of base point preserving continuous maps provided with the compact - open topology called. If the unit circle S1 is taken for A, we obtain as a special case that the reduced device to attach Σ left adjoint to the loop space, .

• Algebraic Topology