Smash product

The smash product describes a topological construction. For two given dotted topological spaces and with base points and consider first the product space with the identification for all and all . The quotient of under this identification is called the smash product of and and is denoted by. It usually depends on the base points chosen. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle x_{0}}$ ${\displaystyle y_{0}}$ ${\displaystyle X\times Y}$ ${\displaystyle (x,y_{0})\sim (x_{0},y)}$ ${\displaystyle x\in X}$ ${\displaystyle y\in Y}$ ${\displaystyle X\times Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\wedge Y}$

If one identifies the space with and with , then and in intersect and their union yields the subspace of . The smash product is then the quotient ${\displaystyle X}$ ${\displaystyle X\times \left\{y_{0}\right\}}$ ${\displaystyle Y}$ ${\displaystyle \left\{x_{0}\right\}\times Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle (x_{0},y_{0})}$ ${\displaystyle X\vee Y}$ ${\displaystyle X\times Y}$

${\displaystyle X\wedge Y=X\times Y/X\vee Y}$.

The smash product is particularly important in homotopy theory , where it makes the homotopy category a symmetrical monoidal category , with the 0 sphere (consisting of two points) as the neutral element . The smash product is commutative except for homeomorphism and associative except for homotopy, i.e. H. and are not necessarily homeomorphic , but equivalent to homotopy . ${\displaystyle X\wedge (Y\wedge Z)}$ ${\displaystyle (X\wedge Y)\wedge Z}$

Examples

• The smash product of two spheres and is homeomorphic to the sphere . The smash product of two circles is therefore a 2-sphere, which is the quotient of a torus . ${\displaystyle S^{m}}$ ${\displaystyle S^{n}}$ ${\displaystyle S^{m+n}}$
• With the Smash product, the so-called reduced suspension can be obtained as:

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

Functorial properties

In the category of dotted topological spaces, the smash product has the following property, which is analogous to the tensor product of modules . The adjunct formula applies to locally compact ${\displaystyle A}$

${\displaystyle \mathrm {Top} _{\bullet }(X\wedge A,Y)\cong \mathrm {Top} _{\bullet }(X,\mathrm {Top} _{\bullet }(A,Y))\,,}$

where the space of the base point-preserving continuous mappings is designated with the compact-open topology . If one takes for the unit circle , the result is a special case that the reduced suspension on the left is adjoint to the loop space . ${\displaystyle Top_{*}(A,Y)}$ ${\displaystyle A}$ ${\displaystyle S^{1}}$ ${\displaystyle \Sigma }$ ${\displaystyle \Omega }$