Smash product explained

In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) and is the quotient of the product space under the identifications for all in and in . The smash product is itself a pointed space, with basepoint being the equivalence class of The smash product is usually denoted or . The smash product depends on the choice of basepoints (unless both X and Y are homogeneous).

X\veeY=(X\amalgY)/{\sim}

. In particular, in is identified with in

X\veeY

, ditto for and . In

X\veeY

, subspaces and intersect in the single point

x0\simy0

. The smash product is then the quotient

X\wedgeY=(X x Y)/(X\veeY).

The smash product shows up in homotopy theory, a branch of algebraic topology. In homotopy theory, one often works with a different category of spaces than the category of all topological spaces. 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.

Examples

As a symmetric monoidal product

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

\begin{align} X\wedgeY&\congY\wedgeX,\\ (X\wedgeY)\wedgeZ&\congX\wedge(Y\wedgeZ). \end{align}

However, for the naive category of pointed spaces, this fails, as shown by the counterexample

X=Y=Q

and

Z=N

found by Dieter Puppe.[1] A proof due to Kathleen Lewis that Puppe's counterexample is indeed a counterexample can be found in the book of Johann Sigurdsson and J. Peter May.[2]

These isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of tensor product in an appropriate category of pointed spaces.

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

(-RA)

is left adjoint to the internal Hom functor

Hom(A,-)

, so that

Hom(XA,Y)\congHom(X,Hom(A,Y)).

In the category of pointed spaces, the smash product plays the role of the tensor product in this formula: if

A,X

are compact Hausdorff then we have an adjunction
Maps*(X\wedge

A,Y)\cong

Maps*(X,Maps*(A,Y))

where

\operatorname{Maps*}

denotes continuous maps that send basepoint to basepoint, and
Maps*(A,Y)
carries the compact-open topology.[3]

In particular, taking

A

to be the unit circle

S1

, we see that the reduced suspension functor

\Sigma

is left adjoint to the loop space functor

\Omega

:
Maps*(\Sigma

X,Y)\cong

Maps*(X,\Omega

Y).

Notes and References

  1. Dieter. Puppe. Dieter Puppe . Homotopiemengen und ihre induzierten Abbildungen. I. . . 69 . 1958. 299–344. 0100265. 10.1007/BF01187411. 121402726 . (p. 336)
  2. Book: J. Peter. May. J. Peter May. Johann. Sigurdsson. Parametrized Homotopy Theory. Mathematical Surveys and Monographs. 132. American Mathematical Society. Providence, RI. 2006. 978-0-8218-3922-5. section 1.5. 2271789.
  3. "Algebraic Topology", Maunder, Theorem 6.2.38c