In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the smash product of two
pointed spaces (i.e.
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s 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).
One can think of and as sitting inside as the
subspaces and These subspaces intersect at a single point: the basepoint of So the union of these subspaces can be identified with the
wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the ...
. In particular, in is identified with in
, ditto for and . In
, subspaces and intersect in the single point
. The smash product is then the quotient
:
The smash product shows up in
homotopy theory, a branch of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. 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

* 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
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
s is a quotient of the
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
homeomorphic to the 2-sphere. That is, it is the quotient space of the torus
by the
figure-8 space . This can be visualized by taking the union of the innermost line of latitude of the torus and a given line of longitude and assuming their intersection is the basepoint. The union of two circles intersecting at a point is homeomorphic to the figure-8 space, which is then collapsed to a single point, resulting in a quotient space homeomorphic to the 2-sphere (see diagram).
* More generally, the smash product of two spheres ''S''
''m'' and ''S''
''n'' is homeomorphic to the sphere ''S''
''m''+''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).
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)
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
s
:
However, for the naive category of pointed spaces, this fails, as shown by the counterexample
and
found by
Dieter Puppe. A proof due to Kathleen Lewis that Puppe's counterexample is indeed a counterexample can be found in the book of
J. Peter May and Johann Sigurdsson.
These
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s 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
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
''R'', the tensor functor
is left adjoint to the internal
Hom functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...
, so that
:
In the
category of pointed spaces, the smash product plays the role of the tensor product in this formula: if
are compact Hausdorff then we have an adjunction
:
where
denotes continuous maps that send basepoint to basepoint, and
carries the
compact-open topology.
[* Reissued in 1980 (]Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, ISBN 0-521-29840-7) and 1996 (Dover Publications, Mineola, New York
Mineola is a Administrative divisions of New York#Village, village and the county seat of Nassau County, New York, Nassau County, on Long Island, Long Island, New York, United States. The population was 20,800 at the time of the 2020 United Stat ...
, ISBN 0-486-69131-4), Theorem 6.2.38c
In particular, taking
to be the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, we see that the reduced suspension functor
is left adjoint to the
loop space functor
:
:
Notes
References
*
{{DEFAULTSORT:Smash Product
Topology
Homotopy theory
Operations on structures