The space
(CW-complex)
where
is the unit interval and the slant line denotes the
operation of identifying a subspace with one point. The
suspension
of a
pointed space
is defined to be the pointed space
This is also known as a
reduced
or
contracted suspension.
A suspension is denoted by
(or sometimes
).
The correspondence
defines a functor from the category of topological (pointed) spaces into itself.
Since the suspension operation is a functor, one can define a homomorphism
,
which is also called the
suspension.
This homomorphism is identical with the composite
of the homomorphism induced by the imbedding
and the
Hurewicz isomorphism
,
where
is the operation of forming loop spaces (cf.
Loop space).
For any
homology theory
(cohomology theory
)
one has an isomorphism
that coincides with the connecting homomorphism of the exact sequence of the pair
,
where
is the
cone
over
.
The image of a class
under this isomorphism is known as the
suspension
of
and is denoted by
(or
).
The
suspension
of a
cohomology operation
is defined to be the cohomology operation whose action on
coincides with
,
and whose action on
coincides with that of
.