# Suspension

of a topological space (CW-complex) $X$

The space (CW-complex)

$$( X \times [ 0, 1]) / [( X \times \{ 0 \} ) \cup ( X \times \{ 1 \} )] ,$$

where $[ 0, 1]$ is the unit interval and the slant line denotes the operation of identifying a subspace with one point. The suspension of a pointed space $( X, x _ {0} )$ is defined to be the pointed space

$$S ^ {1} \wedge X =$$

$$= \ ( X \times [ 0, 1]) / [ ( X \times \{ 0 \} ) \cup ( X \times \{ 1 \} ) \cup ( x _ {0} \times [ 0, 1])].$$

This is also known as a reduced or contracted suspension. A suspension is denoted by $SX$( or sometimes $\Sigma X$). The correspondence $X \mapsto SX$ defines a functor from the category of topological (pointed) spaces into itself.

Since the suspension operation is a functor, one can define a homomorphism $\pi _ {n} ( X) \rightarrow \pi _ {n + 1 } ( SX)$, which is also called the suspension. This homomorphism is identical with the composite of the homomorphism induced by the imbedding $X \rightarrow \Omega SX$ and the Hurewicz isomorphism $\pi _ {n} ( \Omega SX) \cong \pi _ {n + 1 } ( SX)$, where $\Omega$ is the operation of forming loop spaces (cf. Loop space). For any homology theory $h _ {*}$( cohomology theory $h ^ {*}$) one has an isomorphism

$$\delta : {\widetilde{h} } {} ^ {n} ( X) \cong \ {\widetilde{h} } {} ^ {n + 1 } ( SX) = \ h ^ {n + 1 } ( CX, X)$$

that coincides with the connecting homomorphism of the exact sequence of the pair $( CX, X)$, where $CX$ is the cone over $X$. The image of a class $x \in h ^ {n} ( X)$ under this isomorphism is known as the suspension of $x$ and is denoted by $\delta x$( or $Sx$).

The suspension of a cohomology operation $a$ is defined to be the cohomology operation whose action on ${\widetilde{h} } {} ^ {*}$ coincides with $\delta ^ {-} 1 a \delta$, and whose action on $h ^ {*} ( pt)$ coincides with that of $a$.

$$\mathop{\rm Top} ( SX, Y) \cong \mathop{\rm Top} ( X, \Omega Y) .$$
The bijection above associates to $f: SX \rightarrow Y$ the mapping $g: X \rightarrow \Omega Y$ which associates the loop $g( x)( t)= f( x, t)$ to $x \in X$. This adjointness is compatible with the homology and thus also defines an adjunction for the category of pointed topological spaces and homotopy classes of mappings.