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

The space ([[CW-complex|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|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|Loop space]]). For any [[Homology theory|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|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|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 $.

====Comments====
The suspension functor and the [[Loop space|loop space]] functor on the category of pointed spaces are adjoint:

$$ 
 \mathop{\rm Top} ( SX, Y)  \cong   \mathop{\rm Top} ( X, \Omega Y) .
$$

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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Switzer,   "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapt. 2</TD></TR></table>