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 $.

Comments

The suspension functor and the 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

[a1]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapt. 2