Simplicial Structures in Topology

32 I Fundamental Concepts
The map ˙kn+1 has the property
˙kn+1(0,x)=˙kn+1(1,x)=˙kn+1(t,e0)=e0
for every t ∈ I and every x ∈ S n ; therefore, it gives rise to a homeomorphism
� kn+1 : ΣS n ∼ = S n+1 ,
as can be seen in Fig. I.7. We may then say that for every y ∈ S n+1 � {e0} there are
Fig. I.7
I × e 0
˙k n+1
a unique element x ∈ S n and a unique t ∈ I � ∂I such that y = t ∧ x.
The functor
Ω : Top ∗ → Top ∗
is defined, on a given object (X,x0) ∈ Top ∗, as the space
ΩX = { f ∈ M(I,X) | f (0)= f (1)=x0}
with the topology induced by the compact-open topology of M(I,X). The morphism
Ω( f ): ΩX → ΩY
corresponding to the morphism f ∈ Top ∗(X,Y) is defined through composition of
maps:
(∀g ∈ ΩX)Ω( f )(g)= fg: I → Y.
The space ΩX is called loop space (with base at x0). The base point of ΩX is
the constant path on x0.
There is a special relation between the functors Ω and Σ as follows. Let f : I ×
X → Y be a map such that f (I ×{x0}∪∂I × X)=y0; its adjoint ˆf : X → M(I,Y ),
being continuous (see Theorem (I.1.37)), is such that
(∀x ∈ X) ˆf (x)(∂I)=y0
and so we are able to construct a function
Φ : M∗(ΣX,Y ) → M∗(X,ΩY).
t∧e 0