Simplicial Structures in Topology

Chapter VI
Homotopy Groups
VI.1 Fundamental Group
The Fundamental Theorem of Surfaces assures us that any connected compact
surface is homeomorphic to one of the following closed surfaces: the twodimensional
sphere, a connected sum of tori, or a connected sum of real projective
planes. We have seen that the homology groups of such closed surfaces are not
isomorphic, and therefore, the surfaces under discussion cannot be homeomorphic.
It is possible to arrive at this same result by computing another algebraic invariant
of the polyhedra, the so-called fundamental group, which is clearly related to the
first homology group. In what follows, we shall study such concepts in detail.
In Sect. I.2, we have defined the concept of homotopy between two based maps
f ,g ∈ Top ∗((X,x0),(Y,y0)) and we have considered the set
[X,Y ]∗ = Top ∗(X,Y )/ ∼;
we now turn our attention to the set [X,Y ]∗ when (X,x0) is the unit circle S1 with
base point e0 =(1,0).
We intend to define a group structure on the set [S1 ,Y]∗; we must, therefore, define
a multiplication in [S1 ,Y ]∗. We start by saying that a map f : (S1 ,e0) → (Y,y0)
is a loop of Y (based at y0); we can now compose two loops f and g to form a loop
( f ∗g), obtained by traveling with double speed first the loop f and then g; formally,
this loop is given by the function
( f ∗ g)(e 2πti �
f (e2π2ti 1 ) 0 ≤ t ≤
)=
2
g(e2π(2t−1)i ) 1 2 ≤ t ≤ 1.
This is an intuitive approach to the multiplication in [S 1 ,Y ]∗. Let us now go over
this definition in such a way that we may prove the following results more systematically.
Let us take the wedge product
S 1 ∨ S 1 = {e0}×S 1 ∪ S 1 ×{e0}⊂S 1 × S 1
D.L. Ferrario and R.A. Piccinini, Simplicial Structures in Topology, 195
CMS Books in Mathematics, DOI 10.1007/978-1-4419-7236-1 VI,
© Springer Science+Business Media, LLC 2011