1. Introduction
The properties of a topological space that were developed so far have been depended on the choice of topology, the collection of open sets. Taking a different tack, we introduce a different structure, algebraic in nature, associated to a space together with a choice of base point
. The structure will allow us to bring to bear the power of algebraic arguments [1] [2] [3]. The fundamental group was introduced by Poincairé in his investigations of the actions of a group on a manifold. In a long paper entitled Analysis Situs, Poincare introduced the concept of the fundamental group of a topological space. The research begun with a heuristic introduction, using functions
(not necessarily single valued) on a manifold defined by equations between coordinates
. It assumed that these functions satisfy certain differential equations, where
are known single-valued differentiable functions of
and
which satisfy certain integrability conditions [4]. Then it considered that the transformations of
produces result if one traces their values along a closed loop. Dugundji, put, for the first time, a topology on fundamental groups of certain spaces and deduced a classification theorem for connected covers of a space.
Furthermore, topologists of the early 20th century dreamed of a generalisation to higher dimensions of the non-abelian fundamental group, for applications to problems in geometry and analysis for which group theory had been successful.
For some decades now, the theorem
is a group, established to be a fundamental group with respect to “
” in the interval [0, 1] [5]. It has been used to prove some mathematical concepts such as connectedness, metric space, isomorphism, Cech homotopy etc. The theorem has led few researchers to work within this interval [0, 1] [6] [7]. Cannon and Conner (2005) used the fact that
is a group to work in one dimensions. Their research examined the fundamental groups of complicated one-dimensional spaces. In attempt to prove some other mathematical concepts, it was established that if X is a space of dimension at most 1, then, the fundamental group is isomorphic to a subgroup of the first Cech homotopy group based on finite open covers. Consequently, for a one-dimensional continuum X, the fundamental group is isomorphic to a subgroup of the first Cech homotopy group [8]. A potentially new approach to homotopy theory derived from the expositions in Brown’s (1968) and Higgins’ (1971), which in effect suggested that most of 1-dimensional homotopy theory can be better expressed in terms of groupoids rather than groups [9].
This led to a search for the uses of groupoids in higher homotopy theory, and in particular for higher homotopy groupoids. The basic intuitive concept was generalising from the usual partial compositions of homotopy classes of paths to partial compositions of homotopy classes (of some form) of complexes. But a search for such constructs proved abortive for some years from 1966 [10].
Recently, the concept of fundamental group was used on the ageing process of human. The homotopy relates the topological shape of the infant to the topological shape of the adult. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. This complex connected cylindrical shape of the body
, described by the homotopic functions
provided the ageing process in vertical interval
. But this work is limited to the interval
[11].
Therefore this research intends to establish the proof that
is a group in other domains other than [0, 1]. In this paper, we defined homotopy and presented some group properties. We then described the fundamental group and its related properties such as group homomorphisms. We also looked at useful definitions and theorems that play an important role in computing fundamental groups, and our main result was actually built upon these theorems. In the end, we established the proof that the equivalent class
is a fundamental group in the interval
.
Throughout this paper we assumed the knowledge of basic algebra and general topology.
2. Preliminaries
2.1. Homotopy
Definition 2.1.1
Let X be a topological space. A path in X from x0 to x1 is a continuous map
such that
and
. We say that x0 is the initial point and x1 the final point [12].
2.2. Homotopic Path Concatenation
Definition 2.2.1
Suppose X is a space and
is a choice of base point in X. The space of based loops in X denoted
, is the subspace of
,
(1)
Composition of loops determines a binary operation
. We restrict the notion of homotopy when applied to the space of based loops in X in order to stay in that space during the deformation.
Definition 2.2.2
Given two based loops
and
, a loop homotopy between them is a homotopy of paths
with
,
and
. That is, for each
, the path
is a loop at
.
Proposition 2.2.3
Continuous mappings
and
induce well-defined functions
and
by
and
for
.
Proof. We need to show that if
, then
and
. Fixing a homotopy
with
and
, then the desired homotopies are
and
.
To a space X we associate a space particularly rich in structure, the mapping space of paths in X,
. Recall that
is the set of continuous mappings
with the compact-open topology. The space
has the following properties:
1) X embeds into
by associating to each point
to the constant path,
for all
.
2) Given a path
, we can reverse the path by composing with
. Let
.
3) Given a pair of paths
for which
, we can compose paths by
(2)
Thus, for certain pairs of paths
and
, we obtain a new path
. Composition of paths is always defined when we restrict to a certain subspace of
[13].
Definition 2.2.4
Let X be a topological space, and
a point in X. The fundamental group of X is the set of path homotopy classes
of loops
based at
, together with the operation “
”. We denote it by
[14].
Definition 2.2.5
Given two loop classes
and
we define:
1)
.
2) The inverse of
is given by
, that is
, where
.
Theorem 2.2.6
If X is a convex subset of
, and if
,
. This defines a homotopy between
and
.
Theorem 2.2.7
The relation
is an equivalence relation on the set,
, of continuous mappings from X to Y.
Proof. Let
be a given mapping. The homotopy
is a continuous mapping
and so
.
If
and
is a homotopy between
and
, then the mapping
given by
is continuous and a homotopy between
and
that is
.
Finally, for
and
, suppose that
is a homotopy between
and
, and
is a homotopy between
and
. Define the homotopy
by
Since
, the piecewise definition of H gives a continuous function. By definition,
and
and so
[15].
Definition 2.2.8
The equivalence classes of maps from X to Y as in the Theorem 2.2.7 are called the homotopy classes ad denoted by [f] as the homotopy class of the map f. While we use the notation
when
is a loop in X based at
.
2.3. Homotopy Classes
Definition 2.3.1
The equivalent classes
determined by homotopy modulo
on the collection
of all closed paths f on S based at
are called homotopy classes of
. The collection of these homotopy classes is denoted by
[12].
3. Main Results
Path Concatenation in [m, n]
Definition 3.1.1
1) If
we define the juxtaposition
of f and g as follows:
Thus
and “
” is a binary operation on
.
2) If
, then let
Proposition 3.1.2
Let
be topological spaces and let A a subset of X. Then
is ann equivalence relation on the set
of maps from X to Y which agree with with a given map on A.
Proof. For notational convenience, drop the subscript A from the notation.
1) Reflexive property
: Define
. This is the composition of f with the projection of
on X. Since it is a composition of two continuous maps, it is continuous.
2) Symmetric property
: Suppose
is a homotomy (relative to A) of f tog. Let
. The
and similarly for
. Also, if
for
, the same is true for
.
is a composition of two continuous maps. What are they?
3) Transitive property
: This is somewhat harder. Let
be a homotopy (relative to A) from f to g, and let
be such a homotopy of g to h. Define
Note that the definitions agree for
. We need to show H is continuous.
Corollary 3.1.3
.
Theorem 3.1.4
(Eckmann-Hilton). Let G be a group space and
be the identity point. Then
is abelian.
Proof. To show
is abelian, we will show that for any two loops
we have
. Indeed for this, we construct a homotopy between the two loops above.
Theorem 3.1.5
(Van Kampen’s theorem). Suppose
where each
contains a green basepoint
so that each
is path connected and each
is path connected. We have homomorphisms
induced by the inclusions
and homomorphisms
induced by the inclusions
.
1) The homomorphism
is surjective.
2) If further each
is path connected, then the kernel of
is the minimal normal subgroup
generated by all the elements of the form
for
so
induces an isomorphism
.
Theorem 3.1.6
is a fundamental group with respect to “
” in the general interval
.
Proof
1) “
” is associative. We need to show that
for
.
and
We define a homotopy between
and
as follows:
Then the following is true:
Thus
and
for all
.
Also
and
for all
.
Hence
.
2) We show that the constant mapping
is such that [c] is the identity element of
with respect to “
”. Thus we must show that
for any
. Let
be defined as follows:
Then
and
if
[i.e.
]. Thus
and
for all
. More so,
and
for all
.
Hence
.
3) Finally we want to show that each homotopy class
has an inverse
such that
. Thus we want to show that if
there exist a
such that
. Let
. Since
,
by definition, we have
We then define homotopy h between
and c as follows:
Since f and g are continuous, h is continuous and we have
and
Thus
and
. Also
for all
. Hence
.
4. Conclusions
This paper, in full accordance with the principles of homotopy, has been able to establish the proof that
is a fundamental group in the general interval
,
.
In general,
depends upon
. However, in the case of an arc-wise-connected space S, we can show that
is independent of
.
Acknowledgements
Sincere thanks to Mrs. Belinda Zigli and Prof. William Obeg-Denteh for their immersed supports. The authors wish to acknowledge the support of their respective Universities and the anonymous referees for their helpful comments towards the improvement of the paper.