In the
mathematical field of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the fundamental group of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is the
group of the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es under
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
of the
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
s contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest
homotopy group. The fundamental group is a
homotopy invariant—topological spaces that are
homotopy equivalent (or the stronger case of
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
) have
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
fundamental groups. The fundamental group of a topological space
is denoted by
.
Intuition
Start with a space (for example, a
surface), and some point in it, and all the loops both starting and ending at this point—
paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second.
Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space.
History
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
defined the fundamental group in 1895 in his paper "
Analysis situs". The concept emerged in the theory of
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s, in the work of
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
, Poincaré, and
Felix Klein. It describes the
monodromy properties of
complex-valued functions, as well as providing a complete topological
classification of closed surfaces.
Definition
Throughout this article, ''X'' is a topological space. A typical example is a surface such as the one depicted at the right. Moreover,
is a point in ''X'' called the ''base-point''. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on ''X'' can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.
Homotopy of loops
Given a topological space ''X'', a ''
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
based at
'' is defined to be a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
(also known as a continuous map)
:
such that the starting point
and the end point
are both equal to
.
A ''
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
'' is a continuous interpolation between two loops. More precisely, a homotopy between two loops
(based at the same point
) is a continuous map
:
such that
*
for all
that is, the starting point of the homotopy is
for all ''t'' (which is often thought of as a time parameter).
*
for all
that is, similarly the end point stays at
for all ''t''.
*
for all