In the mathematical
field of algebraic topology
, the fundamental group of a topological space
is the group
of the equivalence class
es under homotopy
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
) have isomorphic
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 together 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.
defined the fundamental group in 1895 in his paper "Analysis situs
". The concept emerged in the theory of Riemann surface
s, in the work of Bernhard Riemann
, Poincaré, and Felix Klein
. It describes the monodromy
properties of complex-valued function
s, as well as providing a complete topological classification of closed surfaces
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
'' is defined to be a continuous function
(also known as a continuous map
such that the starting point
and the end point
are both equal to
'' is a continuous interpolation between two loops. More precisely, a homotopy between two loops
(based at the same point
) is a continuous map
that is, the starting point of the homotopy is
for all ''t'' (which is often thought of as a time parameter).
that is, similarly the end point stays at
for all ''t''.