In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops 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 fundamental groups. The fundamental group of a topological space X {\displaystyle X} is denoted by π 1 ( X ) {\displaystyle \pi _{1}(X)} .
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 defined the fundamental group in 1895 in his paper "Analysis situs". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, 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 {\displaystyle X} is a topological space. A typical example is a surface such as the one depicted at the right. Moreover, x 0 {\displaystyle x_{0}} is a point in X {\displaystyle 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 {\displaystyle 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 {\displaystyle X} , a loop based at x 0 {\displaystyle x_{0}} is defined to be a continuous function (also known as a continuous map)
γ : [ 0 , 1 ] → X {\displaystyle \gamma \colon [0,1]\to X}
such that the starting point γ ( 0 ) {\displaystyle \gamma (0)} and the end point γ ( 1 ) {\displaystyle \gamma (1)} are both equal to x 0 {\displaystyle x_{0}} .
A homotopy is a continuous interpolation between two loops. More precisely, a homotopy between two loops γ , γ ′ : [ 0 , 1 ] → X {\displaystyle \gamma ,\gamma '\colon [0,1]\to X} (based at the same point x 0 {\displaystyle x_{0}} ) is a continuous map
h : [ 0 , 1 ] × [ 0 , 1 ] → X , {\displaystyle h\colon [0,1]\times [0,1]\to X,}
such that
h ( 0 , t ) = x 0 {\displaystyle h(0,t)=x_{0}} for all t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} , that is, the starting point of the homotopy is x 0 {\displaystyle x_{0}} for all t {\displaystyle t} (which is often thought of as a time parameter).
h ( 1 , t ) = x 0 {\displaystyle h(1,t)=x_{0}} for all t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} , that is, similarly the end point stays at x 0 {\displaystyle x_{0}} for all t {\displaystyle t} .
h ( r , 0 ) = γ ( r ) {\displaystyle h(r,0)=\gamma (r)} , h ( r , 1 ) = γ ′ ( r ) {\displaystyle h(r,1)=\gamma '(r)} for all r ∈ [ 0 , 1 ] {\displaystyle r\in [0,1]} . If such a homotopy h {\displaystyle h} exists, γ {\displaystyle \gamma } and γ ′ {\displaystyle \gamma '} are said to be homotopic. The relation " γ {\displaystyle \gamma } is homotopic to γ ′ {\displaystyle \gamma '} " is an equivalence relation so that the set of equivalence classes can be considered:
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