Fundamental group - Imported Knowlpedia Content

Definition

Let $X$ be a topological space with basepoint $x_0$ . A based loop is a path $\gamma:[0,1]\to X$ with $\gamma(0)=\gamma(1)=x_0$ . Two based loops are equivalent when they are joined by a continuous homotopy through based loops, keeping both endpoints fixed throughout. The fundamental group

\pi_1(X,x_0)

is the set of these equivalence classes, with multiplication induced by concatenating loops. The constant loop is the identity, and reversing a loop gives its inverse. These operations are well defined on homotopy classes and make $\pi_1(X,x_0)$ a group.

Basepoint dependence

If $x_0$ and $x_1$ lie in the same path component, a path from $x_0$ to $x_1$ induces an isomorphism

\pi_1(X,x_0)\cong\pi_1(X,x_1).

Different choices of path can change this isomorphism by an inner automorphism. Hence a path-connected space has a fundamental group well defined up to noncanonical isomorphism, while a specific basepoint and connecting paths matter for functorial constructions.

Functoriality

A continuous map $f:X\to Y$ satisfying $f(x_0)=y_0$ induces a group homomorphism

f_*:\pi_1(X,x_0)\to\pi_1(Y,y_0), \qquad [\gamma]\longmapsto[f\circ\gamma].

Homotopic pointed maps induce the same homomorphism. In particular, a homotopy equivalence induces an isomorphism of fundamental groups, subject to the usual basepoint choices.

Examples

Every nonempty convex subset of $\mathbb R^n$ has trivial fundamental group.
The circle satisfies $\pi_1(S^1,1)\cong\mathbb Z$ ; the integer records winding number.
The $n$ -sphere has trivial fundamental group for $n\ge2$ .
A bouquet of $r$ circles has free fundamental group on $r$ generators.

Interpretation

The fundamental group measures the obstruction to continuously contracting based loops. It also governs connected covering spaces: under standard local hypotheses, connected coverings of $X$ correspond to conjugacy classes of subgroups of $\pi_1(X,x_0)$ . Because it can be nonabelian, it retains information that first homology discards.

References

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. Author-hosted chapter record. Relevant: Chapter 1, the fundamental group and covering spaces.
Edwin H. Spanier, Algebraic Topology, Springer, 1966. DOI record. Relevant: fundamental groups and covering-space theory.