Fundamental group

Mathematical group of the homotopy classes of loops in a topological space

This generalizes the above observations since the figure eight is the wedge sum of two circles.

This includes the torus, being the case of genus 1, whose fundamental group is

is a homotopy equivalence and therefore yields an isomorphism of their fundamental groups.

As was mentioned above, computing the fundamental group of even relatively simple topological spaces tends to be not entirely trivial, but requires some methods of algebraic topology.

As the universal covering group suggests, there is an analogy between the fundamental group of a topological group and the center of a group; this is elaborated at Lattice of covering groups.

If E happens to be path-connected and simply connected, this sequence reduces to an isomorphism