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Topological groups

The fundamental group of a topological group X (with respect to the base point being the neutral element) is always commutative. In particular, the fundamental group of a Lie group is commutative. In fact, the group structure on X endows π 1 (X) {\displaystyle \pi _{1}(X)} with another group structure: given two loops γ {\displaystyle \gamma } and γ ′ {\displaystyle \gamma '} in X, another loop γ ⋆ γ ′ {\displaystyle \gamma \star \gamma '} can defined by using the group multiplication in X:

This binary operation ⋆ {\displaystyle \star } on the set of all loops is a priori independent from the one described above. However, the Eckmann–Hilton argument shows that it does in fact agree with the above concatenation of loops, and moreover that the resulting group structure is abelian.

An inspection of the proof shows that, more generally, π 1 (X) {\displaystyle \pi _{1}(X)} is abelian for any H-space X, i.e., the multiplication need not have an inverse, nor does it have to be associative. For example, this shows that the fundamental group of a loop space of another topological space Y, X = Ω (Y), {\displaystyle X=\Omega (Y),} is abelian. Related ideas lead to Heinz Hopf 's computation of the cohomology of a Lie group.