|Algebraic structure → Group theory|
In its most general form a loop group is a group of continuous mappings from a manifold M to a topological group G.
equipped with the compact-open topology. An element of LG is called a loop in G. Pointwise multiplication of such loops gives LG the structure of a topological group. Parametrize S1 with θ,
and define multiplication in LG by
Associativity follows from associativity in G. The inverse is given by
and the identity by
The space LG is called the free loop group on G. A loop group is any subgroup of the free loop group LG.
An important example of a loop group is the group
of based loops on G. It is defined to be the kernel of the evaluation map
and hence is a closed normal subgroup of LG. (Here, e1 is the map that sends a loop to its value at .) Note that we may embed G into LG as the subgroup of constant loops. Consequently, we arrive at a split exact sequence
The space LG splits as a semi-direct product,
We may also think of ΩG as the loop space on G. From this point of view, ΩG is an H-space with respect to concatenation of loops. On the face of it, this seems to provide ΩG with two very different product maps. However, it can be shown that concatenation and pointwise multiplication are homotopic. Thus, in terms of the homotopy theory of ΩG, these maps are interchangeable.