String group

Article summary

In topology, a branch of mathematics, a string group is an infinite-dimensional group \operatorname{String}(n) introduced by as a 3 -connected cover of a spin group. A string manifold is a manifold with a lifting of its frame bundle to a string group bundle. This means that in addition to being able to define holonomy along paths, one can also define holonomies for surfaces going between strings. There is a short exact sequence of topological groups 0\rightarrow{\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname{String}(n)\rightarrow \operatorname{Spin}(n)\rightarrow 0 where K(\mathbb{Z},2) is an Eilenberg–MacLane space and \operatorname{Spin}(n) is a spin group. The string group is an entry in the Whitehead tower (dual to the notion of Postnikov tower) for the orthogonal group: \cdots\rightarrow \operatorname{Fivebrane}(n) \to \operatorname{String}(n)\rightarrow \operatorname{Spin}(n)\rightarrow \operatorname{SO}(n) \rightarrow \operatorname{O}(n) It is obtained by killing the \pi_3 homotopy group for \operatorname{Spin}(n) , in the same way that \operatorname{Spin}(n) is obtained from \operatorname{SO}(n) by killing \pi_1 . The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional compact Lie groups have a non-vanishing \pi_3 . The fivebrane group follows, by killing \pi_7 .…

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