# String group

In topology, a branch of mathematics, a string group is an infinite-dimensional group String(n) introduced by Stolz (1996) 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

${\displaystyle 0\rightarrow K(Z,2)\rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow 0}$

where K(Z, 2) is an Eilenberg–MacLane space and Spin(n) is a spin group.

The string group is an entry in the Postnikov tower for the orthogonal group:

${\displaystyle \ldots \rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow {\text{SO}}(n)\rightarrow {\text{O}}(n)}$

It is preceded by the fivebrane group in the tower. It is obtained by killing the ${\displaystyle \pi _{3}}$ homotopy group for ${\displaystyle {\text{Spin}}(n)}$, in the same way that ${\displaystyle {\text{Spin}}(n)}$ is obtained from ${\displaystyle {\text{SO}}(n)}$ by killing ${\displaystyle \pi _{1}}$. The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional Lie groups have a non-vanishing ${\displaystyle \pi _{3}}$. The fivebrane group follows, by killing ${\displaystyle \pi _{7}}$.

More generally, the construction of the Postnikov tower via short exact sequences starting with Eilenberg–MacLane spaces can be applied to any Lie group G, giving the string group String(G).