String group

In <a href="/facts/Topology/2EqW47cU">topology</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a string group is an infinite-dimensional group 
 
 
 
 String
 ⁡
 (
 n
 )
 
 
 {\displaystyle \operatorname {String} (n)}
 
 introduced by Stolz (1996) as a 
 
 
 
 3
 
 
 {\displaystyle 3}
 
-connected cover of a <a href="/facts/Spin_group/tfvR9cqM">spin group</a>. A string manifold is a <a href="/facts/Manifold/rXPERJtu">manifold</a> with a lifting of its <a href="/facts/Frame_bundle/Gugt3qyL">frame bundle</a> to a string group bundle. This means that in addition to being able to define <a href="/facts/Holonomy/GnxeTqI4">holonomy</a> along paths, one can also define holonomies for surfaces going between strings. There is a short <a href="/facts/Exact_sequence/leUJnwJb">exact sequence</a> of <a href="/facts/Topological_group/cAgNMMRU">topological groups</a><blockquote>
 
 
 
 0
 →
 
 
 K
 (
 
 Z
 
 ,
 2
 )
 
 
 →
 String
 ⁡
 (
 n
 )
 →
 Spin
 ⁡
 (
 n
 )
 →
 0
 
 
 {\displaystyle 0\rightarrow {\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow 0}
 
</blockquote>where 
 
 
 
 K
 (
 
 Z
 
 ,
 2
 )
 
 
 {\displaystyle K(\mathbb {Z} ,2)}
 
 is an <a href="/facts/Eilenberg%E2%80%93MacLane_space/mrux6oMJ">Eilenberg–MacLane space</a> and 
 
 
 
 Spin
 ⁡
 (
 n
 )
 
 
 {\displaystyle \operatorname {Spin} (n)}
 
 is a spin group. The string group is an entry in the <a href="/facts/Whitehead_tower/wdHYnz7F">Whitehead tower</a> (dual to the notion of <a href="/facts/Postnikov_tower/wdHYnz7F">Postnikov tower</a>) for the <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a>:<blockquote>
 
 
 
 ⋯
 →
 Fivebrane
 ⁡
 (
 n
 )
 →
 String
 ⁡
 (
 n
 )
 →
 Spin
 ⁡
 (
 n
 )
 →
 SO
 ⁡
 (
 n
 )
 →
 O
 ⁡
 (
 n
 )
 
 
 {\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}
 
</blockquote>It is obtained by killing the 
 
 
 
 
 π
 
 3
 
 
 
 
 {\displaystyle \pi _{3}}
 
 <a href="/facts/Homotopy_group/xjogL95Y">homotopy group</a> for 
 
 
 
 Spin
 ⁡
 (
 n
 )
 
 
 {\displaystyle \operatorname {Spin} (n)}
 
, in the same way that 
 
 
 
 Spin
 ⁡
 (
 n
 )
 
 
 {\displaystyle \operatorname {Spin} (n)}
 
 is obtained from 
 
 
 
 SO
 ⁡
 (
 n
 )
 
 
 {\displaystyle \operatorname {SO} (n)}
 
 by killing 
 
 
 
 
 π
 
 1
 
 
 
 
 {\displaystyle \pi _{1}}
 
. The resulting manifold cannot be any finite-dimensional <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>, since all finite-dimensional compact Lie groups have a non-vanishing 
 
 
 
 
 π
 
 3
 
 
 
 
 {\displaystyle \pi _{3}}
 
. The fivebrane group follows, by killing 
 
 
 
 
 π
 
 7
 
 
 
 
 {\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 <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> G, giving the string group String(G).

String group open-in-new

String group