We know that the Spin group is quite a useful concept in physics. For example, Spin$(3)=SU(2)$ (and Spin$(6)=SU(4)$) that describe gauge groups in the Standard model and the isospin symmetry in the quarks or in the mesons or even approximately in the baryon (proton/neutron). Spin$(3)$ is also a double cover of $SO(n)$.
question: Now, what is the physical use/application of Postnikov tower [constructing a topological space from its homotopy groups]? And what is the physical application the topological groups arise in Postnikov tower, say the Fivebrane($n$) and String($n$)?
There is a short exact sequence of topological groups $$ 0\rightarrow K(Z,2)\rightarrow \text{String}( n)\rightarrow \text{Spin}( n)\rightarrow 0 $$
The spin group appears in a Postnikov tower anchored by the orthogonal group: $$ { \ldots \rightarrow {\text{Fivebrane}}(n)\rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow {\text{SO}}(n)\rightarrow {\text{O}}(n)} $$ The string group is an entry in the Postnikov tower for the orthogonal group, preceded by the fivebrane group in the tower.
${\text{Spin}(n)}$ is obtained from ${\text{SO}(n)}$ by killing $\pi _{1}$
String($n$) is obtained by killing the $\pi _{3}$ homotopy group for ${\text{Spin}(n)}$.
Thus,
- Fivebrane($n$) is obtained by killing the $\pi _{5}$ homotopy group for ${\text{String}(n)}$, correct? The tower is constructed by killing higher homotopy group $\pi _{2k+1}$?