The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513

Woodward proposed a classification of $\mathrm{PU}(n)={\mathrm{U}(n)}/{Z(\mathrm{U}(n))}=\mathrm{SU}(n)/\mathbb{Z}_n$ principle bundles over a 4-complex $X$ in terms of two characteristic classes $q \in H^4(X,\mathbb{Z})$ and $t \in H^2(X,\mathbb{Z}_n)$. These classes are related as $\rho_{2n}q=(n+1)\mathfrak{P}(t)$ for $n$ even (a similar expression holds for $n$ odd). Here $\rho_{r}$ is the $\mod r$-reduction of a integral class and $\mathfrak{P}:H^2(X,\mathbb{Z}_{2s})\rightarrow H^2(X,\mathbb{Z}_{4s})$ is the Pontraygin square. The class $t$ can be interpreted as the obstruction to define an $\mathrm{SU}(n)$ bundle and $q$ as an instanton/Chern number.

I have two questions:

  1. For $n=4$, we have $\mathrm{PSU}(4)=\mathrm{PSO}(6)$. What are the conditions on $t$ and $q$ to be able to define an $\mathrm{SO}(6)$ bundle? If such a bundle can be defined, what are the relations between $q$, $t$ of the $\mathrm{PSU}(n)$ bundle and the characteristic classes of the $\mathrm{SO}(6)$ bundle (i.e., the Stiefel-Whitney classes $w_{2,4}$ and the Pontryagin class $p_1$). I believe there should be relation as $2p_1=q$.

  2. Let $P(n,mn)=\mathrm{SU}(mn)/\mathbb{Z}_n$, where $\mathbb{Z}_n$ is a normal subgroup of the center of $\mathrm{SU}(mn)$. Is there a similar result for $P(n,mn)$ bundles? (i.e., classification, relation between the characteristic classes, specially with the instanton number). My intuition tells me that there should still be classes $\tilde{q}\in H^4(X,\mathbb{Z})$ and $\tilde{t}\in H^2(X,\mathbb{Z}_n)$ together with some new classes in some $H^4(X,\mathbb{Z}_{l})$ because of the $n=4$, $m=2$ case $\mathrm{SU}(4)/\mathbb{Z}_2=\mathrm{Spin}(6)/\mathbb{Z}_2=\mathrm{SO}(6)$.

I know there is some work on the classifying space of $\mathrm{P}(n,mn)$ by Xing Gu, e.g. On Topological Brauer Classes over 8-Complexes with Periods Divisible by 4 (arXiv:1803.05100), and The Topological Period-Index Problem over 8-Complexes (arXiv:1709.00787), but I don't know how to proceed from there.


Please be warned that this answer might lack absolute rigour. Here it goes.

Fix a finite 4-dim CW complex $X$.

First we study the homotopy classes of maps from $X$ to $K(\mathbb{Z}/n,2)$, the $3$rd stage of the Postnikov tower of $\mathbf{B}P(n,mn)$, the classifying space of $P(n,mn)$, which involves the k-invariant $\kappa_3$. The class $\tilde{t}$ is to be chosen such that its composite with $\kappa_3$ vanishes.

Then there are (homotopy classes of) maps from X to $\mathbf{B}P(n,mn)[5]$ (the $5$th stage of the Postnikov tower) lifting $\tilde{t}$. Roughly speaking, to choose one of them amounts to a choice of $\tilde{q}$.


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