Topology of weak $G$-bundles via Coulomb gauges in critical dimensions

The transition maps for a Sobolev $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle $P$ is equipped with a Sobolev connection $A$, then one can associate a topological isomorphism class to the pair $(P,A),$ which is invariant under Sobolev gauge changes and coincides with the usual notions for regular bundles and connections. This is based on a regularity result which says any bundle in the critical dimension in which a Sobolev connection is in Coulomb gauges are actually $C^{0,\alpha }$ for any $\alpha <1.$ We also show any such pair can be strongly approximated by smooth connections on smooth bundles. Finally, we prove that for sequences $(P^{\nu },A^{\nu })$ with uniformly bounded $n\mathrm{/}2$-Yang-Mills energy, the topology stabilizes if the $n\mathrm{/}2$ norm of the curvatures are equiintegrable. This implies a criterion to detect topological flatness in Sobolev bundles in critical dimensions via $n\mathrm{/}2$-Yang-Mills energy.