  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 $\left( P, A\right),$ 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 $n/2$-Yang-Mills energy, the
topology stabilizes if the $n/2$ norm of the curvatures are equiintegrable.
This implies a criterion to detect topological flatness in Sobolev bundles in
critical dimensions via $n/2$-Yang-Mills energy.
