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The analysis for Yang-Mills functional and in general, problems related
to higher dimensional gauge theory, often requires one to work with
notions of Sobolev principal bundles and Sobolev connections on them.
The bundle transition functions for a Sobolev principal $G$-bundle are
not continuous in the critical dimension and thus the usual notion of
topology does not make sense.
In this talk, we shall see that if 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. In stark contrast to classical notions,
this notion of `bundle topology' is \emph{not} independent of the
connection. However, for more regular bundles and connections, this
coincides with the usual notion. On the other hand, we shall see that
this notion behaves well with respect to passage to the limit of
sequences with control on $n/2$-Yang-Mills energies and is thus more
suitable to capture the change of topology in the limit due to
concentration of curvatures. |