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In a lecture I attended today, I heard the use of gerbes in the cobordism theory.

Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group generators.

My understanding was that I can simply use characteristic classes, fiber bundles, topological invariants (Arf, ABK, $\eta$, etc.) etc to write down the cobordism invariants.

Why do we need gerbes in the cobordism theory? What are some good recommended lecture notes on this? Is this concept really helpful here? (Thanks so much for answering my naive inquiry.)

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    $\begingroup$ Codimension 3 submanifolds naturally give rise to gerbes (see for example arxiv.org/abs/math/9907034 example 1.3). Conversely, one can take holonomy of a gerbe on a surface (arxiv.org/abs/math/0312175) and then cobordisms between surfaces related the holonomies. $\endgroup$
    – David Roberts ♦
    Commented Nov 1, 2018 at 0:47
  • $\begingroup$ @David Roberts, +1, thanks very much for the Refs $\endgroup$
    – wonderich
    Commented Nov 1, 2018 at 3:48
  • $\begingroup$ Are you still studying gerbes and/or cobordism? $\endgroup$
    – Praphulla Koushik
    Commented Mar 3, 2019 at 0:31
  • $\begingroup$ @Praphulla Koushik, why you ask? whats the matter? thanks! $\endgroup$
    – wonderich
    Commented Mar 4, 2019 at 1:18
  • $\begingroup$ I am also working on gerbes/stacks and looking for some one who is working (definitely a PhD student, not a well established researcher :) :) ) with whom I can discuss something... That is why I asked. :) $\endgroup$
    – Praphulla Koushik
    Commented Mar 4, 2019 at 2:01

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