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The mapping class group of a manifold is the group $\pi_0 Diff(M)$ of components of the diffeomorphism group. There are several variations: oriented manifolds and orientation preserving diffeomorphism …

In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some …

I looked at Wall's paper Classification problems in differential topology. V On certain 6-manifolds. In theorem 3 of that paper Wall describes some invariants of 6-mainfolds, and the relation between …

This MO answer by Arun Debray gives an example in the unoriented case where two specific homeomorphic manifolds can be distinguished by a specific TFT of this kind. In general all these constructions …

Let us say that a (closed, connected) manifold has a symmetry if it admits a non-trivial action by a finite group. Note that I am not asking the action to be free. So for example rotating the 2-sphere …

There is a stupid answer which is equivalence classes of G-bundles with connection on M are the same as homotopy classes of maps $M \to BG$. That is as long as two G-bundles with connection are consid …