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Given a smooth manifold $M$ of dimension $n$ and a diffeomorphism $\phi: M \to M$, we can construct a smooth cobordism of dimension $(n+1)$ from $M$ to $M$ by gluing $M \times [0,1]$ with itself by $\phi$. However, smooth cobordism of dimension $(n+1)$ is almost the same as surgery theory of dimension $n$. Therefore, I expect a correspondence among

  • Surgery theory of dimension $n$ (in particular, its higher Kirby diagrams)
  • (Smooth) mapping class group theory for $n$-dimensional manifolds

Has any theory in this line of thought been developed (any reference)?

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    $\begingroup$ Your cobordism is always diffeomorphic to the trivial cobordism by sending $(x,t)$ to $(x,t/2)$ if $t \leq 1$ and to $(\phi^{-1}(x),t/2)$ if $t \geq 1$. You are not wrong to say there is a connection though between surgery theory and mapping class groups, though I think it is much easier to study the components of $\operatorname{Diff(M)}$ with it than the actual mapping class group. Quinn formulated this approach in his paper on surgery spaces. $\endgroup$
    – Connor Malin
    Commented Oct 4, 2021 at 17:40
  • $\begingroup$ Which paper of Quinn did you mean? $\endgroup$
    – Student
    Commented Oct 4, 2021 at 20:56
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    $\begingroup$ Sorry, I meant to include it: "A geometric formulation of surgery". It is difficult to read in my opinion, so I might recommend reading the intro to section 2 of arxiv.org/pdf/2002.04647.pdf . The point being that Quinn's structure space $S(M)$ can be written as a quotient of homotopy automorphisms modulo block diffeomorphisms, both of which are easier to study than $\operatorname{Diff}(M)$ with homotopy theory. $\endgroup$
    – Connor Malin
    Commented Oct 4, 2021 at 21:47
  • $\begingroup$ I don't know how related this is to what you had in mind, but Kreck established something rougly called "bordisms of diffeomorphisms" and these are closely related to mapping tori (i.e. taking $M\times [0,1]$ and gluing the ends together via your diffeomorphism) $\endgroup$
    – ThorbenK
    Commented Oct 20, 2021 at 12:11

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