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In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U \simeq N \amalg \partial V$.

In the case that the category is $\{\text{(unoriented) compact smooth manifolds (with boundary)}\}$, then two closed objects $M,N$ are bordant if and only if there is some $W$ for which $M \amalg N \simeq \partial W$, the more familiar definition of bordism.

I'm confused, though, about how this extends to bordism of manifolds with structure. The simplest case is orientation, but more generally we have some fibration $f \colon B \to BO$ and an "$f$-structure" on $M$ will be a choice of lift of the stable normal bundle $\nu_M \colon M \to BO$ to $B$ (up to suitable homotopy). There are induced $f$-structures on boundaries.

In this situation, for $M$ closed we have $\partial(M \times I) \simeq M \amalg -M$ where $-M$ has the same underlying manifold but possibly distinct $f$-structure (e.g. opposite orientation).

I want to show now that when $M,N$ are bordant, $M \amalg -N \simeq \partial W$, but I'm doing something wrong. Can anyone help in clearing this up?

Thank you.

Backgroud: In the unoriented case I know how to show this. The slightly harder direction is to show the first implies the second, which goes like: If $M \amalg \partial U \simeq N \amalg \partial V$ then define $W_1 := M \times I \amalg U$ and $W_2 := N \times I \amalg V$. Then $\partial W_1 \simeq M \amalg M \amalg \partial U$ and $\partial W_2 \simeq N \amalg N \amalg \partial V$. Then we can glue $W_1$ and $W_2$ along $M \amalg \partial U \simeq N \amalg \partial V$ to form $W$ with $\partial W \simeq M \amalg N$.

In the argument above, for the case of orientation/structure, I now have $\partial W_1 \simeq M \amalg -M \amalg \partial U$ and similarly for $W_2$. But it seems like I can still glue them along $M \amalg \partial U \simeq N \amalg \partial V$ (the $f$-structure on the pushout should be one on each factor agreeing on the intersection, and the isomorphism $M \amalg U \simeq N \amalg \partial V$ should say precisely that the $f$-structures will agree on this intersection). And then I'll end up with $-M \amalg -N \simeq \partial W$.

In the example of orientation I think this has something to do with the fact that for some reason it seems I can glue two things with the same orientation when I should be gluing them with opposite orientations (incidentally I also don't fully understand why that is). Or maybe there's something I'm missing about the induced orientation on the boundary.

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    $\begingroup$ Related: math.stackexchange.com/q/410917 $\endgroup$ – Grigory M Dec 17 '14 at 19:46
  • $\begingroup$ It is not true in general that $M\coprod \partial U \simeq N\coprod \partial V$ in the oriented case. You need a minus on either $M$ or $N$. $\endgroup$ – Espen Nielsen Jan 4 '15 at 23:45

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