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The computation of the unoriented bordism group of the point $N_*=\Omega_*^O$ is a classic result.

I would like to know a related bordism group, where we specify the twisted fundamental class $[M]\in H_d(M,\mathbb{Z}^w)$ as part of the data. More precisely, I would like to consider the pairs $$ (M, [M]\in H_d(M,\mathbb{Z}^w)) $$ where $\mathbb{Z}^w$ is the coefficient system twisted by $w_1(M)$; I then call two $(M,[M])$ and $(M',[M'])$ bordant when there is $(N,[N]')$ such that $\partial N=M \sqcup M'$ where the twisted orientation of $N$ induces that of $M$ and $M'$, as in the case of oriented bordism.

Most probably this is well known to the experts...

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  • $\begingroup$ It may well be, but I'm not certain if any of those experts are regulars here. $\endgroup$ – Ryan Budney May 7 at 5:19
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    $\begingroup$ Could you suggest the names of those experts? I might pester them by emailing them. $\endgroup$ – Yuji Tachikawa May 7 at 6:23
  • $\begingroup$ Could you address Oscar's concerns? I wasn't certain of what kind of bordism you want to consider. It looks like you are perhaps desiring a kind of bordism theory for the orientation covers of manifolds. $\endgroup$ – Ryan Budney May 7 at 22:11
  • $\begingroup$ @RyanBudney It turned out that what I needed was really the fundamental class as explained by Oscar, and not a twisted version of orientation. So, as Oscar kindly explained, this was a non-issue. Thank you for your time. $\endgroup$ – Yuji Tachikawa May 8 at 5:32
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I think this theory is the same as unoriented bordism, when one tries to make sense of it.

As it stands I don't think it makes sense, because I think that the expression "$\mathbb{Z}^w$" does not describe a local system on $M$, but rather an isomorphism class of local systems, and hence $H_d(M ; \mathbb{Z}^w)$ is an isomorphism class of abelian group, and hence it is not meaningful to talk about an element of it.

A manifold $M$ has an "orientation" local system(=locally constant sheaf). This is the locally constant sheaf with values $\mathcal{O}_M(U) = H_d(M, M \setminus U; \mathbb{Z})$ on all balls $U \cong \mathbb{R}^d$ in $M$. I am sure that $\mathcal{O}_M$ is isomorphic to whatever might be meant by $\mathbb{Z}^w$, but $\mathcal{O}_M$ is an actual local system.

Any $d$-dimensional closed manifold $M$ has a unique fundamental class $[M] \in H_d(M ; \mathcal{O}_M)$, where $\mathcal{O}_M$ is the "orientation" local system of $M$, and fundamental class means that it restricts to the canonical generator of $$H_d(\mathbb{R}^d, \mathbb{R}^d\setminus \{0\} ; \mathcal{O}_{\mathbb{R}^d}) \cong_{UCT} H_d(\mathbb{R}^d, \mathbb{R}^d\setminus \{0\} ; \mathbb{Z}) \otimes H_d(\mathbb{R}^d, \mathbb{R}^d\setminus \{0\} ; \mathbb{Z})$$ for every ball $\mathbb{R}^d \cong U \subset M$. Note that this rank 1 $\mathbb{Z}$-module indeed has a preferred generator as it is the tensor square of a rank 1 $\mathbb{Z}$-module. That such a fundamental class is unique if it exists, and that it exists, is by the usual proof of existence of fundamental classes.

Made precise in this way the question is vacuous, because $[M]$ is seen to not be a choice of data.

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  • $\begingroup$ I always thought $\mathbb Z^w$ was defined to be the orientation local system that you called $\mathcal O_M$, not just the isomorphism class of that system. Perhaps I've been sloppy with notation. $\endgroup$ – Arun Debray May 7 at 19:37
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    $\begingroup$ @ArunDebray: Well, that is what it should be. But I expect that some might mean something like fixing a basepoint $m \in M$ and considering local systems as $\mathbb{Z}[\pi_1(M,m)]$-modules, where this one is given by the first Stiefel--Whitney class as a $\pi_1(M,m) \to \mathbb{Z}^\times$. $\endgroup$ – Oscar Randal-Williams May 7 at 20:21
  • $\begingroup$ Thank you for the clarification. I guess I was confused even in the orientable and oriented case: the orientation is not the choice of the fundamental class (which is a canonical generator of $H_d(M,\mathcal{O}_M)$) but the choice of a generator of $H_d(M,\mathbb{Z})$ (which does not have a canonical choice.) Am I right? $\endgroup$ – Yuji Tachikawa May 8 at 2:43
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    $\begingroup$ From the point of view taken here, I would say that an orientation is a choice of trivialisation of the local system $\mathcal{O}_M$. This is equivalent to a class in $H_d(M; Z)$ which restricts to a generator of the local homology of each point, as this is an invertible global section of $\mathcal{O}_M$. $\endgroup$ – Oscar Randal-Williams May 8 at 7:09

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