Reversing a Spin^c structure

Question

I was wondering if anyone might be able to shed some light on the notion of a "reverse" Spin$^c$ structure, as described on pages 11-12 of https://arxiv.org/pdf/1604.03502.pdf. This is used to define additive inverses in the Baum-Douglas geometric K-homology group $K_0$.

The definition of a Spin$^c$ structure for an $n$-dimensional manifold $M$) given in the paper (page 6) is a certain kind of map from a principal Spin$^c(n)$ bundle to the frame bundle $GL(TM)$.

The problem is that the authors then define the reverse of this Spin$^c$-structure to be a bundle $P^-$ over $M$ with structure group Spin$^c(n)$ and fibre the double cover of $O^-(n)$ (denoting matrices of determinant $-1$), together with a map from $P^-$ to another bundle $F_{SO}^-(TM)$, whose fibre is $O^-$. Does such a map somehow define a Spin$^c$ structure on $M$ in the sense above?

Could someone elaborate a little on how this works?

Answer 1

Another good reference for this kind of thing is this paper: https://arxiv.org/pdf/math/0701484.pdf

In my experience, which is admittedly fairly limited, it seems as though the important thing that one gets from a $spin^c$-structure is the spinor bundle. This is what implements the Thom Isomorphism in K-theory. In the paper I linked they briefly mention how a spinor bundle implements a reduction of the structure group in remark 4.5. Just above that they show how a spinor bundle for $V$ yields an orientation for $V$, and from that it is somewhat more clear that the opposite $spin^c$-structure is the same spinor bundle but with the $\mathbb{Z}_2$-grading reversed. The reason for this also comes up in the notion of bordism.