Index theorems and orientability

Question

Given a Dirac operator $D$ acting on some Clifford bundle $\mathcal{E}$ over a compact, even-dimensional, oriented manifold $M$, the Atiyah-Singer index theorem states that its index is given by pairing some characteristic class (which are elements of $H^n(M, \mathbb{R})$) with the fundamental class of the manifold, i.e. $$ \mathrm{ind}(D) = \langle \hat{A}(M) \wedge \mathrm{ch}_{\mathcal{E}/S}(\mathcal{E}), [M] \rangle $$ in the case of a Dirac operator. In this case, we could use Chern-Weil theory, such that the characteristic classes are (equivalence classes) of differential forms, which can be integrated over the manifold, as it is oriented.

However, the theorem is true in the non-oriented case as well. In fact, the characteristic class above can be interpreted as a volume density, which can be integrated over $M$.

Of course, I am aware that one can just go to the oriented double cover and use that fact that the index is multiplicative with respect to Riemannian coverings, so the "non-oriented Atiyah-Singer" follows easily from the regular one. However, I find this somewhat unsatisfactory.

Isn't there some (co-)homology theory in which the terms on the right side of the equation above can be expressed? I know that one often uses $\mathbb{Z}_2$-valued (co-)homology to deal with non-oriented manifolds; however, the index can be any integer, not just zero or one, so that this does not seem useful here.

Answer 1

The trick, rather, is to write $\text{Ind}(D)$ in terms of the (co-)homology of (the total space of) $T^\ast M$, which is always orientable, viz, $$ \text{Ind}(D) = \int_{T^\ast M} \text{ch}[\sigma_m(D)] \smile \text{Td}(T^\ast M \otimes \mathbb{C}) $$ where $\text{ch}[\sigma_m(D)] \in H^{\text{even}}(T^\ast M)$ is the Chern character of the symbol class $\sigma_M(D) \in K(T^\ast M)$ of $D$ and $\text{Td}(T^\ast M \otimes \mathbb{C})$ is the Todd genus of $T^\ast M \otimes \mathbb{C}$. For an explanation, see

Intuitive explanation for the Atiyah-Singer index theorem

and especially Paul Siegel's excellent answer. In particular, written this way, the Atiyah--Singer index theorem can be viewed as the translation into (co-)homological terms of a purely $K$-theoretic statement.