If I have a fibration, perhaps with twisting data respecting the fibration, is there a Serre spectral sequence computing cobordism of the total space?

An example that I'm particularly interested in is twisted spin cobordism for the fibration of classifying spaces of the groups $\mathbb{Z}/2 \to \mathbb{Z}/4 \to \mathbb{Z}/2$ with twisting bundle given by the sign representation on the base, its pullback on the total space, and the trivial twisting bundle on the fiber.

By twisted spin cobordism I mean I want to consider (unoriented) manifolds with a map to the space along with a spin structure on the tangent bundle direct sum the pullback of the twisting bundle.


It might not be quite of the form that you're looking for, but for a fibre sequence $F \to E \to B$, there is an Atiyah-Hirzebruch-Serre spectral sequence of the form

$$H^*(B; R^*(F)) \implies R^*E$$

for a generalized cohomology theory $R$. Here $H^*(B; R^*(F))$ is the singular cohomology of $B$ with coefficients in the local system $R^*(F)$. In the case that you mention, you get a group cohomology calculation

$$H^*(\mathbb{Z} / 2; R^*(B\mathbb{Z} / 2)) \implies R^*(B\mathbb{Z} / 4).$$

I'm not sure I totally understand what you mean by "twisting bundle" here, but if you mean that you want to modify this by some sort of local system (or Thom isomorphism) $L$ on the base (and its pullback to the total space), that works, too:

$$H^*(\mathbb{Z} / 2; R^*(B\mathbb{Z} / 2; L)) \implies R^*(B\mathbb{Z} / 4; p^* L).$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.