Some fivebrane bordism groups at the prime 2

Hassan Abdallah, Wayne State University

Given a smooth n-dimensional manifold M, a tangential G-structure on M is a lift of the classifying map of the tangent bundle of M to BG, the classifying space of G. If G=SO(n) or G=Spin(n), this gives the familiar notion of an orientation or spin structure. A higher analog of orientations and spin structures is that of a fivebrane structure (G=Fivebrane(n) cong O(n)), named based on its relationship to ideas from physics.

While the bordism groups of orientable and spin manifolds are well-understood, very little is known about the bordism groups of fivebrane manifolds. The fivebrane bordism groups are isomorphic, via the Pontryagin-Thom construction, to the homotopy groups of the Thom spectrum MO, which has connections to deep ideas in stable homotopy theory.

In this talk, I will present new calculations of some fivebrane bordism groups at the prime 2 using the Adams spectral sequence.