On Poincare cobordism spectra and multiplicative signature maps

I will introduce a proposed notion of categorical symmetric prespectra as well as an unstable extension of the category of Poincaré categories which we call that of bundled categories. Using this formalism, we construct the commutative ring cobordism spectrum of Poincaré spaces equipped with Pontryagin-Thom maps, as well as TQFT-like index and signature maps to hermitian K-theory. I will also discuss further applications such as a spectral refinement of Ranicki's Poincaré surgery square involving normal L-theory via some Goodwillie calculus and the Weiss-Williams index theorem for topological manifold bundles. This reports on joint work-in-progress with Bianchi, Kirstein, and Kremer, and additionally with Hebestreit, Land, Nikolaus, and Steimle.