Tensors of Thom spectra with spaces: generalizing Mahowald's Thom isomorphism

The topological Hochschild homology (THH) of a commutative ring spectrum R can be expressed as a tensor with a circle. More generally, one can tensor R with any space X. In this talk we will introduce this construction, then look at the case where R is a Thom spectrum Mf of an E_infty map G\to Pic(R) of E_infty groups. We will give a formula for the tensor of Mf with X where X is based. The cornerstone is a careful analysis of the interaction between different tensors, e.g. over based or unbased spaces. We will explain how this result generalizes Mahowald's version of the Thom isomorphism theorem. Joint with Nima Rasekh and Gabriel Valenzuela.