Motivic strict ring spectra representing semi-topological cohomology theories

TL;DR

This paper extends detection functors to motivic symmetric spectra and constructs strict ring spectra representing various semi-topological cohomology theories, linking semi-topological cobordism to complex cobordism.

Contribution

It generalizes Shipley's detection functor to motivic spectra and constructs new motivic strict ring spectra for semi-topological cohomology theories.

Findings

Semi-topological cobordism relates to semi-topological K-theory via a Conner-Floyd isomorphism.

Inverting a specific element makes semi-topological cobordism isomorphic to periodic complex cobordism.

Construction of motivic strict ring spectra for morphic cohomology, semi-topological K-theory, and cobordism.

Abstract

We show that Shipley's "detection functor" for symmetric spectra generalizes to motivic symmetric spectra. As an application, we construct motivic strict ring spectra representing morphic cohomology, semi-topological $K$-theory, and semi-topological cobordism for complex varieties. As a further application to semi-topological cobordism, we show that it is related to semi-topological $K$-theory via a Conner-Floyd type isomorphism and that after inverting a lift of the Friedlander-Mazur $s$-element in morphic cohomology, semi-topological cobordism becomes isomorphic to periodic complex cobordism.