The real Betti realization of motivic Thom spectra and of very effective Hermitian K-theory

Abstract

Real Betti realization is a symmetric monoidal functor from the category of motivic spectra to that of topological spectra, extending the functor that associates to a scheme over $\mathbb{R}$ the space of its real points. In this article, we prove some results about the real Betti realizations of certain motivic ${\mathcal{E}}_1$- and ${\mathcal{E}}_{\infty }$-rings. We show that the motivic Thom spectrum functor and the topological one correspond to each other, as symmetric monoidal functors, under real (and complex) realization. In particular, we obtain equivalences of ${\mathcal{E}}_{\infty }$-rings between the real realizations of the variants $\mathsf{MGL}$, $\mathsf{MSL}$, and $\mathsf{MSp}$ of algebraic cobordism, and the variants $\mathsf{MO}$, $\mathsf{MSO}$, and $\mathsf{MU}$ of topological cobordism, respectively. Using this, we identify the ${\mathcal{E}}_1$-ring structure on the real realization of $\mathsf{ko}$, the very effective cover of Hermitian K-theory, by an explicit 2-local fracture square, as being equivalent to $\mathsf{L}(\mathbb{R}{)}_{\ge 0}$, the connective L-theory spectrum of $\mathbb{R}$.

57 pages. In version 3, the case of the complex realization of motivic Thom spectra has been added