The real Betti realization of motivic Thom spectra and of very effective Hermitian K-theory
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912979459506176 |
|---|---|
| author | Bannwart, Julie |
| author_facet | Bannwart, Julie |
| contents | 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 $\mathsf{L}(\mathbb{R})_{\geq 0}$, the connective L-theory spectrum of $\mathbb{R}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The real Betti realization of motivic Thom spectra and of very effective Hermitian K-theory Bannwart, Julie Algebraic Topology K-Theory and Homology 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 $\mathsf{L}(\mathbb{R})_{\geq 0}$, the connective L-theory spectrum of $\mathbb{R}$. |
| title | The real Betti realization of motivic Thom spectra and of very effective Hermitian K-theory |
| topic | Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2505.07297 |