arXiv subjects
Publications and source records attributed to Liam Keenan.
We develop homological Real trace methods, an approach to understanding the continuous mod two Bredon homology of Real topological periodic homology and Real topological negative cyclic homology, generalizing prior work of Bruner and Rognes. We use this new approach to do several computations, including the continuous mod two Bredon homology of the Real topological negative cyclic homology of Real bordism.
Classically, the Pontryagin-Thom isomorphism asserts that the multiplicative cohomology theory given by (structured) geometric cobordism is isomorphic to the cohomology theory determined by an associated Thom spectrum. We construct a presentably symmetric monoidal stable $\infty$-category of homotopy invariant sheaves with transfers on smooth manifolds whose unit is precisely (structured) geometric cobordism. We show the endomorphism ring of the unit can be canonically identified with the associated $\mathbb{E}_\infty$-Thom ring spectrum, i.e., we provide an $\mathbb{E}_\infty$-lift of the Pontryagin-Thom isomorphism.
Given a symmetric monoidal $\infty$-category $\mathscr{E}$, compatible with finite colimits, we show that the functor sending a simplicial object in $\mathscr{E}$ to its skeletal filtration is canonically lax symmetric monoidal. This monoidal structure is the analogue of the one induced by the Eilenberg-Zilber homomorphism from the Dold-Kan correspondence. To accomplish this, we establish some new results around $\mathscr{O}$-promonoidal $\infty$-categories for any $\infty$-operad $\mathscr{O}$; most notably, we show that it is possible to localize $\mathscr{O}$-promonoidal $\infty$-categories in the same way one localizes symmetric monoidal $\infty$-categories.
In this note, we establish a vanishing result for telescopically localized $\mathrm{TR}$. More precisely, we prove that $T(k)$-local $\mathrm{TR}$ vanishes on connective $L_n^{p,f}$-acyclic $\mathbf{E}_1$-rings for every $1 \leq k \leq n$ and deduce consequences for connective Morava K-theory and the Thom spectra $y(n)$. The proof relies on the relationship between $\mathrm{TR}$ and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive $\infty$-categories which was recently established by Córdova Fedeli.
In this paper, we prove that both topological Hochschild homology and topological cyclic homology are sheaves for the fpqc topology on connective commutative ring spectra, by exploiting the May filtration on topological Hochschild homology.