Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2306.03817

Smash powers and traces on parametrized spectra: An application of rigidity

Cary Malkiewich·Kate Ponto
View original source ↗Read source document ↗Export metadata ↓

Abstract

The bicategory of parametrized spectra has a remarkably rich structure. We can take traces in this bicategory, giving classical invariants that count fixed points. We can also take $C_n$-equivariant external smash powers and equivariant traces, which give significant generalizations of the classical invariants that count periodic points. Unfortunately, the existence of these smash powers and traces for parametrized spectra depends on technical statements about the bicategory that can be difficult to verify directly, especially if one wants the construction to have a direct geometric interpretation. In this paper, we demonstrate the effectiveness of two tools -- rigidity and deformable functors -- by using them to establish formal structures on this bicategory directly from point-set level data.

Explore related subjects

math.ATmath.CT

Keep this discovery

BibTeXRIS

Cary Malkiewich, Kate Ponto. 2023-06-06. Smash powers and traces on parametrized spectra: An application of rigidity. https://arxiv.org/abs/2306.03817

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Homogeneous Milnor fibers and Kato--Matsumoto bounds via simplicial multiwedges

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato--Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

math.AT↗
arXiv2026

The homotopy types of directed path and trace spaces

We construct a saturated directed space with a Hausdorff $\Delta$-generated underlying space and two distinct points such that the trace space between them is homeomorphic to a square, whereas the directed path space has a nontrivial fundamental group. In particular, the canonical quotient map is not a weak homotopy equivalence. The same conclusion holds for regular directed paths modulo increasing homeomorphisms.

math.AT↗
arXiv2026

Moduli spaces of geometric functorial field theories

We develop tools to compute moduli spaces of geometric functorial field theories as mapping spaces of equivariant simplicial presheaves. Given a d-dimensional geometric structure F, presented as a presheaf on the site of smooth families of d-manifolds, we define its Cartesian realization, which is an O(d)-equivariant simplicial presheaf on the site of Cartesian spaces. We use Cartesian realizations to present the moduli space of functorial field theories with geometric structure F as a mapping space between O(d)-equivariant simplicial presheaves. In a companion paper, we use this result to compute the moduli space of smooth one-dimensional oriented Riemannian functorial field theories valued in an arbitrary smooth symmetric monoidal infinity-category.

math.AT↗