arXiv subjects

Sarah Yeakel

Publications and source records attributed to Sarah Yeakel.

7 recordsLinked to original sources

arXiv2025

An isovariant Blakers--Massey theorem

An isovariant map is an equivariant map between $G$-spaces which strictly preserves isotropy groups. In this paper, we lay the groundwork for the study of isovariant stable homotopy theory when $G$ is a finite group. We prove an isovariant Blakers--Massey theorem and its $n$-cubical generalization, define a suitable notion of suspension (by a trivial representation sphere) in the isovariant category, and prove an isovariant Freudenthal suspension theorem.

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arXiv2023

Isovariant homotopy theory and fixed point invariants

An isovariant map is an equivariant map between $G$-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein--Williams equivariant intersection theory for a finite group $G$. We prove that under certain reasonable dimension and codimension conditions on $H$-fixed subspaces (for $H\leq G$), the fixed points of a self-map of a compact smooth $G$-manifold can be removed isovariantly if and only if the equivariant Reidemeister trace of the map vanishes. In doing so, we study isovariant maps between manifolds up to isovariant homotopy, yielding an isovariant Whitehead's theorem. In addition, we speculate on the role of isovariant homotopy theory in distinguishing manifolds up to homeomorphism.

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arXiv2022

An isovariant Elmendorf's theorem

An isovariant map between spaces with a group action is an equivariant map which preserves isotropy groups. In this paper, we show that for a finite group $G$, the category of $G$-spaces with isovariant maps has a Quillen model structure. We prove a Piacenza-style model theoretic proof of an isovariant Elmendorf's theorem, showing that this model structure is Quillen equivalent to a model category of diagrams.

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arXiv2018

A Monoidal Model for Multilinearization

Using the category of finite sets and injections, we construct a new model for the multilinearization of multifunctors between spaces that appears in the derivatives of Goodwillie calculus. We show that this model yields a lax monoidal functor from the category of symmetric functor sequences to the category of symmetric sequences of spaces after evaluating at the unit.

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arXiv2017

Inverting operations in operads

We construct a localization for operads with respect to one-ary operations based on the Dwyer-Kan hammock localization. For an operad O and a sub-monoid of one-ary operations W we associate an operad LO and a canonical map O to LO which takes elements in W to homotopy invertible operations. Furthermore, we give a functor from the category of O-algebras to the category of LO-algebras satisfying an appropriate universal property.

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arXiv2017

Infinite loop spaces from operads with homological stability

Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces. The recent, strong homological stability results of Galatius and Randal-Williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability. As a consequence the map to $K$-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.

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arXiv2014

Unbased calculus for functors to chain complexes

Recently, the Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from an unbased simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a discrete calculus tower for functors from an unbased simplicial model category to chain complexes over a fixed commutative ring. Much of the construction of the Taylor tower for functors to spectra carries over to this context. However, one of the essential steps in the construction requires proving that a particular functor is part of a cotriple. For this, one needs to prove that certain identities involving homotopy limits hold up to isomorphism, rather than just up to weak equivalence. As the target category of chain complexes is not a simplicial model category, the arguments for functors to spectra need to be adjusted for chain complexes. In this paper, we take advantage of the fact that we can construct an explicit model for iterated fibers, and prove that the functor is a cotriple directly. We use related ideas to provide concrete infinite deloopings of the first terms in the resulting Taylor towers when evaluated at the initial object in the source category.

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