arXiv subjects

Cary Malkiewich

Publications and source records attributed to Cary Malkiewich.

At least 19 recordsLinked to original sources

arXiv2026

Scissors automorphism groups II: Solomon-Tits theorems

The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic subspaces of Euclidean, hyperbolic, or spherical geometry, assuming the collection is generated either by points or by hyperplanes. In the third paper of this series of papers, we will combine this with the homological stability theorems from the first paper to compute the homology of groups of scissors automorphisms in these geometries.

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arXiv2025

Higher Spherical Scissors Congruence I: Hopf Algebra

In the study of the generalization of Hilbert's Third Problem to spherical geometry, Sah constructed a Hopf algebra of spherical polytopes with product given by join and coproduct given by a generalized Dehn invariant. Using Zakharevich's reinterpretation of scissors congruence via algebraic K-theory, we lift the Sah algebra to an $(E_\infty, E_1)$-Hopf algebra spectrum whose $\pi_0$ is the classical Sah algebra. As an application, we show that the reduced spherical scissors congruence $K$-theory groups $\widetilde K_{2n}\big(\mathcal{P}^{S^{2k+1}}_{O(2k+2)}\big)$ are nonzero for all nonnegative integers $n$ and $k$.

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arXiv2025

From Hopf Algebras in Model Categories to Hopf Algebras in $\infty$-categories

We show that algebra objects in model categories can be transferred to algebra objects in $\infty$-categories, without any cofibrancy or fibrancy assumptions on the algebra. We furthermore show under some mild extra assumptions that this correspondence extends to commutative bialgebras and to commutative Hopf algebras.

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arXiv2024

Scissors automorphism groups and their homology

In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. We apply this to various families of groups, such as interval exchange groups and Brin--Thompson groups, recovering results of Szymik--Wahl, Li, and Tanner, and obtaining new results as well.

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arXiv2024

A concise proof of the stable model structure on symmetric spectra

It is well-known that the stable model structure on symmetric spectra cannot be transferred from the one on sequential spectra through the forgetful functor. We use the fibrant transfer theorem of Guetta--Moser--Sarazola--Verdugo to show it can be transferred between fibrant objects, providing a new, short and conceptual proof of its existence.

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arXiv2023

A trace map on higher scissors congruence groups

Cut-and-paste $K$-theory has recently emerged as an important variant of higher algebraic $K$-theory. However, many of the powerful tools used to study classical higher algebraic $K$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $K$-theory. In this paper we address the particular case of the $K$-theory of polyhedra, also called scissors congruence $K$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $K$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason's general framework of $K$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $K$-theory.

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arXiv2023

Coherence for indexed symmetric monoidal categories

Indexed symmetric monoidal categories are an important refinement of bicategories -- this structure underlies several familiar bicategories, including the homotopy bicategory of parametrized spectra, and its equivariant and fiberwise generalizations. In this paper, we extend existing coherence theorems to the setting of indexed symmetric monoidal categories. The most central theorem states that a large family of operations on a bicategory defined from an indexed symmetric monoidal category are all canonically isomorphic. As a part of this theorem, we introduce a rigorous graphical calculus that specifies when two such operations admit a canonical isomorphism.

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arXiv2023

The Transfer is Functorial

We prove that the Becker-Gottlieb transfer is functorial up to homotopy, for all fibrations with finitely dominated fibers. This resolves a lingering foundational question about the transfer, which was originally defined in the late 1970s in order to simplify the proof of the Adams conjecture. Our approach differs from previous attempts in that we closely emulate the geometric argument in the case of a smooth fiber bundle. This leads to a "multiplicative'" description of the transfer, different from the standard presentation as the trace of a diagonal map.

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arXiv2023

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

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.

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arXiv2023

Parametrized spectra, a low-tech approach

We give an alternative treatment of the foundations of parametrized spectra, with an eye toward applications in fixed-point theory. We cover most of the central results from the book of May and Sigurdsson, sometimes with weaker hypotheses, and give a new construction of the bicategory $\mathcal Ex$ of parametrized spectra. We also give a careful account of coherence results at the level of homotopy categories. This is an expository work, akin to a set of lecture notes, encompassing and extending the more novel material in the paper "A convenient category of parametrized spectra." Our primary goal is to give careful and explicit explanations for the standard facts surrounding the bicategory of parametrized spectra and the Reidemeister trace. In particular, our proof of Costenoble-Waner duality gives a more explicit account of how the concrete index formulas of Dold are related to the abstract theory of bicategorical duality from May and Sigurdsson, Ponto, and Shulman.

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arXiv2023

A convenient category of parametrized spectra

We describe a point-set category of parametrized orthogonal spectra, a model structure on this category, and a separate, more geometric class of cofibrant-and-fibrant objects. The structures we describe are "convenient" in that they are preserved by the most common operations. They allow us to reduce sophisticated statements about the homotopy category to straightforward claims at the point-set level. We use this framework to give a construction of the bicategory of parametrized spectra, one that is far more direct than earlier approaches. This gives a clean bridge between the concrete index theory pioneered by Dold, and the formal bicategorical theory developed by May and Sigurdsson, Ponto, and Shulman.

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arXiv2023

On the multiplicativity of the Euler characteristic

In this short paper, we give two proofs that the Euler characteristic is multiplicative, for fiber sequences of finitely dominated spaces. This is equivalent to proving that the Becker-Gottlieb transfer is functorial on $π_0$.

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arXiv2023

On the functoriality of the space of equivariant smooth $h$-cobordisms

We construct an $(\infty,1)$-functor that takes each smooth $G$-manifold with corners $M$ to the space of equivariant smooth $h$-cobordisms ${\mathcal H}_{\mathrm{Diff}}(M)$. We also give a stable analogue ${\mathcal H}^{\mathcal U}_{\mathrm{Diff}}(M)$ where the manifolds are stabilized with respect to representation discs. The functor structure is subtle to construct, and relies on several new ideas. In the non-equivariant case $G=e$, our $(\infty,1)$-functor agrees with previous constructions of the smooth $h$-cobordism space as a functor to the homotopy category.

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arXiv2022

On higher scissors congruence

We solve the higher version of Hilbert's Third Problem for one-dimensional geometries, and in higher dimensions we reduce the problem to a computation in group homology. Our central result concerns the scissors congruence $K$-theory spectrum of Zakharevich, whose homotopy groups are the correct higher version of the classical scissors congruence groups. We prove that this spectrum is a Thom spectrum, whose base space is the homotopy orbit space of a Tits complex. The relevant computations quickly follow from this more foundational result.

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arXiv2021

Periodic points and topological restriction homology

We answer in the affirmative two conjectures made by Klein and Williams. First, in a range of dimensions, the equivariant Reidemeister trace defines a complete obstruction to removing $n$-periodic points from a self-map $f$. Second, this obstruction defines a class in topological restriction homology. We prove these results using duality and trace for bicategories. This allows for immediate generalizations, including a corresponding theorem for the fiberwise Reidemeister trace.

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arXiv2021

Coherence for bicategories, lax functors, and shadows

Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This perspective makes the approach of Mac Lane's proof very amenable to generalization. We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors.

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arXiv2021

The equivariant parametrized $h$-cobordism theorem, the non-manifold part

We construct a map from the suspension $G$-spectrum $Σ_G^\infty M$ of a smooth compact $G$-manifold to the equivariant $A$-theory spectrum $A_G(M)$, and we show that its fiber is, on fixed points, a wedge of stable $h$-cobordism spectra. This map is constructed as a map of spectral Mackey functors, which is compatible with tom Dieck style splitting formulas on fixed points. In order to synthesize different definitions of the suspension $G$-spectrum as a spectral Mackey functor, we present a new perspective on spectral Mackey functors, viewing them as multifunctors on indexing categories for "rings on many objects" and modules over such. This perspective should be of independent interest.

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arXiv2020

$K$-theoretic torsion and the zeta function

We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.

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