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Mona Merling

Publications and source records attributed to Mona Merling.

At least 19 recordsLinked to original sources
arXiv2026

Topological symmetric and braid homologies

We identify topological symmetric homology as the free $\mathbb{E}_\infty$-algebra on an $\mathbb{E}_1$-algebra and topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra. In this way, topological symmetric homology and topological braid homology can be regarded as variants of $1$-dimensional representation homology. In order to identify topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra, we prove that the $\mathbb{E}_2$-monoidal envelope of the associative operad can be identified with the braided crossed simplicial group. Using this, we also compute the topological braid homology of grouplike $\mathbb{E}_1$-spaces. Further, we develop computational tools for topological symmetric and braid homologies. These tools allow us to perform low-degree computations of topological symmetric homology and prove that it is not Morita invariant. We also compute the topological $\Delta \mathbf{G}$-homology of Thom spectra in general and produce explicit formulas in the case of topological symmetric and braid homologies.

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arXiv2025

Homotopy Type of the Space of Fibrations of the Three-sphere by Simple Closed Curves

We show that the moduli space of all smooth fibrations of a three-sphere by simple closed curves has the homotopy type of a disjoint union of a pair of two-spheres if the fibers are oriented, and of a pair of real projective planes if unoriented, the same as for its finite-dimensional subspace of Hopf fibrations by parallel great circles. This moduli space is the quotient of the diffeomorphism group of the three-sphere (a Fr\'echet Lie group) by its subgroup of automorphisms of the Hopf fibration, which we show is a smooth Fr\'echet submanifold of the diffeomorphism group. Then we show that the moduli space, already known to be a Fr\'echet manifold by [HKMR12], can be modeled on the concrete Fr\'echet space of vector fields on the three-sphere which are "horizontal" and "balanced" with respect to a given Hopf fibration, and see how the structure of this moduli space helps us to determine its homotopy type.

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arXiv2025

Parametrized scissors congruence $K$-theory of manifolds and cobordism categories

We introduce a parametrized version of scissors congruence $K$-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence $K$-theory of oriented manifolds as a special case. We examine the relation of this $K$-theory spectrum with cut-and-paste invariants, the (parametrized) cobordism category and with (bivariant) algebraic $K$-theory of spaces. We show that the scissors congruence $K$-theory of oriented manifolds agrees on $\pi_0$ with a version of the oriented cobordism category where we allow cobordisms to have free boundaries. Lastly, we show that the spectrum level refinement of the Euler characteristic from the scissors congruence $K$-theory to $K(\mathbb{Z})$ detects on $\pi_1$ the Kervaire semicharacteristic.

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arXiv2025

Scissors congruence K-theory for equivariant manifolds

We introduce a scissors congruence $K$-theory spectrum which lifts the equivariant scissors congruence groups for compact $G$-manifolds with boundary, and we show that on $\pi_0$ this is the source of a spectrum level lift of the Burnside ring valued equivariant Euler characteristic of a compact $G$-manifold. We also show that the equivariant scissors congruence groups for varying subgroups assemble into a Mackey functor, which is a shadow of a conjectural higher genuine equivariant structure.

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arXiv2024

Topological $\Delta G$ homology of rings with twisted $G$-action

We construct topological $\Delta G$-homology for rings with twisted $G$-action. Here a ring with twisted $G$-action is a common generalization of a ring with anti-involution and a ring with $G$-action. This construction recovers as special cases topological Hochschild homology (THH) of rings, with its $S^1$-action, and Real topological Hochschild homology (THR) of rings with anti-involution, with its $O(2)$-action. A new example of this construction is quaternionic topological Hochschild homology (THQ) of rings with twisted $C_4$-action, which carries a $Pin(2)$-action. We prove that THQ of a loop space with twisted $C_4$-action can be $Pin(2)$-equivariantly identified with a twisted free loop space. Other new examples of interest are topological symmetric homology and topological hyperoctrahedral homology and more generally topological twisted symmetric homology. We prove a homotopical version of results of Fiedorowicz, Ault, and Graves computing these new topological homology theories on loop spaces with twisted $G$-action. A key step of independent interest in this program is the construction of a new family of crossed simplicial groups, which correspond to operads that encode the structure of rings with twisted $G$-action.

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arXiv2023

Algebraic $K$-theory for squares categories

In this paper we introduce a new formalism for $K$-theory, called squares $K$-theory. This formalism allows us to simultaneously generalize the usual three-term relation $[B] = [A] + [C]$ for an exact sequence $A \hookrightarrow B \twoheadrightarrow C$ or for a subtractive sequence $A\hookrightarrow B \leftarrow C$, by defining $K_0$ of a squares category to satisfy a four-term relation $[A]+[D]= [C] + [B]$ for a ``good'' square diagram with these corners. Examples that rely on this formalism are $K$-theory of smooth manifolds of a fixed dimension and $K$-theory of (smooth and) complete varieties. Another application we give of this theory is the construction of a derived motivic measure taking value in the $K$-theory of homotopy sheaves.

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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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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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arXiv2021

Multiplicative equivariant $K$-theory and the Barratt-Priddy-Quillen theorem

We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in arXiv:1207.3459. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal $G$-spectra from symmetric monoidal $G$-categories. The new machine produces highly structured associative ring and module $G$-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal $G$-categories to the multicategory of orthogonal $G$-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of $G$-spectra in arXiv:1110.3571.

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arXiv2020

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

We construct a map from the suspension $G$-spectrum $\Sigma_G^\infty M$ 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

Cut and paste invariants of manifolds via algebraic K-theory

Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic $K$-theory, and applying these tools to studying the Grothendieck ring of varieties. In this paper we give a new application of their framework: we construct a $K$-space that recovers the classical $\mathrm{SK}$ ("schneiden und kleben," German for "cut and paste") groups for manifolds on $\pi_0$, and we construct a derived version of the Euler characteristic.

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arXiv2019

Coassembly is a homotopy limit map

We prove a claim by Williams that the coassembly map is a homotopy limit map. As an application, we show that the homotopy limit map for the coarse version of equivariant $A$-theory agrees with the coassembly map for bivariant $A$-theory that appears in the statement of the topological Riemann-Roch theorem.

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arXiv2018

Symmetric monoidal G-categories and their strictification

We give an operadic definition of a genuine symmetric monoidal G-category, and we prove that its classifying space is a genuine E_\infty G-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power, and Lack, to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal G-categories to genuine permutative G-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When G is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal G-categories as input to an equivariant infinite loop space machine that gives genuine G-spectra as output.

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arXiv2018

The user's guide project: giving experiential context to research papers

This paper was written in 2015, and published in the Journal of Humanistic Mathematics. This paper announces the first issue (2015) of Enchiridion: Mathematics User's Guides, a project to produce peer-reviewed User's Guides as companions to published papers. These User's Guides are meant to explain the key insights and organizing principles in their companion papers, the metaphors and imagery used by the authors, the story of the development of the companion papers, and a colloquial summary appropriate for a non-mathematical audience. Examples of User's Guides can be found at https://mathusersguides.com/

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arXiv2017

A symmetric monoidal and equivariant Segal infinite loop space machine

In [MMO] (arXiv:1704.03413), we reworked and generalized equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. In this paper, we construct a new variant of the equivariant Segal machine that starts from the category $\scr{F}$ of finite sets rather than from the category ${\scr{F}}_G$ of finite $G$-sets and which is equivalent to the machine studied by Shimakawa and in [MMO]. In contrast to the machine in [MMO], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of $\scr{F}$-$G$-spaces to the symmetric monoidal category of orthogonal $G$-spectra. We relate it multiplicatively to suspension $G$-spectra and to Eilenberg-MacLane $G$-spectra via lax symmetric monoidal functors from based $G$-spaces and from abelian groups to $\scr{F}$-$G$-spaces. Even non-equivariantly, this gives an appealing new variant of the Segal machine. This new variant makes the equivariant generalization of the theory essentially formal, hence is likely to be applicable in other contexts.

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arXiv2017

Equivariant infinite loop space theory, the space level story

We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $\Omega$-$G$-spectra for any topological group $G$, but our focus is on the construction of genuine $\Omega$-$G$-spectra when $G$ is finite. We also show what is and is not true when $G$ is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$ but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature.

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arXiv2016

Motivic Homotopical Galois Extensions

We establish a formal framework for Rognes's homotopical Galois theory and adapt it to the context of motivic spaces and spectra. We discuss examples of Galois extensions between Eilenberg-MacLane motivic spectra and between the Hermitian and algebraic K-theory spectra.

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