Mona Merling

Research Interests

My research is in the area of algebraic topology, more specifically in homotopy theory. I have been particularly interested in developing and studying equivariant algebraic K-theory and A-theory, namely starting with a ring, a space or appropriate category with G-action, encoding the naive action as a genuine G-spectrum and then studying this object. This had led me to work on different projects in equivariant stable homotopy theory needed for laying the foundations of equivariant algebraic K-theory.

My research has been supported in part by NSF grant DMS-1709461/1850644, NSF CAREER grant DMS-1943925 and NSF FRG grant DMS-2052988.

Publications and preprints

Algebraic K-theory for squares categories (with J. Campbell, J. Kuijper, and I. Zakharevich), arXiv:2310.02852, 28 pages. See abstract.

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 $A \rightarrow B \rightarrow C$ or for a subtractive sequence $A\rightarrow B \leftarrow C$, by defining $K_0$ of exact and subtractive categories 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.

On the functoriality of the space of equivariant smooth $h$-cobordisms (with T. Goodwillie, K. Igusa and C. Malkiewich), submitted, arXiv:2303.14892, 65 pages. See abstract.

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

A trace map on higher scissors congruence groups (with A.M. Bohmann, T. Gerhardt, C. Malkiewich, and I. Zakharevich), submitted, arXiv:2303.08172, 32 pages. See abstract.

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. .

Deformation retraction of the group of strict contactomorphisms of the three-sphere to the unitary group (with D. DeTurck, H. Gluck, L. Lichtenfelz, J. Yang and Y. Wang), submitted, arXiv:2108.08961, 43 pages. See abstract.

We prove that the group of strict contactomorphisms of the standard tight contact structure on the three-sphere deformation retracts to its unitary subgroup $U(2)$. .

Multiplicative equivariant $K$-theory and the Barratt-Priddy-Quillen theorem , (with B. Guillou, J.P. May and A. Osorno), Advances in Mathematics, Volume 414 (2023), arXiv:2001.05563, 91 pages. See abstract.

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 $G$-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.

Equivariant infinite loop space theory, the space level story, (with J.P. May and A. Osorno), to appear in Memoirs of the AMS, arXiv:1704.03413, 121 pages. See abstract.

We rework the May and Segal equivariant infinite loop space machines, and show that given equivalent input, they yield equivalent genuine G-spectra. The proof of the nonequivariant uniqueness theorem, due to May and Thomason, fails equivariantly; our proof is a direct comparison of the two machines.

The equivariant parametrized $h$-cobordism theorem, the non-manifold part, (with C. Malkiewich), Advances in Mathematics, Volume 399 (2022), arXiv:2001.05563, 41 pages. See abstract.

We construct a map from the suspension $G$-spectrum $\Sigma^\infty_G M_+$ 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 $h$-cobodism 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.

Cut and paste invariants of manifolds via algebraic K-theory, (with with R. Hoekzema, L. Murray, C. Rovi, J. Semikina), Topology and its Applications, Volume 316 (2022), arXiv:2001.00176, 22 pages. See abstract.

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 spectrum that recovers the classical $SK$ ("schneiden und kleben," German for "cut and paste") groups for manifolds on $\pi_0$, and we construct a derived version of the Euler characteristic.

Coassembly is a homotopy limit map, (with C. Malkiewich), Annals of K-theory, Volume 5 Issue 3 (2020), 373-394 arXiv:1904.05858. See abstract.

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.

Symmetric monoidal G-categories and their strictification, (with B. Guillou, J.P. May and A. Osorno), Quarterly Journal of Mathematics, Volume 71 Issue 1 (2020), 207–246 arXiv:1809.03017. See abstract.

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.

Equivariant A-theory, (with C. Malkiewich), Documenta Mathematica, Volume 24 (2019), 815-855 arXiv:1609.03429. See abstract.

We give a general construction that produces a genuine $G$-spectrum from a Waldhausen category with $G$-action, for a finite group $G$. For the category $R(X)$ of retractive spaces over a $G$-space $X$, this produces an equivariant lift of Waldhausen's functor $\mathbf A(X)$, whose $H$-fixed points agree with the bivariant $A$-theory of the fibration $X_{hH} \to BH$. We then use the framework of spectral Mackey functors to produce a second equivariant refinement $\mathbf A_G(X)$ whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized stable $h$-cobordism theorem.

A symmetric monoidal and equivariant Segal machine, (with B. Guillou, J.P. May and A. Osorno), Journal of Pure and Applied Algebra, Volume 226 (6) (2018), 2425-2454, arXiv:1711.09183. See abstract.

In [MMO], 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 $\sF$ of finite sets rather than from the category $\sF_G$ of finite $G$-sets and which is equivalent to the machine studied by Shimakawa and in [MMO]. In contrast to the machine studied by Shimakawa and in [MMO], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of $\sF$-$G$-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 $\sF$-$G$-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.

Motivic homotopical Galois extensions, (with A. Beaudry, K. Hess, M. Kedziorek, and V. Stojanoska), Topology and its Applications, Volume 235 (2018), 290-338. arXiv:1611.00382. See abstract.

We develop a formal framework in which to study a homotopical version of Galois theory, generalizing Rognes's Galois theory of commutative ring spectra. We apply this to the categories of motivic spaces and motivic spectra and we compute a series of first examples of motivic Galois extensions.

Categorical models for equivariant classifying spaces, (with B. Guillou and J.P. May), Algebraic and Geometric Topology, 17-5 (2017), 2565-2602, arXiv:1201.5178. See abstract.

We give simple categorical models of universal principal equivariant bundles and their classifying spaces. The motivation for having these models is twofold: they give E-infinity operads in GCat, and they provide an equivariant generalization of the plus construction definition of the algebraic K-theory of a ring.

Equivariant algebraic K-theory of G-rings, Mathematische Zeitschrift, 285(3) (2017), 1205-1248. arXiv:1505.07562. See abstract.

A group action on the input ring or category induces an action on the algebraic K-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic K-theory is, for example, that the map of spectra with G-action induced by a G-map of G-rings is not equivariant. We define a version of equivariant algebraic K-theory which encodes a group action on the input in a functorial way to produce a genuine algebraic K-theory G-spectrum for a finite group G. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological K-theory, Atiyah's Real K-theory, and existing statements about algebraic K-theory spectra with G-action. We recover the map from the Quillen-Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory. We also give a definition of an equivariant version of Waldhausen's A-theory of a G-space.

Unbased calculus for functors to chain complexes, (with M. Basterra, K. Bauer, A. Beaudry, R. Eldred, B. Johnson and S. Yeakel), Contemporary Mathematics, Vol. 641 (2015), arXiv:1409.1553v2 See abstract.

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.

Function Fields With Class Number Indivisible by a Prime l, (with M. Daub, J. Lang, A. Pacelli, N. Pitiwan and M. Rosen), Acta Arithmetica, 150 (2011), 339-359, arXiv:0906.3728. See abstract.

It is known that infinitely many number fields and function fields of any degree m have class number divisible by a given integer n. However, significantly less is known about the indivisibility of class numbers of such fields. While it is known that there exist infinitely many quadratic number fields with class number indivisible by a given prime, the fields are not constructed explicitly, and nothing appears to be known for higher degree extensions. Pacelli and Rosen explicitly constructed an infinite class of function fields of any degree m, where 3 does not divide m, over Fq(T) with class number indivisible by 3, generalizing a result of Ichimura for quadratic extensions. We generalize that result, constructing, for an arbitrary prime l, and positive integer m greater than 1, infinitely many function fields of degree m over the rational function field, with class number indivisible by l.

Gassmann Equivalent Dessins, (with R. Perlis), Communications in Algebra, Vol. 38, Issue 6 (2010), 2129-2137. See abstract.

We study pairs of Grothendieck dessins d'enfants that arise from a Gassmann triple of groups (G,H,H') together with a pair of elements in G. We show that the two resulting dessins have isomorphic monodromy groups, have the same branching data and the same number of components. Moreover, the sums of the genera of the components of the two dessins are the same. However, we give an example where the individual genera of the components of the first dessin differ from the genera of the components of the second dessin.

Oberwolfach reports

G-manifolds and algebraic K-theory, Oberwolfach reports, Report No. 31 (2018), 37-40.

Moduli spaces of equivariant $h$-cobordisms, Oberwolfach reports, Report No. 34 (2023), 11-14.

Volumes edited

New directions in homotopy theory, (co-edited with N. Kitchloo, J. Morava, E. Riehl, S. Wilson), Contemporary Mathematics, Volume 707, (2018).

Other publications

I was part of the first iteration of the User's guides project, a project meant to make research mathematics more accessible by having authors provide "user's guides" to their published papers.

The User's Guide Project: Giving Experiential Context to Research Papers, (with C. Malkiewich, D. White, L. Wolcott, C. Yarnall), Journal for Humanistic Mathematics, Vol.5, Issue 2 (2015). See abstract.

This article is an announcement and decription of our User's Guide Project.

A user's guide: Categorical models for equivariant classifying spaces, Enchiridion, Vol. 1 (2015).