arXiv subjects

Irakli Patchkoria

Publications and source records attributed to Irakli Patchkoria.

At least 19 recordsLinked to original sources

arXiv2026

Chromatic Euler characteristics and duality for infinite groups

We study a family of generalizations of the notion of Euler characteristic of discrete groups (or of orbifolds, depending on one's perspective) indexed on the natural numbers. For $n=0$, this is the classical orbifold Euler characteristic as studied by Wall and Serre, whereas for $n \geq 1$ and finite groups, this is the chromatic cardinality as studied by Ben-Moshe--Carmeli--Schlank--Yanovski. For general $n$, we show that our generalized Euler characteristic admits a natural interpretation in terms of the Morava $E$-theories. Our work involves showing that the generalized cohomology of infinite groups $G$ with finite universal space for proper actions $\underline{E}G$ has a good theory of duality, as expressed by a new duality functor on the category of proper $G$-equivariant spectra. In particular, for such groups we prove the vanishing of Klein's generalized Farrell--Tate cohomology with $T(n)$-local coefficients. We compute our generalized orbifold Euler characteristics in a large number of examples. This includes many mapping class groups, where the classical calculation is a result of Harer--Zagier, and many arithmetic groups, whose classical orbifold Euler characteristics were computed by Harder.

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arXiv2025

On the Farrell--Tate $K$-theory of $\text{Out}(F_n)$

Using L\"uck's Chern character isomorphism we obtain a general formula in terms of centralisers for the $p$-adic Farrell--Tate $K$-theory of any discrete group $G$ with a finite classifying space for proper actions. We apply this formula to $\text{Out}(F_n)$. The case $n=p+1$ turns out to be especially interesting for the following reason: Up to conjugacy there is exactly one order $p$ element in $\text{Out}(F_{p+1})$ which does not lift to an order $p$ element in $\text{Aut}(F_{p+1})$. We compute the rational cohomology of the centraliser of this element and as a consequence obtain a full calculation of the $p$-adic Farrell--Tate $K$-theory of $\text{Out}(F_{p+1})$ for any prime $p \geq 5$. Our arguments provide an infinite family of $\mathbb{Q}_p$ summands in $K^1(B \text{Out}(F_n)) \otimes_\mathbb{Z} \mathbb{Q}$, with no need for computer calculations: the first such summand is in $K^1(B \text{Out}(F_{12})) \otimes_\mathbb{Z} \mathbb{Q}$.

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arXiv2024

Character theory and Euler characteristic for orbispaces and infinite groups

Given a discrete group $G$ with a finite model for $\underline{E}G$, we study $K(n)^*(BG)$ and $E^*(BG)$, where $K(n)$ is the $n$-th Morava $K$-theory for a given prime and $E$ is the height $n$ Morava $E$-theory. In particular we generalize the character theory of Hopkins, Kuhn and Ravenel who studied these objects for finite groups. We give a formula for a localization of $E^*(BG)$ and the $K(n)$-theoretic Euler characteristic of $BG$ in terms of centralizers. In certain cases these calculations lead to a full computation of $E^*(BG)$, for example when $G$ is a right angled Coxeter group, and for $G=SL_3(\mathbb{Z})$. We apply our results to the mapping class group $\Gamma_\frac{p-1}{2}$ for an odd prime $p$ and to certain arithmetic groups, including the symplectic group $Sp_{p-1}(\mathbb{Z})$ for an odd prime $p$ and $SL_2(\mathcal{O}_K)$ for a totally real field $K$.

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arXiv2024

Chromatic congruences and Bernoulli numbers

For every natural number $n$ and a fixed prime $p$, we prove a new congruence for the orbifold Euler characteristic of a group. The $p$-adic limit of these congruences as $n$ tends to infinity recovers the Brown-Quillen congruence. We apply these results to mapping class groups and using the Harer-Zagier formula we obtain a family of congruences for Bernoulli numbers. We show that these congruences in particular recover classical congruences for Bernoulli numbers due to Kummer, Voronoi, Carlitz, and Cohen.

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arXiv2023

Witt vectors with coefficients and TR

We give a new construction of $p$-typical Witt vectors with coefficients in terms of ghost maps and show that this construction is isomorphic to the one defined in terms of formal power series from the authors' previous paper. We show that our construction recovers Kaledin's polynomial Witt vectors in the case of vector spaces over a perfect field of characteristic $p$. We then identify the components of the $p$-typical TR with coefficients, originally defined by Lindenstrauss and McCarthy and later reworked by the second and third authors in joint work with McCandless, with the $p$-typical Witt vectors with coefficients. This extends a celebrated result of Hesselholt and Hesselholt-Madsen relating the components of TR with the Witt vectors. As an application, we given an algebraic description of the components of the Hill-Hopkins-Ravenel norm for cyclic $p$-groups in terms of $p$-typical Witt vectors with coefficients.

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arXiv2023

On the cyclic homology of certain universal differential graded algebras

Let $p$ be an odd prime and $R$ a $p$-torsion-free commutative $\mathbb{Z}_{(p)}$-algebra. We compute the periodic cyclic homology over $R$ of the universal differential graded algebra $R//p$ which is obtained from $R$ by universally killing $p$. We furthermore compute the cyclic and negative cyclic homologies of $R//p$ over $R$ in infinitely many degrees.

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arXiv2023

Comparison of equivariant cohomological dimensions

We compare three definitions of the equivariant cohomological dimension of a group with operators, coming from Takasu, Adamson and Bredon relative group cohomologies, giving examples of strict inequality in all cases where it can occur. We prove and make use of Stallings--Swan type results which characterise the groups of equivariant cohomological dimension one. Some of our examples are relevant to Farber's problem which asks for an algebraic characterisation of the topological complexity of discrete groups. In particular, the topological complexity of a group is not in general given by a relative cohomological dimension of the product relative to the diagonal subgroup.

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arXiv2021

Adams spectral sequences and Franke's algebraicity conjecture

To any well-behaved homology theory we associate a derived $\infty$-category which encodes its Adams spectral sequence. As applications, we prove a conjecture of Franke on algebraicity of certain homotopy categories and establish homotopy-coherent monoidality of the Adams filtration.

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arXiv2021

On the geometric fixed-points of real topological cyclic homology

We give a formula for the geometric fixed-points spectrum of the real topological cyclic homology of a bounded below ring spectrum, as an equaliser of two maps between tensor products of modules over the norm. We then use this formula to carry out computations in the fundamental examples of spherical group-rings, perfect $\mathbb{F}_p$-algebras, and $2$-torsion free rings with perfect modulo $2$ reduction. Our calculations agree with the normal L-theory spectrum in the cases where the latter is known, as conjectured by Nikolaus.

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arXiv2020

The spectrum of derived Mackey functors

We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum of derived Mackey functors can be understood as the space obtained from the spectrum of the Burnside ring by "ungluing" closed points. In order to compute the spectrum, we provide a new description of Kaledin's category, as the derived category of an equivariant ring spectrum, which may be of independent interest. In fact, we clarify the relationship between several different categories, establishing symmetric monoidal equivalences and comparisons between the constructions of Kaledin, the spectral Mackey functors of Barwick, the ordinary derived category of Mackey functors, and categories of modules over certain equivariant ring spectra. We also illustrate an interesting feature of the ordinary derived category of Mackey functors that distinguishes it from other equivariant categories relating to the behavior of its geometric fixed points.

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arXiv2020

On the de Rham-Witt complex over perfectoid rings

Fix an odd prime $p$. The results in this paper are modeled after work of Hesselholt and Hesselholt-Madsen on the $p$-typical absolute de Rham-Witt complex in mixed characteristic. We have two primary results. The first is an exact sequence which describes the kernel of the restriction map on the de Rham-Witt complex over $A$, where $A$ is the ring of integers in an algebraic extension of $\mathbb{Q}_p$, or where $A$ is a $p$-torsion-free perfectoid ring. The second result is a description of the $p$-power torsion (and related objects) in the de Rham-Witt complex over $A$, where $A$ is a $p$-torsion-free perfectoid ring containing a compatible system of $p$-power roots of unity. Both of these results are analogous to results of Hesselholt and Madsen. Our main contribution is the extension of their results to certain perfectoid rings. We also provide algebraic proofs of these results, whereas the proofs of Hesselholt and Madsen used techniques from topology.

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arXiv2020

Witt vectors with coefficients and characteristic polynomials over non-commutative rings

For a not-necessarily commutative ring R we define an abelian group W(R;M) of Witt vectors with coefficients in an R-bimodule M. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that W(R) := W(R;R) is Morita invariant in R. For an R-linear endomorphism f of a finitely generated projective R-module we define a characteristic element $\chi_f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto \chi_f$ induces an isomorphism between a suitable completion of cyclic K-theory and W(R).

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arXiv2019

Equivariant dimensions of groups with operators

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\sf cd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and the equivariant Lusternik-Schnirelmann category ${\sf cat}_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $\pi\rtimes G$ consisting of sub-conjugates of $G$. When $G$ is finite, we extend theorems of Eilenberg-Ganea and Stallings-Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$-group $\pi$ with ${\sf cat}_G(\pi)={\sf cd}_G(\pi)=2$ and ${\sf gd}_G(\pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings-Swan type result for families of subgroups which do not contain all finite subgroups.

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arXiv2019

Proper equivariant stable homotopy theory

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective `genuine' indicates that the theory comes with appropriate transfers and Wirthm\"uller isomorphisms, and the resulting equivariant cohomology theories support the analog of an $RO(G)$-grading. Our model for genuine proper $G$-equivariant stable homotopy theory is the category of orthogonal $G$-spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of $G$. This class of $\pi_*$-isomorphisms is part of a symmetric monoidal stable model structure and the associated tensor triangulated homotopy category is compactly generated. Every orthogonal $G$-spectrum represents an equivariant cohomology theory on the category of $G$-spaces, depending only on the `proper $G$-homotopy type', tested by fixed points under all compact subgroups. An important special case are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties, in the sense of geometric group theory; for example, the $G$-sphere spectrum is a compact object in the equivariant homotopy category if the universal space for proper $G$-actions has a finite $G$-CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper $G$-CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by $G$-vector bundles. Via this description, we can identify the previously defined $G$-cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal $G$-spectra.

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arXiv2019

Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most $p-1$. This extra functoriality allows us to extend the $p$-typical Witt vectors functor from commutative rings to $\mathbb{Z}/2$-Tambara functors, for odd primes $p$. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of the real topological Hochschild homology spectrum at odd primes.

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arXiv2017

Real topological Hochschild homology

This paper interprets Hesselholt and Madsen's real topological Hochschild homology functor THR in terms of the multiplicative norm construction. We show that THR satisfies cofinality and Morita invariance, and that it is suitably multiplicative. We then calculate its geometric fixed points and its Mackey functor of components, and show a decomposition result for group-algebras. Using these structural results we determine the homotopy type of THR($\mathbb{F}_p$) and show that its bigraded homotopy groups are polynomial on one generator over the bigraded homotopy groups of $H\mathbb{F}_p$. We then calculate the homotopy type of THR($\mathbb{Z}$) away from the prime $2$, and the homotopy ring of the geometric fixed-points spectrum $\Phi^{\mathbb{Z}/2}$THR($\mathbb{Z}$).

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arXiv2017

Comparing cyclotomic structures on different models for topological Hochschild homology

The topological Hochschild homology $THH(A)$ of an orthogonal ring spectrum $A$ can be defined by evaluating the cyclic bar construction on $A$ or by applying B\"okstedt's original definition of $THH$ to $A$. In this paper, we construct a chain of stable equivalences of cyclotomic spectra comparing these two models for $THH(A)$. This implies that the two versions of topological cyclic homology resulting from these variants of $THH(A)$ are equivalent.

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arXiv2017

Rigidity and exotic models for $v_1$-local $G$-equivariant stable homotopy theory

We prove that the $v_1$-local $G$-equivariant stable homotopy category for $G$ a finite group has a unique $G$-equivariant model at $p=2$. This means that at the prime $2$ the homotopy theory of $G$-spectra up to fixed point equivalences on $K$-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for $K$-local spectra of the second author with the equivariant rigidity result for $G$-spectra of the first author. Further, when the prime $p$ is at least $5$ and does not divide the order of $G$, we provide an algebraic exotic model as well as a $G$-equivariant exotic model for the $v_1$-local $G$-equivariant stable homotopy category, showing that for primes $p \ge 5$ equivariant rigidity fails in general.

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