pith. sign in

arxiv: 2606.28107 · v1 · pith:X4243WSNnew · submitted 2026-06-26 · 🧮 math.AT · math.GR· math.KT

Chromatic Euler characteristics and duality for infinite groups

Gijs Heuts , Irakli Patchkoria This is my paper

Pith reviewed 2026-06-29 01:50 UTC · model grok-4.3

classification 🧮 math.AT math.GRmath.KT
keywords chromatic Euler characteristicMorava E-theoryproper equivariant spectraduality functorFarrell-Tate cohomologyorbifold Euler characteristicmapping class groupsarithmetic groups
0
0 comments X

The pith

Generalized Euler characteristics for groups with finite proper universal spaces admit a Morava E-theory interpretation and a duality theory on equivariant spectra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a family of generalized Euler characteristics for discrete groups, indexed by natural numbers n. For n=0 this recovers the classical orbifold Euler characteristic, while for finite groups and n >=1 it matches the chromatic cardinality. The central result is that for general n this invariant can be interpreted using Morava E-theories. The authors establish a duality theory for the generalized cohomology of infinite groups that have a finite universal space for proper actions, via a new duality functor on proper G-equivariant spectra, which implies the vanishing of a generalized Farrell-Tate cohomology with T(n)-local coefficients. They also provide explicit computations for many mapping class groups and arithmetic groups.

Core claim

We study generalizations of the Euler characteristic indexed by n, showing that for general n it admits a natural interpretation in terms of Morava E-theories. For groups G with finite EG we construct a new duality functor on proper G-equivariant spectra that expresses a good theory of duality for generalized cohomology, and prove the vanishing of Klein's generalized Farrell-Tate cohomology with T(n)-local coefficients.

What carries the argument

A new duality functor on the category of proper G-equivariant spectra that encodes duality for the generalized cohomology when EG is finite.

If this is right

  • For groups with finite EG the generalized Farrell-Tate cohomology vanishes with T(n)-local coefficients.
  • The generalized Euler characteristics can be computed for many mapping class groups and arithmetic groups.
  • The chromatic cardinality for finite groups extends naturally to infinite groups via Morava E-theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the duality functor extends to more general groups without finite EG, it could apply to a broader class of infinite groups.
  • The interpretation via Morava E-theories may connect to other chromatic invariants in equivariant homotopy theory.
  • Computations suggest that these characteristics stabilize or have patterns in arithmetic groups similar to classical cases.

Load-bearing premise

The groups under consideration possess a finite universal space for proper actions.

What would settle it

Finding a group G with finite EG where the generalized Farrell-Tate cohomology with T(n)-local coefficients does not vanish would falsify the vanishing result.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces a family of generalized Euler characteristics χ_n(G) for discrete groups G (or orbifolds), indexed by n ∈ ℕ. For n=0 this recovers the classical orbifold Euler characteristic of Wall and Serre; for finite G and n≥1 it agrees with the chromatic cardinality of Ben-Moshe–Carmeli–Schlank–Yanovski. The authors give a natural interpretation of χ_n(G) for general n in terms of Morava E-theories. For groups G admitting a finite model for the proper universal space EG, they construct a new duality functor on the category of proper G-equivariant spectra that yields Poincaré duality for generalized cohomology theories; as a corollary they prove vanishing of Klein’s generalized Farrell–Tate cohomology with T(n)-local coefficients. Explicit computations are provided for many mapping class groups (extending Harer–Zagier) and arithmetic groups (extending Harder).

Significance. If the results hold, the work supplies a chromatic refinement of classical group invariants that is directly computable for important classes of infinite groups with finite EG. The duality functor on proper G-spectra and the resulting vanishing theorem are potentially useful tools in equivariant homotopy theory. The explicit link to Morava E-theories and the large collection of computed examples (mapping class groups, arithmetic groups) strengthen the contribution. The paper correctly conditions the duality and vanishing statements on the finite-EG hypothesis, which is standard and verifiable for the cited families.

minor comments (3)
  1. [Abstract / §2] The abstract states that the generalized Euler characteristic admits an interpretation via Morava E-theories for general n, but the manuscript should clarify in §2 or §3 whether this interpretation requires the finite-EG hypothesis or holds unconditionally.
  2. [Introduction] Notation for the generalized Euler characteristic (presumably χ_n(G) or similar) and for the new duality functor should be introduced consistently at first use and listed in a notation table if the paper is long.
  3. [Computations section] The computations for mapping class groups and arithmetic groups are presented as extensions of Harer–Zagier and Harder; the manuscript should include a short table comparing the classical (n=0) values with the new χ_n values for at least two representative groups.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines a family of generalized Euler characteristics indexed by n, with the n=0 case recovering the classical orbifold Euler characteristic and n≥1 recovering chromatic cardinality for finite groups. It then proves an interpretation in terms of Morava E-theories and establishes a duality functor on proper G-equivariant spectra for groups admitting a finite model for the proper universal space, leading to a vanishing result for generalized Farrell-Tate cohomology. These steps are presented as new constructions and theorems rather than reductions to fitted parameters, self-definitions, or load-bearing self-citations. Example computations (mapping class groups, arithmetic groups) are applications of the new theory, not inputs that define it. No step reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; ledger populated from stated conditions in the abstract. The central claims rest on the existence of finite models for proper actions and standard properties of Morava E-theories and equivariant spectra.

axioms (2)
  • domain assumption Groups G admit a finite universal space for proper actions ∎G
    Required for the duality functor and vanishing result; stated as the setting for the main theorems.
  • standard math Standard properties of Morava E-theories and T(n)-localization hold in the equivariant setting
    Invoked for the interpretation of the generalized Euler characteristic.

pith-pipeline@v0.9.1-grok · 5755 in / 1376 out tokens · 40960 ms · 2026-06-29T01:50:24.700707+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 8 canonical work pages

  1. [1]

    [Ash89] A

    arXiv:2505.21803, to appear in Forum of Mathe- matics, Sigma. [Ash89] A. Ash. Farrell cohomology of GL(n,Z).Israel J. Math., 67(3):327–336,

    work page arXiv

  2. [2]

    Ben-Moshe

    [BM24] S. Ben-Moshe. Higher semiadditivity in transchromatic homotopy theory.arXiv:2411.00968,

    work page arXiv

  3. [3]

    [Bro92] S. A. Broughton. Normalizers and centralizers of elementary abelian subgroups of the mapping class group. InTopology ’90 (Columbus, OH, 1990), volume 1 ofOhio State Univ. Math. Res. Inst. Publ., pages 77–89. de Gruyter, Berlin,

    1990

  4. [4]

    57 [BS73] A

    Corrected reprint of the 1982 original. 57 [BS73] A. Borel and J.-P. Serre. Corners and arithmetic groups.Comment. Math. Helv., 48:436–491,

    1982

  5. [5]

    Carmeli, B

    [CCRY25] S. Carmeli, B. Cnossen, M. Ramzi, and L. Yanovski. Characters and transfer maps via categorified traces. arXiv:2210.17364v1, to appear in Forum of Mathematics, Sigma,

    work page arXiv

  6. [6]

    Cnossen, T

    arXiv:2505.19192. [Cno23] B. Cnossen. Twisted ambidexterity in equivariant homotopy the- ory

    work page arXiv

  7. [7]

    [Coh07] H

    arXiv:2303.00736, to appear in Journal of Topology. [Coh07] H. Cohen.Number theory. Vol. II. Analytic and modern tools, volume 240 ofGraduate Texts in Mathematics. Springer, New York,

    work page arXiv

  8. [8]

    [Far78] F. T. Farrell. An extension of Tate cohomology to a class of infinite groups.J. Pure Appl. Algebra, 10(2):153–161, 1977/78. [FM12] B. Farb and D. Margalit.A primer on mapping class groups, vol- ume 49 ofPrinceton Mathematical Series. Princeton University Press, Princeton, NJ,

    1977

  9. [9]

    Master thesis, University of Copen- hagen, arXiv:2505.09150,

    work page arXiv

  10. [10]

    L¨ uck, I

    60 [LPS24] W. L¨ uck, I. Patchkoria, and S. Schwede. Character the- ory and Euler characteristic for orbispaces and infinite groups. arXiv:2410.14510,

    work page arXiv

  11. [11]

    Patchkoria

    [Pat24] I. Patchkoria. Chromatic congruences and bernoulli numbers. arXiv:2406.17705,

    work page arXiv

  12. [12]

    61 [Ser71] J.-P. Serre. Cohomologie des groupes discrets. InS´ eminaire Bour- baki, 23` eme ann´ ee (1970/1971), Exp. No. 399, pages 337–350. Lec- ture Notes in Math., Vol

    1970

  13. [13]

    [Sta40] K

    Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation. [Sta40] K. G. C. Staudt. Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend.J. Reine Angew. Math., 21:372–374,

    1980

  14. [14]

    [Xia92] Y. Xia. Thep-torsion of the Farrell-Tate cohomology of the map- ping class group Γ (p−1)/2. InTopology ’90 (Columbus, OH, 1990), volume 1 ofOhio State Univ. Math. Res. Inst. Publ., pages 391–

    1990