Rational Cohomology Research Articles

On the Chow and cohomology rings of moduli spaces of stable curves

In this paper, we ask: For which (g, n) is the rational Chow or cohomology ring of \bar{\mathcal{M}}_{g,n} generated by tautological classes? This question has been fully answered in genus 0 by Keel (the Chow and cohomology rings are tautological for all n (1992)) and genus 1 by Belorousski (the rings are tautological if and only if n \leq 10 (1998)). For g \geq 2 , work of van Zelm (2018) shows the Chow and cohomology rings are not tautological once 2g + n \geq 24 , leaving finitely many open cases. Here, we prove that the Chow and cohomology rings of \bar{\mathcal{M}}_{g,n} are isomorphic and generated by tautological classes for g = 2 and n \leq 9 and for 3 \leq g \leq 7 and 2g + n \leq 14 . For such (g, n) , this implies that the tautological ring is Gorenstein and \bar{\mathcal{M}}_{g,n} has polynomial point count.

On the Picard Group of the Moduli Space of Curves via $r$-Spin Structures

In this paper, we obtain explicit expressions for Pandharipande-Pixton-Zvonkine relations in the second rational cohomology of $\overline{\mathcal{M}}_{g,n}$ and comparing the result with Arbarello-Cornalba's theorem we prove Pixton's conjecture in this case.

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

AbstractWe compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the rational homology of the unit tangent bundles of the surfaces. These coefficients define a covariant functor over the classical category associated to mapping class groups, rather than a contravariant one, and are thus out of the scope of the traditional framework to study twisted cohomological stability. A remarkable property is that the computed stable twisted cohomology is not free as a module over the stable cohomology algebra with constant rational coefficients. For comparison, we also compute the stable cohomology group with coefficients in the first rational cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

On unstable twisted rational cohomology groups of the automorphism groups of free groups

In this paper, we consider low-dimensional twisted rational cohomology groups of the automorphism groups of free groups. In particular, we study behaviors of Kawazumi’s cocycles by observing their restrictions to the IA-automorphism groups. We show the nontrivialities of certain unstable twisted cocycles constructed from Kawazumi’s cocycles, and compute the second and the third cohomology groups of the automorphism group of the free group of rank three with coefficients in the exterior powers of the abelianization of the free group.

Finitistic spaces with the orbit space 𝔽ℙn × 𝕊m

Let [Formula: see text], [Formula: see text] or [Formula: see text], act freely on a finitistic connected space [Formula: see text]. This paper gives the cohomology classification of [Formula: see text] if the mod 2 or rational cohomology of the orbit space [Formula: see text] is isomorphic to the product of a projective space and sphere [Formula: see text], where [Formula: see text] or [Formula: see text], respectively. For a free involution on [Formula: see text], a lower bound of covering dimension of the coincidence set of a continuous map [Formula: see text] is also determined.

On the codimension-two cohomology of [formula omitted]

Borel–Serre proved that SLn(Z) is a virtual duality group of dimension (n2) and the Steinberg module Stn(Q) is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to SLn(Q). We determine the “relations among the relations” of this Steinberg module. That is, we construct an explicit partial resolution of length two of the SLn(Z)-module Stn(Q). We use this partial resolution to show the codimension-2 rational cohomology group H(n2)−2(SLn(Z);Q) of SLn(Z) vanishes for n≥3. This resolves a case of a conjecture of Church–Farb–Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of SLn(Z).

On the top-weight rational cohomology of 𝒜g

On the top-weight rational cohomology of 𝒜g

The second variation of the Hodge norm and higher Prym representations

AbstractLet denote a rational cohomology class, and let denote its Hodge norm. We recover the result that is a plurisubharmonic function on the Teichmüller space , and characterize complex directions along which the complex Hessian of vanishes. Moreover, we find examples of such that is not strictly plurisubharmonic. As part of this construction, we find an unbranched covering such that the subgroup of generated by lifts of simple curves from is strictly contained in . Finally, combining the characterization theorem with the Riemann–Roch, and the Li–Yau [Invent. Math. 69 (1982), no. 2, 269–291] gonality estimate, we show that geometrically uniform covers of satisfy the Putman–Wieland Conjecture about the induced Higher Prym representations.

Note on the \(F_{0}\)-spaces

A rationally elliptic space \(X\) is called an \(F_{0}\)-space if its rational cohomology is concentrated in even degrees. The aim of this paper is to characterize such a space in terms of the homotopy groups of its skeletons as well as the rational cohomology of its Postnikov sections.

Equivariant formality of the isotropy action on (ℤ2 ⊕ ℤ2)-symmetric spaces

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalization, it is even known that their isotropy action is equivariantly formal. In this paper, we show that [Formula: see text]-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.

A new approach to twisted homological stability with applications to congruence subgroups

AbstractWe introduce a new method for proving twisted homological stability, and use it to prove such results for symmetric groups and general linear groups. In addition to sometimes slightly improving the stable range given by the traditional method (due to Dwyer), it is easier to adapt to nonstandard situations. As an illustration of this, we generalize to of many rings a theorem of Borel that says that passing from of a number ring to a finite‐index subgroup does not change the rational cohomology. Charney proved this generalization for trivial coefficients, and we extend it to twisted coefficients.

The space of commuting elements in a Lie group and maps between classifying spaces

Let $\pi$ be a discrete group, and let $G$ be a compact-connected Lie group. Then, there is a map $\Theta \colon \mathrm {Hom}(\pi,G)_0\to \mathrm {map}_*(B\pi,BG)_0$ between the null components of the spaces of homomorphisms and based maps, which sends a homomorphism to the induced map between classifying spaces. Atiyah and Bott studied this map for $\pi$ a surface group, and showed that it is surjective in rational cohomology. In this paper, we prove that the map $\Theta$ is surjective in rational cohomology for $\pi =\mathbb {Z}^m$ and the classical group $G$ except for $SO(2n)$ , and that it is not surjective for $\pi =\mathbb {Z}^m$ with $m\ge 3$ and $G=SO(2n)$ with $n\ge 4$ . As an application, we consider the surjectivity of the map $\Theta$ in rational cohomology for $\pi$ a finitely generated nilpotent group. We also consider the dimension of the cokernel of the map $\Theta$ in rational homotopy groups for $\pi =\mathbb {Z}^m$ and the classical groups $G$ except for $SO(2n)$ .

Moment maps and cohomology of non-reductive quotients

Let H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document} be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document}. Given an ample linearisation of the action and an associated Fubini–Study Kähler form which is invariant for a maximal compact subgroup Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q$\\end{document} of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}, we define a notion of moment map for the action of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}, and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} introduced in (Bérczi et al. in J. Topol. 11(3):826–855, 2018) in terms of this moment map. Using this description we derive formulas for the Betti numbers of X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} and express the rational cohomology ring of X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} in terms of the rational cohomology ring of the GIT quotient X//TH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/T^{H}$\\end{document}, where TH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$T^{H}$\\end{document} is a maximal torus in H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}. We relate intersection pairings on X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} to intersection pairings on X//TH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/T^{H}$\\end{document}, obtaining a residue formula for these pairings on X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} analogous to the residue formula of (Jeffrey and Kirwan in Topology 34(2):291–327, 1995). As an application, we announce a proof of the Green–Griffiths–Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree.

The Euler characteristic of the moduli space of graphs

The moduli space of rank n graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like −e−1/4(n/e)n/(nlog⁡n)2 as n goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with n.

Cohomology algebra of mapping spaces between quaternion Grassmannians

Let Gk,n(ℍ) for 2≤k<n denote the quaternion Grassmann manifold of k-dimensional vector subspaces of ℍn. In this paper we compute, in terms of the Sullivan models, the rational cohomology algebra of the component of the inclusion i: Gk,n(ℍ) → Gk,n+r(ℍ) in the space of mappings from Gk,n(ℍ) to Gk,n+r(ℍ) for r≥1 and, more generally, we show that the cohomology of Map(Gk,n(ℍ),Gk,n+r(ℍ);i) contains a truncated algebra ℚ[x]x4r+n+k^{2}-nk for n≥4.

Moduli spaces of Riemann surfaces as Hurwitz spaces

We consider the moduli space Mg,n of Riemann surfaces of genus g≥0 with n≥1 ordered and directed marked points. For d≥2g+n−1 we show that Mg,n is homotopy equivalent to a component of the simplicial Hurwitz space HurΔ(Sdgeo) associated with the partially multiplicative quandle Sdgeo. As an application, we give a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces. We also provide a combinatorial model for the infinite loop space Ω∞−2MTSO(2) of Hurwitz flavour.

Integral p-Adic Cohomology Theories

Abstract In this paper, we show the non-existence of finitely generated integral $p$-adic cohomology that satisfies finite étale descent and the associated rational cohomology coincides with rigid cohomology.

On the Cohomology of Torelli Groups. II

Abstract We describe the ring structure of the rational cohomology of the Torelli groups of the manifolds $\#^g S^n \times S^n$ in a stable range, for $2n \geq 6$. Some of our results are also valid for $2n=2$, where they are closely related to unpublished results of Kawazumi and Morita.

Homology Representations of Compactified Configurations on Graphs Applied to 𝓜2,n

We obtain new calculations of the top weight rational cohomology of the moduli spaces M2,n , equivalently the rational homology of the tropical moduli spaces , as a representation of Sn . These calculations are achieved fully for all , and partially—for specific irreducible representations of Sn —for . We also present conjectures, verified up to n = 22, for the multiplicities of the irreducible representations and . We achieve our calculations via a comparison with the homology of compactified configuration spaces of graphs. These homology groups are equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. In this paper, we construct an efficient free resolution for these homology representations, from which we extract calculations on irreducible representations one at a time, simplifying the calculation of these homology representations.