Ordinary Cohomology Research Articles

The elliptic Weyl character formula

AbstractWe calculate equivariant elliptic cohomology of the partial flag variety$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where$H\subseteq G$are compact connected Lie groups of equal rank. We identify the${\rm RO}(G)$-graded coefficients${\mathcal{E}} ll_G^*$as powers of Looijenga’s line bundle and prove that transfer along the map$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$is calculated by the Weyl–Kac character formula. Treating ordinary cohomology,$K$-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N. Ganter and A. Ram,Elliptic Schubert calculus, in preparation].

Harer stability and orbifold cohomology

In this paper we review the combinatorics of the twisted sectors of $\mathcal{M}_{g,n}$, and we exhibit a formula for the age of each of them in terms of the combinatorial data. Then we show that orbifold cohomology of $\mathcal{M}_{g,n}$ when $g \to \infty$ reduces to its ordinary cohomology. We do this by showing that the twisted sector with minimal age is always the hyperelliptic twisted sector with all markings in the Weierstrass points; the age of the latter moduli space is just half its codimension in $\mathcal{M}_{g,n}$.

Becker-Gottlieb transfer for Hochschild cohomology

Let G G be a finite group. Over any finite G G -poset P \mathcal {P} we may define a transporter category G ∝ P G\propto \mathcal {P} as the corresponding Grothendieck construction. There exists a Becker-Gottlieb transfer from the ordinary cohomology of G ∝ P G\propto \mathcal {P} to that of G G . We shall construct it using module-theoretic methods and then extend it to a transfer from the Hochschild cohomology of k ( G ∝ P ) k(G\propto \mathcal {P}) to that of k G kG , where k k is a base field.

Bredon cohomological finiteness conditions for generalisations of Thompson groups

We define a family of groups that generalises Thompson’s groups T and G , and also those of Higman, Stein and Brin. For groups in this family we describe centralisers of finite subgroups and show that for a given finite subgroup Q there are finitely many conjugacy classes of finite subgroups isomorphic to Q . We consider groups of type quasi- \underline{\rm F}_\infty . This is a property slightly weaker than possessing a finite type model for the classifying space for proper actions \underline{E}G . We give criteria for the T versions of our groups to be of type quasi- \underline{\rm F}_\infty . We also generalise some well-known properties of ordinary cohomology to Bredon cohomology.

Balance in complete cohomology

The balance of complete cohomology for modules that admit complete resolutions has been established by Christensen and Jorgensen (2013), as well as by Enochs, Estrada and Iacob (2012), by using two types of constructions on a given bicomplex. In this paper, we show that these constructions are closely related to each other. We also present an alternative proof of balance, which is based on the corresponding assertion for ordinary cohomology.

Projective resolutions up to bounded torsion and bounds for the orders of the finite subgroups

We prove that group (co)homology can be computed up to bounded torsion using certain generalized projective resolutions. We investigate the notion of dimension obtained using this kind of resolutions and its relationship with the existence of torsion in both ordinary and complete cohomology groups.

Automorphic symbols, p -adic L -functions and ordinary cohomology of Hilbert modular varieties

We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general $p$-adic $L$-functions to nearly ordinary Hilbert automorphic forms. Then we establish an exact control theorem for the $p$-adically completed cohomology of a Hilbert modular variety localized at a suitable nearly ordinary maximal ideal of the Hecke algebra. We also show its freeness over the corresponding Hecke algebra which turns out to be a universal deformation ring. In the last part of the paper we combine the above results to construct $p$-adic $L$-functions for Hida families of Hilbert automorphic forms in universal deformation rings of Galois representations.

Some Results on the Finiteness of Generalized Local Cohomology Modules Using Serre Classes

Let R be a commutative Noetherian ring with identity and \(\mathfrak{a}\) be an ideal of R. In this paper, we investigate the conditions under which the set of associated primes of generalized local cohomology modules is finite. For this aim, we study some relations between generalized local cohomology modules, ideal transform, Ext and ordinary local cohomology modules in the Serre classes of R-modules. Such relations provided a common language for expressing some results about the generalized local cohomology R-modules, that has appeared in some papers.

Lifting problems and transgression for non-abelian gerbes

We discuss lifting and reduction problems for bundles and gerbes in the context of a Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen’s long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties for this transgression map. For instance, it reduces–in case of a Lie 2-group with a single object–to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: first, we obtain a new comparison theorem for different notions of string structures. Second, our transgression map establishes a direct relation between string structures and spin structures on the loop space.

THE PRODUCT STRUCTURE OF THE EQUIVARIANT K-THEORY OF THE BASED LOOP GROUP OF SU(2)

Let G=SU(2) and let Ω G denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H*(Ω G) is a divided polynomial algebra Γ[x]. The algebra Γ[x] can be described as an inverse limit as k→∞ of the symmetric subalgebra in Λ(x1, …, xk), where Λ(x1, …, xk) is the usual exterior algebra in the variables x1, …, xk. We compute the R(G)-algebra structure of the G-equivariant K-theory K*G(Ω G) which naturally generalizes the classical computation of H*(Ω G) as Γ[x]. Specifically, we prove that K*G(Ω G) is an inverse limit of the symmetric (S2r-invariant) subalgebra (K*G((ℙ1)2r))S2r of K*G((ℙ1)2r), where the symmetric group S2r acts in the natural way on the factors of the product (ℙ1)2r and G acts diagonally via the standard action on each factor.

${\mathcal{B}$-BOUNDED COHOMOLOGY AND APPLICATIONS

A discrete group with word-length (G, L) is [Formula: see text]-isocohomological for a bounding classes [Formula: see text] if the comparison map from [Formula: see text]-bounded cohomology to ordinary cohomology (with coefficients in ℂ) is an isomorphism; it is strongly [Formula: see text]-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong [Formula: see text]-isocohomologicality. In particular, we show strong [Formula: see text]-isocohomologicality for an FP∞ group G if all of the weighted G-sensitive Dehn functions are [Formula: see text]-bounded. Such groups include all [Formula: see text]-asynchronously combable groups; moreover, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. We also provide examples where the comparison map fails to be injective, as well as surjective, and give an example of a solvable group with quadratic first Dehn function, but exponential second Dehn function. Finally, a relative theory of [Formula: see text]-bounded cohomology of groups with respect to subgroups is introduced. Relative isocohomologicality is determined in terms of a new notion of relative Dehn functions and a relativeFP∞ property for groups with respect to a collection of subgroups. Applications for computing [Formula: see text]-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.

Topological quantum field theory structure on symplectic cohomology

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic cohomology. These constructions yield new applications in symplectic topology relating to the Arnol'd chord conjecture and to exact contact embeddings. We prove that if a Liouville domain M admits an exact embedding into an exact convex symplectic manifold X, and the boundary of M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian lying in the boundary. The TQFT respects the isomorphism between the symplectic cohomology of a cotangent bundle and the homology of the free loop space, so it recovers the TQFT of string topology. Finally, we use the TQFT to prove that symplectic cohomology vanishes iff Rabinowitz Floer cohomology vanishes.

DEFORMATION OF SINGULARITIES AND THE HOMOLOGY OF INTERSECTION SPACES

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface. Regardless of monodromy, the middle degree homology of intersection spaces is always a subspace of the homology of the deformation, yet itself contains the middle intersection homology group, the ordinary homology of the singular space, and the ordinary homology of the regular part.

Dyer–Lashof operations on Tate cohomology of finite groups

Let k be the field with p>0 elements, and let G be a finite group. By exhibiting an E-infinity-operad action on Hom(P,k) for a complete projective resolution P of the trivial kG-module k, we obtain power operations of Dyer-Lashof type on Tate cohomology H*(G; k). Our operations agree with the usual Steenrod operations on ordinary cohomology. We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups G. We also show that the operations in negative degree are non-trivial. As an application, we prove that at the prime 2 these operations can be used to determine whether a Tate cohomology class is productive (in the sense of Carlson) or not.

Witten–Hodge theory for manifolds with boundary and equivariant cohomology

We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and corresponding vector field XM on M, one defines Wittenʼs inhomogeneous coboundary operator dXM=d+ιXM:ΩG±→ΩG∓ (even/odd invariant forms on M) and its adjoint δXM. Witten (1982) [18] showed that the resulting cohomology classes have XM-harmonic representatives (forms in the null space of ΔXM=(dXM+δXM)2), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the set N(XM) of zeros of XM. Our principal purpose is to extend these results to manifolds with boundary. In particular, we define relative (to the boundary) and absolute versions of the XM-cohomology and show the classes have representative XM-harmonic fields with appropriate boundary conditions. To do this we present the relevant version of the Hodge–Morrey–Friedrichs decomposition theorem for invariant forms in terms of the operators dXM and δXM. We also elucidate the connection between the XM-cohomology groups and the relative and absolute equivariant cohomology, following work of Atiyah and Bott. This connection is then exploited to show that every harmonic field with appropriate boundary conditions on N(XM) has a unique XM-harmonic field on M, with corresponding boundary conditions. Finally, we define the XM-Poincaré duality angles between the interior subspaces of XM-harmonic fields on M with appropriate boundary conditions, following recent work of DeTurck and Gluck.

Self-intersections of immersions and Steenrod operations

We present a formula describing the action of a generalised Steenrod operation of Z2-type (14) on the cohomology class represented by a proper self-transverse immersion f : M X. Our formula depends only on the Umkehr map, the characteristic classes of the normal bundle, and the class rep- resented by the double point immersion of f . This generalises a classical result of R. Thom (13): If α ∈ H k (X; Z2) is the ordinary cohomology class represented by f : M X ,t henSq i (α )= f∗wi(νf ).

The Chen–Ruan cohomology of moduli of curves of genus 2 with marked points

In this work we describe the Chen–Ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points. In the first half of the paper we compute the additive structure of the Chen–Ruan cohomology ring for the moduli stack of stable n-pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen–Ruan cohomology ring as an algebra on the ordinary cohomology.

The Chow ring of the classifying space [formula omitted

We compute the Chow ring of the classifying space BSO ( 2 n , C ) in the sense of Totaro using the fibration Gl ( 2 n ) / SO ( 2 n ) → BSO ( 2 n ) → BGl ( 2 n ) and a computation of the Chow ring of Gl ( 2 n ) / SO ( 2 n ) in a previous paper. We find this Chow ring is generated by Chern classes and a characteristic class defined by Edidin and Graham which maps to 2 n − 1 times the Euler class under the usual class map from the Chow ring to ordinary cohomology. Moreover, we show this class represents 1 / 2 n − 1 ( n − 1 ) ! times the nth Chern class of the representation of SO ( 2 n ) whose highest weight vector is twice that of the half-spin representation.

Schubert calculus and singularity theory

Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces, it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K-theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note, we show that there is, in some sense, the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of Lerche Vafa and Warner. The main conjecture these authors made was that the classical cohomology of a Hermitian symmetric homogeneous manifold is a Jacobi ring of an appropriate potential. We extend this conjecture and provide a simple proof. Namely, we show that the cohomology of the Hermitian symmetric space is a Jacobi ring of a certain potential and the equivariant and the quantum cohomology and the K-theory is a Jacobi ring of a particular deformation of this potential. This suggests to study the most general deformations of the Frobenius algebra of cohomology of these manifolds by considering the versal deformation of the appropriate potential. The structure of the Jacobi ring of such potential is a subject of well developed singularity theory. This gives a potentially new way to look at the classical, the equivariant, the quantum and other flavors of Schubert calculus.

Cohomology of Products and Coproducts of Augmented Algebras

We show that the ordinary cohomology functor $\Lambda \mapsto {\operatorname{Ext}} ^* _\Lambda (k,k)$ from the category of augmented k-algebras to itself exchanges coproducts and products, then that Hochschild cohomology is close to sending coproducts to products if the factors are self-injective. We identify the multiplicative structure of the Hochschild cohomology of a product, modulo a certain ideal, in terms of the cohomology of the factors.