Equivariant Cohomology Research Articles

On the Cohomology of Calogero–Moser Spaces

We compute the equivariant cohomology of smooth Calogero-Moser spaces and some associated symplectic resolutions of symplectic quotient singularities.

On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms

Let $$\mathrm {R}$$ be a real closed field. We prove that for any fixed d, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $$\mathrm {R}^k$$ defined by polynomials of degrees bounded by d vanishes in dimensions d and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $$d^{O(d)} s^d k^{\lfloor d/2 \rfloor -1} $$ on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $$\mathrm {R}^k$$ defined by quantifier-free formulas involving s symmetric polynomials of degrees bounded by d, where $$1 < d \ll s,k$$ . This bound is tight up to a factor depending only on d. These results significantly improve upon those obtained previously in Basu and Riener (Adv Math 305:803–855, 2017) which were proved using different techniques. Our new methods are quite general, and also yield bounds on the equivariant Betti numbers of certain special classes of symmetric definable sets (definable sets symmetrized by pulling back under symmetric polynomial maps of fixed degree) in arbitrary o-minimal structures over $$\mathrm {R}$$ . Finally, we utilize our new approach to obtain an algorithm with polynomially bounded complexity for computing these equivariant Betti numbers. In contrast, the problem of computing the ordinary Betti numbers of (not necessarily symmetric) semi-algebraic sets is considered to be an intractable problem, and all known algorithms for this problem have doubly exponential complexity.

Residue formulas for push-forwards in equivariant cohomology: a symplectic approach

The aim of this paper is to describe how to obtain residue-type formulas for push-forwards in equivariant cohomology, using the Jeffrey-Kirwan nonabelian localization theorem and the related result of Guillemin and Kalkman. This paper relates the description of equivariant push-forwards for homogeneous spaces of classical Lie groups with the action of the maximal torus with symplectic geometry, using the example of projective spaces and classical Grassmannians.

Equivariant quantum cohomology of the Grassmannian via the rim hook rule

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of k-planes in complex n-space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule then gives an effective algorithm for computing all equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights modulo n, suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.

Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we show a correspondence of the Gelfand-Tsetlin bases (resp. the standard bases) to the fixed point classes (resp. the stable classes) in the equivariant elliptic cohomology $E_T(X)$ of the cotangent bundle $X$ of the partial flag variety. As a result we obtain a geometric representation of the elliptic quantum group on $E_T(X)$.

Canonical bases for the equivariant cohomology and K-theory rings of symplectic toric manifolds

Let M be a symplectic toric manifold acted on by a torus T. In this work we exhibit an explicit basis for the equivariant K-theory ring KT(M) which is canonically associated to a generic component of the moment map. We provide a combinatorial algorithm for computing the restrictions of the elements of this basis to the fixed point set; these, in turn, determine the ring structure of KT(M). The construction is based on the notion of local index at a fixed point, similar to that introduced by Guillemin and Kogan in (GK). We apply the same techniques to exhibit an explicit basis for the equivari- ant cohomology ring HT(M;Z) which is canonically associated to a generic component of the moment map. Moreover we prove that the elements of this basis coincide with some well-known sets of classes: the equivariant Poincare duals to the closures of unstable manifolds, and also the canonical classes in- troduced by Goldin and Tolman in (GT), which exist whenever the moment map is index increasing.

Equivariant cohomology of ${(\mathbb Z_{2})^{r}}$-manifolds and syzygies

Equivariant cohomology of ${(\mathbb Z_{2})^{r}}$-manifolds and syzygies

HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for finite-dimensional smooth manifolds [Quillen, ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math., 7 (1971)]. Quillen stresses the fact that this construction allows the definition of Gysin maps for ‘oriented’ proper maps. For finite-dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincaré duality, hence it is not clear how to extend it to infinite-dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite-dimensional space $BG$, which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite-dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$.

The FKMM-invariant in low dimension

In this paper we investigate the problem of the cohomological classification of "Quaternionic" vector bundles in low-dimension ($d\leqslant 3$). We show that there exists a characteristic classes $\kappa$, called the FKMM-invariant, which takes value in the relative equivariant Borel cohomology and completely classifies "Quaternionic" vector bundles in low-dimension. The main subject of the paper concerns a discussion about the surjectivity of $\kappa$.

La théorie de Hodge des bimodules de Soergel (d'après Soergel et Elias-Williamson)

Soergel bimodules are certain bimodules over polynomial algebras, associated with Coxeter groups, and introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras. Even though their definition is algebraic and rather elementary, some of their crucial properties were known until recently only in the case of crystallographic Coxeter groups, where these bimodules can be interpreted in terms of equivariant cohomology of Schubert varieties. In recent work Elias and Williamson have proved these properties in full generality by showing that these bimodules possess Hodge type properties. These results imply positivity of Kazhdan-Lusztig polynomials in full generality, and provide an algebraic proof of the Kazhdan-Lusztig conjecture.

Homotopy of planar Lie group equivariant presheaves

We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.

On the equivalence of integral Tk-cohomology Chern numbers and Tk-K-theoretic Chern numbers

This paper mainly deals with the question of equivalence between equivariant cohomology Chern numbers and equivariant K-theoretic Chern numbers when the transformation group is a torus. By using the equivariant Riemann-Roch relation of Atiyah-Hirzebruch type, it is proved that the vanishing of equivariant cohomology Chern numbers is equivalent to the vanishing of equivariant K-theoretic Chern numbers.

Localization of Chern–Simons type invariants of Riemannian foliations

We prove an Atiyah–Bott–Berline–Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern–Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the union of closed leaves. Various examples are given to illustrate our method.

Cyclic homology and group actions

In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with algebraic crossed-products associated with group actions on unital algebras over any ring k⊃Q. In the second part, we extend the results to actions on locally convex algebras. We then deal with crossed-products associated with group actions on manifolds and smooth varieties. For the finite order components, the results are expressed in terms of what we call “mixed equivariant cohomology”. This “mixed” theory mediates between group homology and de Rham cohomology. It is naturally related to equivariant cohomology, and so we obtain explicit constructions of cyclic cycles out of equivariant characteristic classes. For the infinite order components, we simplify and correct the misidentification of Crainic (1999). An important new homological tool is the notion of “triangular S-module”. This is a natural generalization of the cylindrical complexes of Getzler–Jones. It combines the mixed complexes of Burghelea–Kassel and parachain complexes of Getzler–Jones with the S-modules of Kassel–Jones. There are spectral sequences naturally associated with triangular S-modules. In particular, this allows us to recover spectral sequences of Feigin–Tsygan and Getzler–Jones and leads us to a new spectral sequence.

Assignments for topological group actions

A polynomial assignment for a continuous action of a compact torus T on a topological space X assigns to each p∈X a polynomial function on the Lie algebra of the isotropy group at p in such a way that a certain compatibility condition is satisfied. The space AT(X) of all polynomial assignments has a natural structure of an algebra over the polynomial ring of Lie(T). It is an equivariant homotopy invariant, canonically related to the equivariant cohomology algebra. In this paper we prove various properties of AT(X) such as Borel localization, a Chang–Skjelbred lemma, and a Goresky–Kottwitz–MacPherson presentation. In the special case of Hamiltonian torus actions on symplectic manifolds we prove a surjectivity criterion for the assignment equivariant Kirwan map corresponding to a circle in T. We then obtain a Tolman–Weitsman type presentation of the kernel of this map.

Springer’s Weyl Group Representation via Localization

AbstractLet G denote a reductive algebraic group over C and x a nilpotent element of its Lie algebra 𝔤. The Springer variety Bx is the closed subvariety of the flag variety B of G parameterizing the Borel subalgebras of 𝔤 containing x. It has the remarkable property that the Weyl group W of G admits a representation on the cohomology of Bx even though W rarely acts on Bx itself. Well-known constructions of this action due to Springer and others use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when x is what we call parabolic-surjective. The idea is to use localization to construct an action of W on the equivariant cohomology algebra H*S (Bx), where S is a certain algebraic subtorus of G. This action descends to H*(Bx) via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type A and, more generally, all nilpotents for which it is known that W acts on H*S (Bx) for some torus S. Our result is deduced from a general theorem describing when a group action on the cohomology of the ûxed point set of a torus action on a space lifts to the full cohomology algebra of the space.

An explicit cycle map for the motivic cohomology of real varieties

We provide a direct construction of a cycle map in the level of representing complexes from the motivic cohomology of real (or complex) varieties to the appropriate ordinary cohomology theory. For complex varieties, this is simply integral Betti cohomology, whereas for real varieties the recipient theory is the bigraded \operatorname{Gal}(\mathbb C/\mathbb R) -equivariant cohomology [19]. Using the finite analytic correspondences from [7] we provide a sheaf-theoretic approach to ordinary equivariant RO(G) -graded cohomology for any finite group G . In particular, this gives a complex of sheaves \mathbb Zp_{\omega} on a suitable equivariant site of real analytic manifolds-with-corner whose construction closely parallels that of the Voevodsky's motivic complexes $$\mathbb Zp_{\mathcal M} . Our cycle map is induced by the change of sites functor that assigns to a real variety X its analytic space X(\mathbb C)$ together with the complex conjugation involution.

Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.

Chern class of Schubert cells in the flag manifold and related algebras

We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra.

Classification and equivariant cohomology of circle actions on 3d manifolds

The classification of Seifert manifolds was given in terms of numeric data by Seifert in 1933, and then generalized by Orlik and Raymond in 1968 to circle actions on closed 3d manifolds. In this paper, we further generalize the classification to circle actions on 3d manifolds with boundaries by adding a numeric parameter and a union of cycle graphs. Then we describe the equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincar\'{e} series for these manifolds and discuss the equivariant formality