Showing papers on "Equivariant map published in 2006"

Parametrized homotopy theory

Abstract: Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The $qf$-model structure on $\mathcal{K}_B$ Equivariant $qf$-type model structures Ex-fibrations and ex-quasifibrations The equivalence between Ho$G\mathcal{K}_B$ and $hG\mathcal{W}_B$ Parametrized equivariant stable homotopy theory: Introduction to Part III Enriched categories and $G$-categories The category of orthogonal $G$-spectra over $B$ Model structures for parametrized $G$-spectra Adjunctions and compatibility relations Module categories, change of universe, and change of groups Parametrized duality theory: Introduction to Part IV Fiberwise duality and transfer maps Closed symmetric bicategories The closed symmetric bicategory of parametrized spectra Costenoble-Waner duality Fiberwise Costenoble-Waner duality Homology and cohomology, Thom spectra, and addenda: Introduction to Part V Parametrized homology and cohomology theories Equivariant parametrized homology and cohomology Twisted theories and spectral sequences Parametrized FSP's and generalized Thom spectra Epilogue: Cellular philosophy and alternative approaches Bibliography Index Index of notation.

The Equivariant Gromov-Witten theory of P**1

Abstract: We express all equivariant Gromov-Witten invariants of the projective line as matrix elements of explicit operators acting in the Fock space. As a consequence, we prove the equivariant theory is governed by the 2-Toda hierarchy of Ueno and Takasaki. This is the second in a sequence of three papers devoted to the Gromov-Witten theory of nonsingular target curves (the first paper of the series is math.AG/0204305).

Les préfaisceaux comme modèles des types d'homotopie

Abstract: Grothendieck introduced in Pursuing Stacks the notion of test category These are by definition small categories on which presheaves of sets are models for homotopy types of CW-complexes A well known example is the category of simplices (the corresponding presheaves are then simplicial sets) Moreover, Grothendieck defined the notion of basic localizer which gives an axiomatic approach to the homotopy theory of small categories, and gives a natural setting to extend the notion of test category with respect some localizations of the homotopy category of CW-complexes This text can be seen as a sequel of Grothendieck's homotopy theory We prove in particular two conjectures made by Grothendieck: any category of presheaves on a test category is canonically endowed with a Quillen closed model category structure, and the smallest basic localizer defines the homotopy theory of CW-complexes Moreover, we show how a local version of the theory allows to consider in a unified setting the equivariant homotopy theory as well The realization of this program goes through the construction and the study of model category structures on any category of presheaves on an abstract small category, as well as the study of the homotopy theory of small categories following and completing the contributions of Quillen, Thomason and Grothendieck

An atiyah-hirzebruch spectral sequence for kr-theory

Abstract: In recent years much attention has been given to a certain spectral sequence relating motivic cohomology to algebraic K-theory [Be, BL, FS, V3]. This spectral sequence takes on the form H(X,Z(− q 2 )) ⇒ K(X), where the H(X ;Z(t)) are the bi-graded motivic cohomology groups, and K(X) denotes the algebraic K-theory of X . It is useful in our context to use topologists’ notation and write K(X) for what K-theorists call K−n(X). The above spectral sequence is the analog of the classical Atiyah-Hirzebruch spectral sequence relating ordinary singular cohomology to complex K-theory, in a way that is explained further below. It is well known that there are close similarities between motivic homotopy theory and the equivariant homotopy theory of Z/2-spaces (cf. [HK1, HK2], for example). In fact there is even a forgetful map of the form (motivic homotopy theory over R) → (Z/2-equivariant homotopy theory),

Equivariant chow cohomology of toric varieties

Abstract: We show that the equivariant Chow cohomology ring of a toric vari- ety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which lo- calization holds in equivariant Chow cohomology with integer coe!cients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions. If X = X(!) is a smooth, complete complex toric variety then the follow- ing rings are canonically isomorphic: the equivariant singular cohomology ring H ! T (X), the equivariant Chow cohomology ring A ! (X), the Stanley-Riesner ring SR(!), and the ring of integral piecewise polynomial functions PP ! (!). If X is simplicial but not smooth then H ! (X) may have torsion and the natural map from SR(!) takes monomial generators to piecewise linear functions with ra- tional, but not necessarily integral, coe"cients. In such cases, these rings are not isomorphic, but they become isomorphic after tensoring with Q. When X is not simplicial, there are still natural maps between these rings, for instance from A ! (X)Q to H ! (X)Q and from H ! T (X) to PP ! (!), but these maps are far

Applications of equivariant cohomology

Abstract: We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of transversally elliptic operators. In particular, we state a conjecture for the index of a transversally elliptic operator linked to a Hamiltonian action. In the last part, we describe algorithms for numerical computations of values of multivariate spline functions and of vector-partition functions of classical root systems.

Supergroupoids, double structures, and equivariant cohomology

Abstract: Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Mackenzie; in particular, we show that Q-groupoids are intermediary objects between Mackenzie's LA-groupoids and double complexes, which include as a special case the simplicial model of equivariant cohomology. There is also a double complex associated to a Q-algebroid, which in the above special case is the BRST model of equivariant cohomology. Other special cases include models for the Drinfel'd double of a Lie bialgebra and Ginzburg's equivariant Poisson cohomology. Finally, a supergroupoid version of the van Est map is used to give a homomorphism from the double complex of a Q-groupoid to that of a Q-algebroid.

Quiver gauge theory of non-Abelian vortices and noncommutative instantons in higher dimensions

Abstract: We construct explicit Bogomolnyi, Prasad, Sommerfeld (BPS) and non-BPS solutions of the Yang-Mills equations on the noncommutative space Rθ2n×S2 which have manifest spherical symmetry. Using SU(2)-equivariant dimensional reduction techniques, we show that the solutions imply an equivalence between instantons on Rθ2n×S2 and non-Abelian vortices on Rθ2n, which can be interpreted as a blowing-up of a chain of D0-branes on Rθ2n into a chain of spherical D2-branes on Rθ2n×S2. The low-energy dynamics of these configurations is described by a quiver gauge theory which can be formulated in terms of new geometrical objects generalizing superconnections. This formalism enables the explicit assignment of D0-brane charges in equivariant K-theory to the instanton solutions.

Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink

Abstract: We prove a motivic analogue of Steenbrink's conjecture [25, Conjecture 2.2] on the Hodge spectrum (proved by M. Saito in [21]). To achieve this, we construct and compute motivic iterated vanishing cycles associated with two functions. We are also led to introduce a more general version of the convolution operator appearing in the motivic Thom-Sebastiani formula. Throughout the article we use the framework of relative equivariant Grothendieck rings of varieties endowed with an algebraic torus action

Periodic orbits of Hamiltonian homeomorphisms of surfaces

Abstract: John Franks and Michael Handel [FH2] have recently proved that any nontrivial Hamiltonian diffeomorphism of a closed surface of genus at least one has periodic orbits of arbitrarily large period. They proved a similar result for a nontrivial area-preserving diffeomorphism of a sphere with at least three fixed points. We extend these results to the case of the homeomorphisms. When the genus is at least one, we prove, moreover, that the periodic orbits may be chosen contractible if the set of contractible fixed points is contained in a disk. When the surface is a sphere, we extend the result to the case of a nontrivial homeomorphism with no wandering points. The proofs make use of an equivariant foliated version of Brouwer's plane translation theorem (see [B, Proposition 2.1]) and some properties of the linking number of fixed points

Equivariant Chern classes of singular algebraic varieties with group actions

Abstract: We define equivariant Chern–Schwartz–MacPherson classes of a possibly singular algebraic G-variety over the base field C, or more generally over a field of characteristic 0. In fact, we construct a natural transformation C G ∗ from the G-equivariant constructible function functor F G to the G-equivariant homology functor H G ∗ or A G ∗ (in the sense of Totaro–Edidin–Graham). This C G ∗ may be regarded as MacPherson’s transformation for (certain) quotient stacks. The Verdier–Riemann–Roch formula takes a key role throughout.

Four positive formulae for type A quiver polynomials

Abstract: We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99 ]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case. Our method begins by identifying quiver polynomials as multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant Chow groups [EG98 ]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85 ]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82 ] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98 ]). Writing double Schubert polynomials in terms of pipe dreams [FK96 ] then provides another geometric formula for double quiver polynomials, via [KM05 ]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials in terms of products of Stanley symmetric functions [Sta84 ]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92 , KM05 ] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80 ]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a , RS98 ], and from there to our formula of Buch–Fulton type.

The Cannon–Thurston map for punctured-surface groups

Abstract: Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.

Rotation equivariant Minkowski valuations

Abstract: The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivial operator with these properties. MSC 2000: 52B45, 52A20

An exotic Deligne-Langlands correspondence for symplectic groups

Abstract: Let G be a complex symplectic group. We introduce a G x (C ^x) ^{l + 1}-variety N_{l}, which we call the l-exotic nilpotent cone. Then, we realize the Hecke algebra H of type C_n ^(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N_2. This enables us to establish a Deligne-Langlands type classification of "non-critical" simple H-modules. As applications, we present a character formula and multiplicity formulas of H-modules.

Gluing affine torus actions via divisorial fans

Abstract: Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a ``proper polyhedral divisor'' introduced in earlier work, we develop the concept of a ``divisorial fan'' and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C*-surfaces and projectivizations of (non-split) vector bundles over toric varieties.

Topological string partition functions as equivariant indices

Abstract: We propose to use the identification of topological string partition functions as equivariant indices on framed moduli spaces of instantons to study the Gopakumar-Vafa conjecture for some local Calabi-Yau geometries.

From Brouwer theory to the study of homeomorphisms of surfaces

Abstract: We will state an equivariant foliated version of the classical Brouwer PlaneTranslation Theorem and will explain how to apply this result to the study of homeomorphisms of surfaces. In particular we will explain why a diffeomorphism of a closed oriented surface of genus . 1 that is the time-one map of a time dependent Hamiltonian vector field has infinitely many periodic orbits. This gives a positive answer in the case of surfaces to a more general question stated by C. Conley. We will give a survey of some recent results on homeomorphisms and diffeomorphisms of surfaces and will explain the links with the improved version of Brouwer�fs theorem.

Rank two quiver gauge theory, graded connections and noncommutative vortices

Abstract: We consider equivariant dimensional reduction of Yang-Mills theory on K?hler manifolds of the form M ? P1 ? P1. This induces a rank two quiver gauge theory on M which can be formulated as a Yang-Mills theory of graded connections on M. The reduction of the Yang-Mills equations on M ? P1 ? P1 induces quiver gauge theory equations on M and quiver vortex equations in the BPS sector. When M is the noncommutative space ?2n both BPS and non-BPS solutions are obtained, and interpreted as states of D-branes. Using the graded connection formalism, we assign D0-brane charges in equivariant K-theory to the quiver vortex configurations. Some categorical properties of these quiver brane configurations are also described in terms of the corresponding quiver representations.

Dirac operators on all Podles quantum spheres

Abstract: We construct spectral triples on all Podles quantum spheres. These noncommutative geometries are equivariant for a left action of $U_q(su(2))$ and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the 2-sphere. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.

Irregularity of hypergeometric systems via slopes along coordinate subspaces

Abstract: We study the irregularity sheaves attached to the $A$-hypergeometric $D$-module $M_A(\beta)$ introduced by Gel'fand et al., where $A\in\mathbb{Z}^{d\times n}$ is pointed of full rank and $\beta\in\mathbb{C}^d$. More precisely, we investigate the slopes of this module along coordinate subspaces. In the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector $L$ on torus equivariant generators. To this end we introduce the $(A,L)$-umbrella, a simplicial complex determined by $A$ and $L$, and identify its facets with the components of the associated graded ring. We then establish a correspondence between the full $(A,L)$-umbrella and the components of the $L$-characteristic variety of $M_A(\beta)$. We compute in combinatorial terms the multiplicities of these components in the $L$-characteristic cycle of the associated Euler-Koszul complex, identifying them with certain intersection multiplicities. We deduce from this that slopes of $M_A(\beta)$ are combinatorial, independent of $\beta$, and in one-to-one correspondence with jumps of the $(A,L)$-umbrella. This confirms a conjecture of Sturmfels and gives a converse of a theorem of Hotta: $M_A(\beta)$ is regular if and only if $A$ defines a projective variety.

Sasaki-Einstein Manifolds and Volume Minimisation

Abstract: We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler-Einstein metric.

Equivariant universal coefficient and künneth spectral sequences

Abstract: This paper constructs hyper-homology spectral sequences of $\mathbb{Z}$-graded and $RO_{G}$-graded Mackey functors that compute $\mathrm{Ext}$ and $\mathrm{Tor}$ over $G$-equivariant $S$-algebras ($A_{\infty}$ ring spectra) for finite groups $G$. These specialize to universal coefficient and K\"unneth spectral sequences.

Equivariant Cohomology and the Cartan Model

Abstract: If a compact Lie group G acts on a manifold M , the space M/G of orbits of the action is usually a singular space. Nonetheless, it is often possible to develop a ’differential geometry’ of the orbit space in terms of appropriately defined equivariant objects on M . In this article, we will be mostly concerned with ’differential forms onM/G’. A first idea would be to work with the complex of ’basic’ forms on M , but for many purposes this complex turns out to be too small. A much more useful complex of equivariant differential forms on M was introduced by H. Cartan in 1950, in [2, Section 6]. In retrospect, Cartan’s approach presented a differential form model for the equivariant cohomology of M , as defined by A. Borel [8] some ten years later. Borel’s construction replaces the quotient M/G by a better behaved (but usually infinite-dimensional) homotopy quotient MG, and Cartan’s complex should be viewed as a model for forms on MG. One of the features of equivariant cohomology are the localization formulas for the integrals of equivariant cocycles. The first instance of such an integration formula was the ’exact stationary phase formula’, discovered by Duistermaat-Heckman [12] in 1980. This formula was quickly recognized, by Berline-Vergne [5] and Atiyah-Bott [3], as a localization principle in equivariant cohomology. Today, equivariant localization is a basic tool in mathematical physics, with numerous applications. In this article, we will begin with Borel’s topological definition of equivariant cohomology. We then proceed to describe H. Cartan’s more algebraic approach, and conclude with a discussion of localization principles. As additional references for the material covered here, we particularly recommend the books by BerlineGetzler-Vergne [4] and Guillemin-Sternberg [17].

Continuous functors as a model for the equivariant stable homotopy category

Abstract: It is a classical observation that a based continuous functor X from the category of finite CW‐complexes to the category of based spaces that takes homotopy pushouts to homotopy pullbacks “represents” a homology theory—the collection of spaces fX.S n /g obtained by evaluating X on spheres yields an ‐prespectrum. Such functors are sometimes referred to as linear or excisive. The main theorem of this paper provides an equivariant analogue of this result. We show that a based continuous functor from finite G ‐CW‐complexes to based G ‐spaces represents a genuine equivariant homology theory if and only if it takes G ‐homotopy pushouts to G ‐homotopy pullbacks and satisfies an additional condition requiring compatibility with Atiyah duality for orbit spaces G=H . Our motivation for this work is the development of a recognition principle for equivariant infinite loop spaces. In order to make the connection to infinite loop space theory precise, we reinterpret the main theorem as providing a fibrancy condition in an appropriate model category of spectra. Specifically, we situate this result in the context of the study of equivariant diagram spectra indexed on the category WG of based G ‐spaces homeomorphic to finite G ‐CW‐complexes for a compact Lie group G . Using the machinery of Mandell‐May‐Schwede‐Shipley, we show that there is a stable model structure on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal G ‐spectra. We construct a second “absolute” stable model structure which is Quillen equivalent to the stable model structure. There is a model-theoretic identification of the fibrant continuous functors in the absolute stable model structure as functors Z such that for A2 WG the collectionfZ.A^ S W /g forms an ‐G ‐prespectrum as W varies over the universe U . Thus, our main result provides a concrete identification of the fibrant objects in the absolute stable model structure. This description of fibrant objects in the absolute stable model structure makes it clear that in the equivariant setting we cannot hope for a comparison between the category of equivariant continuous functors and equivariant A ‐spaces, except when G is finite. We provide an explicit analysis of the failure of the category of equivariant A ‐spaces to model connective G ‐spectra, even for GD S 1 .

Positivity in equivariant quantum Schubert Calculus

Abstract: A conjecture of D. Peterson, proved by W. Graham, states that the structure constants of the (T-)equivariant cohomology of a homogeneous space G/P satisfy a certain positivity property. In this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology. 1. Introduction. It is well known that the (integral) cohomology of the homogeneous space X = G/P (for G a connected, semisimple, complex Lie group and P a parabolic subgroup) satisfies a positivity property: its structure constants are nonnegative integers equal to the number of intersection points of three Schubert varieties, in general position, whose codimensions add up to the dimension of X. Recently, Graham (Gr) has proved a conjecture of Peterson (P), asserting that H � (X), the T-equivariant cohomology of X, where T � (C � ) r is a maximal torus in G, enjoys a more general positivity property.