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In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. We also make some remarks on general methods of calculating certain pullbacks of rings which typically occur in calculations of equivariant cobordism. [ABSTRACT FROM AUTHOR]

A generalisation of the equivariant Dixmier-Douady invariant is constructed as a second-degree cohomology class within a new semi-equivariant Cech cohomology theory. This invariant obstructs liftings of semi-equivariant principal bundles that are associated to central exact sequences of structure groups in which each structure group is acted on by the equivariance group. The results and methods described can be applied to the study of complex vector bundles equipped with linear/anti-linear actions, such as Atiyah's Real vector bundles. [ABSTRACT FROM AUTHOR]

For finite dimensional free Cp-spaces, the calculation of the Bredon cohomology ring as an algebra over the cohomology of S0 is used to prove the non-existence of certain Cp-maps. These are related to Borsuk-Ulam type theorems, and equivariant maps related to the topological Tverberg conjecture. For certain finite dimensional Cp-spaces which are formed out of representations, it is proved that the cohomology is a free module over the cohomology of a point. All the calculations are done for the cohomology with constant coefficients Z/p. [ABSTRACT FROM AUTHOR]

In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type. [ABSTRACT FROM AUTHOR]

In this note we present an analogue of equivariant formal- ity in K-theory and show that it is equivalent to equivariant formality à la Goresky-Kottwitz-MacPherson. We also apply this analogue to give alternative proofs of equivariant formality of conjugation action on compact Lie groups, left translation action on generalized ag manifolds, and compact Lie group actions with maximal rank isotropy subgroups. [ABSTRACT FROM AUTHOR]

In a paper of Braam and Austin, SU ( 2 ) magnetic monopoles in hyperbolic space H 3 with half-integer mass and maximal symmetry breaking, were shown to be the same as solutions to matrix-valued difference equations called the discrete Nahm equations. Here, I discover the ( N − 1 ) -interval discrete Nahm equations and show that their solutions are equivalent to SU ( N ) hyperbolic monopoles of integer or half-integer mass, and maximal symmetry breaking. These discrete time evolution equations on an interval feature a jump in matrix dimensions at certain points in the evolution, which are given by the mass data of the corresponding monopole. I prove the correspondence with higher rank hyperbolic monopoles using localisation and Chern characters. I then prove that the monopole is determined up to gauge transformations by its “holographic image” of U ( 1 ) fields at the asymptotic boundary of H 3 . [ABSTRACT FROM AUTHOR]

The localization theorem is known for compact G-spaces, where G is a compact Lie group. In this study, we show that the localization theorem remains true for finite-dimensional compact group actions, and Borel's fixed point theorem holds not only for torus actions but for arbitrary finite-dimensional compact connected abelian group actions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The classification of Seifert manifolds was given in terms of numeric data by Seifert (1933), and then generalized by Raymond (1968) and Orlik and Raymond (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 graph of cycles. Then, we describe the rational equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincaré series for these manifolds and discuss the equivariant formality. [ABSTRACT FROM AUTHOR]

We show that for any cohomogeneity-one continuous action of a compact connected Lie group G on a closed topological manifold the equivariant cohomology equipped with its canonical H ⁎ ( B G ) -module structure is Cohen–Macaulay. The proof relies on the structure theorem for these actions recently obtained by Galaz-García and Zarei. We generalize in this way our previous result concerning smooth actions. [ABSTRACT FROM AUTHOR]

The cohomology of the configuration space of n points in R 3 is isomorphic to the regular representation of the symmetric group, which acts by permuting the points. We give a new proof of this fact by showing that the cohomology ring is canonically isomorphic to the associated graded of the Varchenko–Gelfand filtration on the cohomology of the configuration space of n points in R 1 . Along the way, we give a presentation of the equivariant cohomology ring of the R 3 configuration space with respect to a circle acting on R 3 via rotation around a fixed line. We extend our results to the settings of arbitrary real hyperplane arrangements (the aforementioned theorems correspond to the braid arrangement) as well as oriented matroids. [ABSTRACT FROM AUTHOR]

We study the existence of S 1 -equivariant characteristic classes on certain natural infinite rank bundles over the loop space L M of a manifold M . We discuss the different S 1 -equivariant cohomology theories in the literature and clarify their relationships. We attempt to use S 1 -equivariant Chern–Weil techniques to construct S 1 -equivariant characteristic classes. The main result is the construction of a sequence of S 1 -equivariant characteristic classes on the total space of the bundles, but these classes do not descend to the base L M . Nevertheless, we conclude by identifying a class of bundles for which the S 1 -equivariant first Chern class does descend to L M . [ABSTRACT FROM AUTHOR]

For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We show the Chern-Weil homomorphism for equivariant vector bundles with connection naturally factors through differential equivariant cohomology. [ABSTRACT FROM AUTHOR]

Associated to any manifold equipped with a closed form of degree >1 is an ‘ L ∞ -algebra of observables’ which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L ∞ -morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L ∞ -algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called ‘string Lie 2-algebra’ arises this way. [ABSTRACT FROM AUTHOR]

In this paper, the numbers of rational curves on general complete intersection Calabi–Yau threefolds in complex projective spaces are computed up to degree six. The results are all in agreement with the predictions made from mirror symmetry. [ABSTRACT FROM AUTHOR]

This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject. [ABSTRACT FROM AUTHOR]

We study what we call quasi-spline sheaves over locally Noetherian schemes. This is done with the intention of considering splines from the point of view of moduli theory. In other words, we study the way in which certain objects that arise in the theory of splines can be made to depend on parameters. In addition to quasi-spline sheaves , we treat ideal difference-conditions , and individual quasi-splines . Under certain hypotheses each of these types of objects admits a fine moduli scheme. The moduli of quasi-spline sheaves are proper, and there is a natural compactification of the moduli of ideal difference-conditions. We include some speculation on the uses of these moduli in the theory of splines and topology, and an appendix with a treatment of the Billera–Rose homogenization in scheme theoretic language. [ABSTRACT FROM AUTHOR]

This work studies the flag varieties of p -compact groups, principally through torus-equivariant cohomology, extending methods and tools of classical Schubert calculus and moment graph theory from the setting of real reflection groups to the broader context of complex reflection groups. In particular we give, for the infinite family of p -compact flag varieties corresponding to the complex reflection groups G ( r , 1 , n ) , a generalized GKM characterization (following Goresky–Kottwitz–MacPherson [8] ) of the torus-equivariant cohomology, building an explicit additive basis and showing its relationship with the polynomial or Borel presentation via the localization map. [ABSTRACT FROM AUTHOR]

Let G be a connected and non-necessarily compact Lie group acting on a connected manifold M . In this short note we announce the following result: for a G -invariant closed differential form on M , the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient. [ABSTRACT FROM AUTHOR]

The paper discusses the structure of the Hopf cyclic homology and cohomology of the algebra of smooth functions on a manifold provided that the algebra is endowed with an action or a coaction of the algebra of Hopf functions on a finite or compact group or of theHopf algebra dual to it. In both cases, an analog of the Connes-Hochschild-Kostant-Rosenberg theorem describing the structure of Hopf cyclic cohomology in terms of equivariant cohomology and other more geometric cohomology groups is proved. [ABSTRACT FROM AUTHOR]

We study some graded geometric constructions appearing naturally in the context of gauge theories. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to the case of arbitrary Q -manifolds introducing thus the concept of equivariant Q -cohomology. Using this concept we describe a procedure for analysis of gauge symmetries of given functionals as well as for constructing functionals (sigma models) invariant under an action of some gauge group. As the main example of application of these constructions we consider the twisted Poisson sigma model. We obtain it by a gauging-type procedure of the action of an essentially infinite dimensional group and describe its symmetries in terms of classical differential geometry. We comment on other possible applications of the described concept including the analysis of supersymmetric gauge theories and higher structures. [ABSTRACT FROM AUTHOR]

For every compact almost complex manifold $$(\mathsf {M},\mathsf {J})$$ equipped with a $$\mathsf {J}$$ -preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that $$\mathsf {M}$$ is symplectic and the action is Hamiltonian. If the manifold satisfies an extra so-called positivity condition, then this algorithm determines a family of vector spaces that contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever $$\dim (\mathsf {M})\le 6$$ and, when $$\dim (\mathsf {M})=8$$ , whenever the $$S^1$$ -action extends to an effective Hamiltonian $$T^2$$ -action, or none of the isotropy weights is $$1$$ . Moreover, there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces. We run the algorithm for $$\dim (\mathsf {M})\le 8$$ , quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for $$\dim (\mathsf {M})=6$$ and, when $$\dim (\mathsf {M})=8$$ , we prove that the equivariant cohomology ring, Chern classes, and isotropy weights agree with those of $${\mathbb {C}}P^4$$ with the standard $$S^1$$ -action (thereby proving the symplectic Petrie conjecture in this setting). [ABSTRACT FROM AUTHOR]

If there exists a set of canonical classes on a compact Hamiltonian-T-space in the sense of R Goldin and S Tolman, we derive some formulas for certain equivariant structure constants in terms of other equivariant structure constants and the values of canonical classes restricted to some fixed points. These formulas can be regarded as a generalization of Tymoczko's results. [ABSTRACT FROM AUTHOR]

Abstract: We show that if is a cohomogeneity one action of a compact connected Lie group G on a compact connected manifold M then is a Cohen–Macaulay module over . Moreover, this module is free if and only if the rank of at least one isotropy group is equal to rank G. We deduce as corollaries several results concerning the usual (de Rham) cohomology of M, such as a new proof of the following obstruction to the existence of a cohomogeneity one action: if M admits a cohomogeneity one action, then if and only if . [Copyright &y& Elsevier]

Abstract: We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus. [Copyright &y& Elsevier]

Abstract: In this paper we study the notion of representation up to homotopy of a Lie groupoid and the resulting derived category, and show that the adjoint representation is well defined as a representation up to homotopy. As an application, we extend Bottʼs spectral sequence converging to the cohomology of classifying spaces of Lie groups to the case of Lie groupoids. We explain the relation of this construction with the models of Cartan and Getzler for equivariant cohomology. Our work is closely related to and inspired by Behrendʼs [3], Bottʼs [4], and Getzlerʼs [9]. [Copyright &y& Elsevier]

The saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong. [ABSTRACT FROM AUTHOR]

Abstract: We give an explicit affine algebraic variety whose coordinate ring is isomorphic (as a W-algebra) with the equivariant cohomology of some Springer fibers. [Copyright &y& Elsevier]

We consider a class of connected oriented (with respect to ℤ/p) closed G-manifolds with a non-empty finite fixed point set, each of which is G-equivariantly formal, where G = ℤ/p and p is an odd prime. Using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a G-manifold in terms of algebra. This makes it possible to determine the number of equivariant cohomology rings (up to isomorphism) of such 2-dimensional G-manifolds. Moreover, we obtain a description of the ring homomorphism between equivariant cohomology rings of such two G-manifolds induced by a G-equivariant map, and show a characterization of the ring homomorphism. [ABSTRACT FROM AUTHOR]

Abstract: 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 on M, one defines Wittenʼs inhomogeneous coboundary operator (even/odd invariant forms on M) and its adjoint . Witten (1982) showed that the resulting cohomology classes have -harmonic representatives (forms in the null space of ), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the set of zeros of . 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 -cohomology and show the classes have representative -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 and . We also elucidate the connection between the -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 has a unique -harmonic field on M, with corresponding boundary conditions. Finally, we define the -Poincaré duality angles between the interior subspaces of -harmonic fields on M with appropriate boundary conditions, following recent work of DeTurck and Gluck. [Copyright &y& Elsevier]

Abstract: In this paper, we compute the -graded equivariant cohomology of rotation groups and Stiefel manifolds with particular involutions. [Copyright &y& Elsevier]

Abstract: In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field on M, Witten defines an inhomogeneous coboundary operator on invariant forms on M. The main purpose is to adapt Belishev–Sharafutdinovʼs boundary data to invariant forms in terms of the operator in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator on invariant forms on the boundary which we call the -DN map and using this we recover the -cohomology groups from the generalized boundary data . This shows that for a Zariski-open subset of the Lie algebra, determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of -cohomology groups from . These results explain to what extent the equivariant topology of the manifold in question is determined by . [Copyright &y& Elsevier]

Abstract: Algebraic topological methods are especially well suited for determining the non-existence of continuous mappings satisfying certain properties. In combinatorial problems it is sometimes possible to define a mapping from a space X of configurations to a Euclidean space in which a subspace, a discriminant, often an arrangement of linear subspaces , expresses a target condition on the configurations. Add symmetries of all these data under a group G for which the mapping is equivariant. If we remove the discriminant from , we can pose the problem of the existence of an equivariant mapping from X to the complement of the discriminant in . Algebraic topology may sometimes be applied to show that no such mapping exists, and hence the image of the original equivariant mapping must meet the discriminant. We introduce a general framework, based on a comparison of Leray–Serre spectral sequences. This comparison can be related to the theory of the Fadell–Husseini index. We apply the framework to: [•] solve a mass partition problem (antipodal cheeses) in , [•] determine the existence of a class of inscribed 5-element sets on a deformed 2-sphere, [•] obtain two different generalizations of the theorem of Dold for the non-existence of equivariant maps which generalizes the Borsuk–Ulam theorem. [Copyright &y& Elsevier]

Abstract: We compute the complete Fadell–Husseini index of the dihedral group acting on for and for coefficients, that is, the kernels of the maps in equivariant cohomology and This establishes the complete cohomological lower bounds, with and with coefficients, for the two-hyperplane case of Grünbaumʼs 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably chosen hyperplanes in ? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of . [Copyright &y& Elsevier]

Abstract: We prove that any continuous map of an N-dimensional simplex with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of to the same point in M: For this we have to assume that , no r vertices of get the same color, and our proof needs that r is a prime. A face of is a rainbow face if all vertices have different colors. This result is an extension of our recent “new colored Tverberg theorem”, the special case of . It is also a generalization of Volovikovʼs 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikovʼs proof, as well as ours, works when r is a prime power. [Copyright &y& Elsevier]

We define degeneracy loci for vector bundles with structure group G2 and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type G2. We include explicit descriptions of the G2 flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, correcting an error in a paper by Edidin and Graham. [ABSTRACT FROM AUTHOR]

Abstract: We prove the following optimal colorful Tverberg–Vrećica type transversal theorem: For prime r and for any colored collections of points in , , , , , there are partitions of the collections into colorful sets such that there is a k-plane that meets all the convex hulls , under the assumption that is even or . Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case (our optimal colored Tverberg theorem (2009) ), calculate the cohomological index for joins of chessboard complexes, and establish a new Borsuk–Ulam type theorem for -equivariant bundles that generalizes results of Volovikov (1996) and Živaljević (1999) . [Copyright &y& Elsevier]

In this paper we use Peter May's algebraic approach to Steenrod operations to construct Steenrod's reduced power operations in simplicial Bredon-Illman cohomology with local coefficients of a one vertex G-Kan complex, G being a discrete group. [ABSTRACT FROM AUTHOR]

We give a classifying theory for LG-bundles, where LG is the loop group of a compact Lie group G, and present a calculation for the string class of the universal LG-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use the caloron correspondence to define (higher) characteristic classes for LG-bundles and to prove a result for characteristic classes for based loop groups for the free loop group. These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions. [ABSTRACT FROM PUBLISHER]

Abstract: For each infinite series of the classical Lie groups of type B, C or D, we construct a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur''s Q- or P-functions defined earlier by Ivanov. [Copyright &y& Elsevier]

Abstract: Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold . We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well. [Copyright &y& Elsevier]

Abstract: We propose Weil and Cartan models for the equivariant cohomology of noncommutative spaces which carry a covariant action of Drinfel’d twisted symmetries. The construction is suggested by the noncommutative Weil algebra of Alekseev and Meinrenken (2000) ; we show how to implement a Drinfel’d twist of their models in order to take into account the noncommutativity of the spaces we are acting on. We also provide basic examples and properties of the twisted noncommutative equivariant cohomology. [Copyright &y& Elsevier]

Abstract: We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of Bredon–Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients. [Copyright &y& Elsevier]

Given a block b in kG where k is an algebraically closed field of characteristic p, there are classes ®Q 2 H2(AutF(Q); k×), constructed by K¨ulshammer and Puig, where F is the fusion system associated to b and Q is an F-centric subgroup. The gluing problem in F has a solution if these classes are the restriction of a class ® H2(Fc; kx). Linckelmann showed that a solution to the gluing problem gives rise to a reformulation of Alperin's weight conjecture. He then showed that the gluing problem has a solution if for every finite group G, the equivariant Bredon cohomology group HG1 (∣¢p(G)∣;A1) vanishes, where ∣¢p(G)∣ is the simplicial complex of the non-trivial p-subgroups of G and A1 is the coefficient functor G/H → Hom(H, kx). The purpose of this note is to show that this group does not anish if G = §p2 where p ⩾ 5. [ABSTRACT FROM AUTHOR]

We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore, the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model. [ABSTRACT FROM AUTHOR]

Let G be a compact Lie group. In 1960, P A Smith asked the following question: "Is it true that for any smooth action of G on a homotopy sphere with exactly two fixed points, the tangent G-modules at these two points are isomorphic?" A result due to Atiyah and Bott proves that the answer is 'yes' for ℤp and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on Sn which are c-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in ℤ-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith's question. [ABSTRACT FROM AUTHOR]

Abstract: Let be a Jordan curve in the plane. It is a simple topological riddle to determine if there is an equilateral triangle with vertices on γ. By reformulating this question in the paradigm of configuration spaces and test maps, we can solve this riddle using a Borsuk–Ulam type theorem obtained using equivariant methods. [Copyright &y& Elsevier]

Abstract: The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are weakly isomorphic. We also prove that quasitoric manifolds, which can be thought of as a topological counterpart to toric manifolds, are equivariantly homeomorphic if and only if their equivariant cohomology algebras are isomorphic. [Copyright &y& Elsevier]

Abstract: The main result of the paper is a Borel type description of the -equivariant cohomology ring of the manifold of all complete flags in . To prove this, we obtain a Goresky–Kottwitz–MacPherson type description of that ring. [Copyright &y& Elsevier]

Abstract: We give a description of equivariant cohomology of grassmannians that places the theory into a general framework for cohomology theories of grassmannians. As a result we obtain a formalism for equivariant cohomology where the basic results of equivariant Schubert calculus, the basis theorem, Pieri''s formula and Giambelli''s formula can be obtained from the corresponding results of the general framework by a change of basis. In order to show that our formalism reflects the geometry of grassmannians we relate our theory to the treatment of equivariant cohomology of grassmannians by A. Knutson and T. Tao. [Copyright &y& Elsevier]