Smooth actions of finite Oliver groups on spheres

In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S, which smooth manifolds F occur as the fixed point sets in S, and which real G-vector bundles ν over F occur as the equivariant normal bundles of F in S? We focus on the case G is an Oliver group and answer both questions under some conditions imposed on G, F, and ν. We construct smooth actions of G on spheres by making use of equivariant surgery, equivariant thickening, and Oliver's equivariant bundle extension method modified by an equivariant wegde sum construction and an equivariant bundle subtraction procedure.     

On equivariant indices of 1-forms on varieties

Given a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group), its equivariant homological index and (reduced) equivariant radial index are defined as elements of the ring of complex representations of the group. We show that these indices coincide on a germ of a smooth complex analytic G-variety. This makes it possible to consider the difference between them as a version of the equivariant Milnor number of a germ of a G-variety with an isolated singular point.     

Equivariant Poincaré series of filtrations and topology

Earlier, for an action of a finite group G on a germ of an analytic variety, an equivariant G-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of G-sets with an additional structure. We discuss to which extent the G-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.     

G-symmetric spectra,semistability and the multiplicative norm

In this paper we develop the basic homotopy theory of G-symmetric spectra (that is, symmetric spectra with a G-action) for a finite group G, as a model for equivariant stable homotopy with respect to a G-set universe. This model lies in between Mandell's equivariant symmetric spectra and the G-orthogonal spectra of Mandell and May and is Quillen equivalent to the two. We further discuss equivariant semistability, construct model structures on module, algebra and commutative algebra categories and describe the homotopical properties of the multiplicative norm in this context.     

Actions of Totally Disconnected Groups and Equivariant Singular Cohomology

In this work, we study the special properties of the equivariant singular cohomology of a G-space X, where G is a totally disconnected, locally compact group. We prove that any short exact sequence of coefficient systems for G, over a ring R, gives a long exact sequence of the associated equivariant singular cohomology modules. We establish the relationship between the ordinary singular cohomology modules and the equivariant singular cohomology modules with the natural contravariant coefficient system. Moreover, under some conditions, we give an isomorphism of the equivariant singular cohomology modules of the G-space X onto the ordinary singular cohomology modules of the orbit space X/G.     

Equivariant autoequivalences for finite group actions

The familiar Fourier-Mukai technique can be extended to an equivariant setting where a finite group G acts on a smooth projective variety X. In this paper we compare the group of invariant autoequivalences Aut(DbG(X)) with the group of autoequivalences of DG(X). We apply this method in three cases: Hilbert schemes on K3 surfaces, Kummer surfaces and canonical quotients.     

Equivariant Nielsen fixed point theory for n-valued maps

We develop an equivariant Nielsen fixed point theory for n-valued G-maps by associating (as in Better (2010) [2]) an abstract simplicial complex to any equivariant n-valued map and defining, in terms of this complex, two n-valued continuous G-homotopy invariants that are lower bounds for the number of fixed points and of orbits in the n-valued continuous G-homotopy class of a given n-valued G-map. We also provide an equivariant Hopf construction for n-valued G-maps as well as a minimality result for the Nielsen numbers introduced in this setting.     

A van Kampen theorem for equivariant fundamental groupoids

The equivariant fundamental groupoid of a G-space X is a category which generalizes the fundamental groupoid of a space to the equivariant setting. In this paper, we prove a van Kampen theorem for these categories: the equivariant fundamental groupoid of X can be obtained as a pushout of the categories associated to two open G-subsets covering X. This is proved by interpreting the equivariant fundamental groupoid as a Grothendieck semidirect product construction, and combining general properties of this construction with the ordinary (non-equivariant) van Kampen theorem. We then illustrate applications of this theorem by showing that the equivariant fundamental groupoid of a G-CW complex only depends on the 2-skeleton and also by using the theorem to compute an example.     

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups $K^{G}_{*}(X)$ , using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the “official” equivariant K-homology groups) and show that these are isomorphisms.     

Orbifold construction for topological field theories

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G?H$ of finite groups assigns in a functorial way to a G-equivariant topological field theory an H-equivariant topological field theory, the pushforward theory. When H is the trivial group, this yields an orbifold construction for G-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization.     

Cohomologie des G-faisceaux en caractéristique positive

Let G be a finite group, and X a noetherian G-scheme defined on an algebraically closed field k, whose characteristic divides the order of G. We define a refinement of the equivariant K-theory of X devoted to give a better account of the information related to modular representation theory. To cite this article: N. Borne, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Equivariant SKK and vector field bordism

We use a notion of equivariant Euler characteristic in order to extend classical results on controllable cutting and pasting, and vector field bordism, to the case of manifolds acted on by an arbitrary finite group G, and modelled on a fixed virtual representation (in the sense of W. Pulikowski and C. Kosniowski). By restricting attention to such G-manifolds, one finds that classical results continue to hold in the oriented and unoriented case. This extends work of several authors.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.     

Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory (basic examples are equivariant Chow groups of Edidin and Graham, Thomason??s equivariant K-theory and equivariant algebraic cobordism). Using the language of equivariant pretheories we generalize the theorem of Karpenko and Merkurjev on G-torsors and rational cycles. As an application, to every G-torsor E and a G-equivariant pretheory we associate a ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information about the motivic J-invariant of E, in the case of Grothendieck??s K ₀ indexes of the respective Tits algebras and in the case of algebraic cobordism ?? it gives a quotient of the cobordism ring of G.     

Equivariant fixed-point indices of iterated maps

We describe an equivariant version (for actions of a finite group G) of Dold’s index theory, [10], for iterated maps. Equivariant Dold indices are defined, in general, for a G-map U → X defined on an open G-subset of a G-ANR X (and satisfying a suitable compactness condition). A local index for isolated fixed-points is introduced, and the theorem of Shub and Sullivan on the vanishing of all but finitely many Dold indices for a continuously differentiable map is extended to the equivariant case. Homotopy Dold indices, arising from the equivariant Reidemeister trace, are also considered.      

Equivariant RO(G)-graded bordism theories

Let G be a finite group. The RO(G)-graded bordism theories of Pulikowski [7] and Kosniowski [3] are studied. Representing equivariant Thom spectra are constructed, and the relevant transversality results proved. New methods for splitting away from the order of G are described, and behavior in the presence of a gap hypothesis is examined.     

Equivariant strong shape

In this paper we propose a construction of the equivariant strong shape for compact metrizable G-spaces using an equivariant version of so-called cotelescopes and the concept of a fibrant G-space.     

Arc Spaces and Equivariant Cohomology

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex G-variety X by its associated arc space J _∞ X, with its induced G-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between $ \mathbb{Z} $ -bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties. We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric $ \mathbb{Z} $ -bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group).     

Equivariant Brauer and Picard groups and a Chase–Harrison–Rosenberg exact sequence

The classical Chase–Harrison–Rosenberg exact sequence relates the Picard and Brauer groups of a Galois extension S of a commutative ring R to the group cohomology of the Galois group. We associate to each action of a locally compact group G on a locally compact space X two groups which we call the equivariant Picard group and the equivariant Brauer group. We then prove an analogue of the Chase–Harrison–Rosenberg exact sequence in the which the roles of the Picard and Brauer groups are played by their equivariant analogues.