Equivariant Gysin maps and pulling back fixed points

We develop a new approach to the pulling back fixed points theorem of W. Browder and use it in order to prove various generalizations of this result.

     

Equivariant Surgery Theory: Deleting-Inserting Theorems of Fixed Point Manifolds on Spheres and Disks

The paper gives a tool to delete and insert fixed point manifolds for smooth actions of finite Oliver groups on spheres and disks. A similar result was already given in a joint article with E. Laitinen and K. Pawaowski for those of finite nonsolvable groups on spheres. It is useful in classifying smooth actions on spheres from the view point of fixed point data. The methods employed in the present paper are equivariant surgery and equivariant connected sum associated with elements in the Burnside ring. The idea of killing surgery obstructions is as follows: Let G be a finite group not of prime power order, C a contractible, finiteG CW complex, and an element in a K theoretic group arising as an obstruction class of geometric object f. It often holds that (1-[C])^m becomes trivial for large integers m where [C] is the element represented byC in the Burnside ring (G). One expects that the algebraic object (1 - [C])^m is realizable as the obstruction class of G connected sum of f's related to (1 - [C])^m. Since it is true for the case here, we can kill the obstruction by taking G connected sum off's     

Equivariant Kasparov Theory and Generalized Homomorphisms

Let G be a locally compact group. We describe elements of KK ^G (A, B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK ^G : It is the universal split exact stable homotopy functor. To describe a Kasparov triple (, , F) for A, B by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form ; and more generally if the group action on A is proper in the sense of Exel and Rieffel.     

EQUIVARIANT SELF EQUIVALENCES OF PRINCIPAL FIBRE BUNDLES

Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known.     

The Reduction of Equivariant Dynamics to the Orbit Space for Compact Group Actions

Symmetry introduces degeneracies in dynamical systems, as well as in bifurcation problems. An “obvious” idea in order to remove these degeneracies is to project the dynamics onto the quotient space obtained by identifying points in phase space which lie in the same group orbits (the so-called orbit space). Unfortunately, several difficulties arise when one tries to implement this idea. First, the orbit space is not, in general a manifold. Second, how does one explicitely realize the orbit space, and how does one compute and analyze the projected dynamics? In this paper I will describe the methods which have been developped in order to answer these questions, and I will show on three examples how they apply. We shall see that, although not always suitable to treat equivariant dynamics, these methods sometimes lead to insightful reductions.     

On the Least Number of Fixed Points of an Equivariant Map

The problem on the least number of fixed points of an equivariant map of a compact polyhedron on which a finite group acts is considered. For such a map, the least number of fixed points and the least number of fixed orbits are estimated in terms of invariants of the type of Nielsen numbers. The estimates obtained are sharp. The results are similar to those of P. Wong, but their assumptions are essentially weaker. Some notations are refined. The proofs are constructive.     

A Strengthening of Resolution of Singularities in Characteristic Zero

Let X be a closed subscheme embedded in a scheme W, smooth overa field k of characteristic zero, and let I (X) be the sheafof ideals defining X. Assume that the set of regular pointsof X is dense in X. We prove that there exists a proper, birationalmorphism, : W_r W, obtained as a composition of monoidal transformations,so that if X_r W_r denotes the strict transform of X W then: (1) the morphism : W_r W is an embedded desingularization ofX (as in Hironaka's Theorem); (2) the total transform of I (X) in factors as a product of an invertible sheaf of ideals L supportedon the exceptional locus, and the sheaf of ideals defining thestrict transform of X (that is, . Thus (2) asserts that we can obtain, in a simple manner, theequations defining the desingularization of X. 2000 MathematicalSubject Classification: 14E15.     

A note on resolution of rational and hypersurface singularities

It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point.

     

Equivariant Casson Invariants and Links of Singularities

This article is a survey of the authors' works over the last few years in trying to understand better various problems about the Casson invariant and Floer homology of links of singularities and, more generally, of graph homology spheres.     

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

Resolution of singularities of real-analytic vector fields in dimension three

Let χ be an analytic vector field defined in a real-analytic manifold of dimension three. We prove that all the singularities of χ can be made elementary by a finite number of blowing-ups in the ambient space. This work has been partially supported by the CNPq/Brasil Grant 205904/2003-5 and Fapesp Grant 02/03769-9.     

Formules de localisation en cohomologie equivariante

In this paper, we develop a method of localization in equivariant cohomology based on the notion of partition of unity cohomology. We apply this method in two cases. In the first case, this method gives a refinement of the localization of Atiyah–Bott and Berline–Vergne (in the frame given by Bismut). After, we consider the Hamiltonian action of a torus, and we realise, following the idea of Witten, the localization on the critical points of the square of the moment map.     

On the Milnor Number of an Equivariant Singularity

Mathematical Notes -     

Equivariant mappings: a new approach in stochastic simulations

Stochastic simulations on manifolds usually are traced back to ⁿ via charts. If a group G is acting on a manifold M and if the respective distribution v is invariant under this group action then in many cases of practical interest there exists a more convenient approach which uses equivariant mappings. The concept of equivariant mappings will be discussed intensively at the instance of the Grassman manifold in which case G equals the orthogonal group. Further advantages of this concept will be demonstrated by applying it to a probabilistic problem from the field of combinatorial geometry.     

D6等变非线性分歧问题的计算

本文研究D6对称群的分歧子群的分类,由此得到相应对称破缺分歧的计算方法,并应用到D6等变的Brusselator反应问题。     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

On the Kernel of the Equivariant Dirac Operator

We prove a vanishing theorem for certain isotypical components of the kernel of the S¹-equivariant Dirac operator with coefficients in an admissible Clifford module. The method is based on changing the metric by a conformal (generally unbounded) factor and studying the effect of this change on the Dirac operator and its kernel. In the cases relevant to S¹-actions we find that the kernel of the new operator is naturally isomorphic to the kernel of the original operator.     

Equivariant Extensions of Categorical Groups

If is a group, then the category of -graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from . In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreiers theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory).Mathematics Subject Classifications (2000) 18D10, 18B40, 20J05, 20J06.Partially supported by MTM2004-01060.     

A Split Exact Sequence of Equivariant K-groups of Virtually Nilpotentt Groups

Let be an extension of a torsion free nilpotent group by a finite group G, of odd order. Then admits a cocompact proper action on ⁿ . This action determines an action of G on a nil-manifold by isometries. In this paper, the equivariant K-theory of the finite group action is studied and it is shown that the forget control map is a split monomorphism.     

On Algorithmic Equi-Resolution and Stratification of Hilbert Schemes

Given an algorithm for resolution of singularities that satisfiescertain conditions (‘a good algorithm’), naturalnotions of simultaneous algorithmic resolution, and of equi-resolution,for families of embedded schemes (parametrized by a reducedscheme T) are defined. It is proved that these notions are equivalent.Something similar is done for families of sheaves of ideals,where the goal is algorithmic simultaneous principalization.A consequence is that given a family of embedded schemes overa reduced T, this parameter scheme can be naturally expressedas a disjoint union of locally closed sets T_j, such that theinduced family on each part T_j is equi-resolvable. In particular,this can be applied to the Hilbert scheme of a smooth projectivevariety; in fact, our result shows that, in characteristic zero,the underlying topological space of any Hilbert scheme parametrizingembedded schemes can be naturally stratified in equi-resolvablefamilies. 2000 Mathematics Subject Classification 14E15, 14D99.