arXiv research

Summing Hamiltonian manifolds with a common submanifold.

Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data…

This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature.

The paper extends Euler class theory to measurable cocycles.

We construct the odd symplectic structure and the equivariant even (pre)symplectic one from it on the space of differential forms on the Riemann manifold. The Poincare -- Cartan like invariants of the second structure define the equivariant generalizations of the Euler classes on the surfaces.

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with some finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. Here we discuss a universal additive topological invariant of V-manifolds: the univers…

We show that J. Lott's equivariant higher analytic torsion for compact group actions depends only on the equivariant Euler characteristic.

Let G be a compact Lie group. Let M be a smooth G-manifold and V --> M be an oriented G-equivariant vector bundle. One defines the spaces of equivariant forms with generalized coefficients on V and M. An equivariant Thom form $θ$ on V is a compactly supported closed equivariant form such that its integral along the fib…

We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced b…

We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fe…

In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem.

Paper uses algebraic methods to prove an equivariant Poincaré-Hopf theorem.

The purpose of this paper is to study finite dimensional equivariant moduli problems from the viewpoint of stratification theory. We show that there exists a stratified obstruction system for a finite dimensional equivariant moduli problem. In addition, we define a coindex for a G-vector bundle which is determined by t…

Develops a new theory of localization in algebraic geometry.

Paper develops equivariant basic cohomology for Lie groupoids.

There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher…

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here a so-called enhanced Burnside ring $\widehat{B}(G)$ of a finite group $G$…

A spine is constructed for a non-orientable surface's decorated Teichmüller space.

Study on cohomology of singular foliations with localization results.

For every abelian compact Lie group A, we prove that the homotopical A-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the A-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism r…

Let G be a finite group and let M be a G-manifold. We introduce the concept of generalized orbifold invariants of M/G associated to an arbitrary group Gamma, an arbitrary Gamma-set, and an arbitrary covering space of a connected manifold Sigma whose fundamental group is Gamma. Our orbifold invariants have a natural and…

We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fiber…

New evidence supports the Euler class one conjecture for tight contact structures.

In this paper, we study Euler classes in groups of homeomorphisms of Seifert fibered 3-manifolds. We show that, in contrast to the familiar Euler class for $\mathrm{Homeo}_0(S^1)^δ$, these Euler classes for $\mathrm{Homeo}_0(M^3)^δ$ are unbounded classes. In fact, we give examples of flat topological M bundles over a g…

We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential Topology/Geometry, including orbibundles, differential forms, integration and (equivariant) …

We prove that the mapping class group $Γ_{g,n}$ for surfaces of negative Euler characteristic has a cofinite universal space $\E$ for proper actions (the resulting quotient is a finite $CW$-complex). The approach is to construct a truncated Teichmueller space $\T_{g,n}(ε)$ by introducing a lower bound for the length of…

Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.

Given a principal bundle G \rightarrow P \rightarrow B (each being compact, connected and oriented) and a G-invariant metric h^{P} on P which induces a volume form μ^{P}, we consider the group of all unimodular automorphisms SAut(P,μ^{P}):={\varphi\in Diff(P) | \varphi^{*}μ^{P}=μ^{P} and \varphi is G-equivariant} of P …

Extends Euler class result to symplectic group.

We study the existence of $S^1$-equivariant characteristic classes on certain natural infinite rank bundles over the loop space $LM$ 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 techniq…

It is shown that in a tower of coverings the regularized determinant of a generalized Laplacian converges to the $L^2$-determinant. This shows generic nontriviality of analytic torsion or regularized determinants since the $L^2$-counterparts are easier to compute. We further have an "Euler product expansion" for regula…

The octonionic flag manifold $Fl(\mathbb{O})$ is the space of all pairs in $\mathbb{O}P^2\times \mathbb{O}P^2$ (where $\mathbb{O}P^2$ denotes the octonionic projective plane) which satisfy a certain "incidence" relation. It comes equipped with the projections $π_1,π_2 : Fl(\mathbb{O})\to \mathbb{O}P^2$, which are $\mat…

Study Euler class of surface bundles with nontrivial results.

The paper proves a criterion for virtual Euler class one in hyperbolic 3-manifolds.

The paper disproves a conjecture about 3D manifolds using even lattice points.

This paper proves a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

The paper defines and proves non-triviality of volume and Euler classes in bounded cohomology of transformation groups.

Using $S^1$-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn-Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of $S^2\times S^3$ a…

Study on Euler class and flux homomorphisms for non-orientable surfaces.

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 t…

A formula calculates the Euler class of foliations using dual graphs.

The paper finds small exotic 4-manifolds with free abelian groups.

An equivariant bundle gerbe à la Meinrenken over a $G$-manifold $M$ is known to be a special type of $S^1$-gerbe over the differentiable stack $[M/G]$. We prove that the natural morphism relating the Cartan and simplicial models of equivariant cohomology in degree 3 maps the Dixmier-Douady class of an equivariant bundl…

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaa…

Extends Euler class formula to general connections with metric.

New techniques create irreducible 4-manifolds with specific properties.

The Whitehead link exterior lacks most Euler class taut foliations.