We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai--Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique equivariant refinement of Hopkins' "theorem of the cube" construction of the ${\rm MString}$-orientation of elliptic cohomology.
In this short paper, we use Tannakian reconstruction techniques to prove a result that explains how to reconstruct the stacky approach to de Rham cohomology from the classical theory algebraic de Rham cohomology via an application of the adjoint functor theorem.
This is a companion paper our previous submission "\infty-categories monoidales rigides et caracteres de Chern", in which we give a comparison between functions on the derived loop space of a smooth scheme of caracteristic zero, and its algebraic de Rham cohomology. As a consequence we obtain functorial and multiplicative versions of the HKR decomposition theorem relating Hochschild homology and Hodge cohomology.
We show that the cohomology group of the equivariant simplicial de Rham complex is isomorphic to the cohomology group of the classifying space of a semi-direct product group.
We show that when a simplicial Lie group acts on a simplicial manifold $\{X_*\}$, we can construct a bisimplicial manifold and the de Rham complex on it. This complex is quasi-isomorphic to the equivariant simplicial de Rham complex on $\{X_*\}$ and its cohomology group is isomorphic to the cohomology group of the fat realization of the bisimplicial manifold. We also exhibit a cocycle in the equivariant simplicial de Rham complex.
In the present paper, we provide the foundation of a $G$-equivariant Cech-de Rham theory for a compact Lie group $G$ by using the Cartan model of equivariant differential forms. Our approach is quite elementary without referring to the Mathai-Quillen framework. In particular, by a direct computation, we give an explicit formula of the $U(l)$-equivariant Thom form of C^l, which deforms the classical Bochnor-Martinelli kernel. Also we discuss a version of equivariant Riemann-Roch formula.
This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.
We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky's h-topology. Our results extend to the algebraic de Rham complex as well. Our main technique is to extend Cech cohomology of hypercovers to arbitrary local acyclic fibrations of simplicial presheaves.
In this note we show that the positivity property of the equivariant signature of the loop space, first observed in [MS1] in the case of the even-dimensional projective spaces, is valid for Picard number 2 toric varieties. A new formula for the equivariant signature of the loop space in the case of a toric spin variety is derived.
In this paper we present the notion of de Rham cohomology with compact support for diffeological spaces. Moreover we shall discuss the existence of three long exact sequences. As a concrete example, we show that long exact sequences exist for the de Rham cohomology of diffeological subcartesian spaces.
We construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincaré Duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth man…
The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincaré duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with …
Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham cohomology in terms of a suitable relative differential graded Ext, defined on the appropriate category of (G,Cg)-modules. The meaning of "relative" is made precise via the dual standard construction associated with the monad involving the aforementioned functor and the associated f…
O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André-Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and \v Cech cohomology, which have been used for instance to prove Pillay's conjecture concerning definably compact groups. In the present paper we elaborate an o-minimal de Rham cohomology theory for abstract-definable $\mathc…
Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the $L^{2}$ de Rham and Hodge cohomology and the intersection cohomology of $X$ associated to some perversities. More precisely, to a kind of metric which we call \emph{quasi edge with weights}, we associate two general perversities in the sense of G. Friedman, $p_{g}$ and its dual $q_{g}$. We then show that the absolute $L^{2}$ Hodge cohomology is isomorphic to the maximal $L^{2}$ de Rham cohomology and this is in turn isomorph…
The purpose of this paper is to present a ``Cech-De Rham'' model for the cohomology of leaf spaces. This model lends itself to the construction of characteristic classes (in the cohomology of classifying spaces) by explicit geometrical constructions which are immediate extensions of the standard constructions for manifolds. In particular we rediscover (and explain) the Thurston formula and the Bott formulas for cocycles on diffeomorphism groups. We also use the Cech-De Rham model to explicitly describe the relation between the cohomology of the classifying space, the basic cohomology, and the …
There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech--de Rham spectral sequence by making use of the {\it factor map} which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus $T_θ$ is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivi…
This work is dedicated to some new exotic homological constructions associated with the different Morse-type inequalities for differential forms and vector fields. It contains also survey of ideas developed by the present author in 1986 for this goal.
For a stratified pseudomanifold $X$, we have the de Rham Theorem $ \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} - \per{p}}{*}{X}, $ for a perversity $\per{p}$ verifying $\per{0} \leq \per{p} \leq \per{t}$, where $\per{t}$ denotes the top perversity. We extend this result to any perversity $\per{p}$. In the direction cohomology $\mapsto$ homology, we obtain the isomorphism $$ \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} -\per{p}}{*}{X,\ib{X}{\per{p}}}, $$ where $ {\displaystyle \ib{X}{\per{p}} = \bigcup\_{S \preceq S\_{1} \atop \per{p} (S\_{1})< 0}S = \bigcup\_{\per{p} (S)< 0}\bar{S}.} $ In the…
Mishchenko-Oliveira proved the piecewise smooth cohomology and Lie algebroid cohomology of a Lie algebroid on a combinatorial compact manifold are isomorphic. In this paper, we describe an application of that result locally trivial Lie groupoids on combinatorial manifolds to obtain an isomorphism between de Rham cohomology and piecewise de Rham cohomology of left invariants forms.
For T an abelian compact Lie group, we give a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology. In the appendix we give applications of this by extending results of Chang-Skjelbred and Goresky-Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory.
Let $\mathbb{F}_{Θ}=G/P_{Θ}$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $% P_{Θ}$. This is a closed subgroup of $G$ determined by a subset $% Θ$ of simple restricted roots of $\mathfrak{g}=Lie(G)$. This paper computes the second de Rham cohomology group of $\mathbb{F}_Θ$. We prove that it zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2(\mathbb{F}_Θ,\mathbb{R})$ through the Weil construction of closed 2-forms as characteristic classes of principal fiber bundles. The starting point is the computatio…
We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form $ω$ (a multi-valued functional). We focus mainly our attention to the cohomology of the de Rham complex of a compact manifold $M^n$ with a deformed differential $d_ω=d +λω$. Using Witten's approach to the Morse theory one can estimate the number of critical points of $ω$ in terms of the cohomology of deformed de Rham complex with sufficiently large values of $λ$ (torsion-free Novikov's inequalities). We show that for an interesting class of solvmanifolds this coh…
We study the dg-algebra $Ω^\bullet_{A|\mathbb{R}}$ of algebraic de Rham forms of a real soft function algebra $A$, i.e., the algebra of global sections of a soft subsheaf of $C_X$, the sheaf of continuous functions on a space $X$. We obtain a canonical splitting $\mathrm H ^n (Ω^\bullet_{A|\mathbb{R}}) \cong \mathrm H ^n (X,\mathbb{R})\oplus V$, where $V$ is some vector space. In particular, we consider the cases $A=C(X)$ for $X$ a compact Hausdorff space and $A = C^\infty (X)$ for $X$ a compact smooth manifold. For the algebra $\mathrm{PPol}_K (|K|)$ of piecewise polynomial functions on a pol…
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.
On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both the ordinary smooth Deligne cohomology and the ordinary equivariant cohomology. Using the cohomology group, we classify equivariant circle bundles with connection, and equivariant gerbes with connection.
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.
