Bredon 层上同调

Sheaf theory for stacks in manifolds and twisted cohomology for S^1-gerbes

This is the first of a series of papers on sheaf theory on smooth and topological stacks and its applications. The main result of the present paper is the characterization of the twisted (by a closed integral three-form) de Rham complex on a manifold. As an object in the derived category it will be related with the push-forward of the constant sheaf from a S^1-gerbe with Dixmier-Douady class represented by the three-form. In order to formulate and prove this result we develop in detail the foundations of sheaf theory for smooth stacks.

环面完全交上的层上同调限制公式

A Sheaf Cohomology Restriction Formula on Toric Complete Intersections

本文针对环面簇完全交上的一类向量丛证明了层上同调限制公式，将完全交上的上同调积转化为外围空间上的计算，并可用于研究经典层上同调环的量子修正。

A new description of equivariant cohomology for totally disconnected groups

We consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the the…

Periodic twisted cohomology and T-duality

The initial motivation of this work was to give a topological interpretation of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary coefficients. To this end we develop a sheaf theory in the context of locally compact topological stacks with emphasis on the construction of the sheaf theory operations in unbounded derived categories, elements of Verdier duality and integration. The main result is the construction of a functorial periodization functor associated to a U(1)-gerbe. As applications we verify the $T$-duality isomorphism in periodic twisted cohomology and in pe…

Twisted equivariant K-theory, groupoids and proper actions

In this paper we define twisted equivariant K-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite CW-complexes with equivariant stable projective bundles. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups.

Godement resolution and operad sheaf homotopy theory

We show how to induce products in sheaf cohomology for a wide variety of coefficients: sheaves of dg commutative and Lie algebras, symmetric Omega-spectra, filtered dg algebras, operads and operad algebras.

A1-invariants in Galois cohomology and a claim of Morel

We establish a variant of the splitting principle of Garibaldi-Merkurjev-Serre for invariants taking values in a strictly homotopy invariant sheaf. As an application, we prove the folklore result of Morel that pi_0 of the motivic localization of the group completion of the stack of finite étale schemes is given by the sheaf of unramified Grothendieck-Witt groups.

Differential cohomology theories as sheaves of spectra

We show that every sheaf on the site of smooth manifolds with values in a stable (infinity,1)-category (like spectra or chain complexes) gives rise to a differential cohomology diagram and a homotopy formula, which are common features of all classical examples of differential cohomology theories. These structures are naturally derived from a canonical decomposition of a sheaf into a homotopy invariant part and a piece which has a trivial evaluation on a point. In the classical examples the latter is the contribution of differential forms. This decomposition suggest a natural scheme to analyse …

The equivariant K-theory of a cohomogeneity-one action

We compute the equivariant complex K-theory ring of a cohomogeneity-one action of a compact Lie group at the level of generators and relations and derive a characterization of K-theoretic equivariant formality for these actions. Less explicit expressions survive for a range of equivariant cohomology theories including Bredon cohomology and Borel complex cobordism. The proof accordingly involves elements of equivariant homotopy theory, representation theory, and Lie theory. Aside from analysis of maps of representation rings and heavy use of the structure theory of compact Lie groups, a more cu…

Hochschild cohomology and Atiyah classes

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification…

Unramified logarithmic Hodge-Witt cohomology and $\mathbb{P}^1$-invariance

Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for any $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers $G$. This generalize a result of Binda-Rülling-Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge-Witt cohomology is a $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers.

Vanishing Theorems for Real Algebraic Cycles

We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of the motivic cohomology of a real variety as homotopy groups of the complex of averaged equidimensional cycles. We also establish an equivariant Poincare dua…