The aim of this paper is to construct and examine three candidates for local-to-global spectral sequences for the cohomology of diagrams of algebras with directed indexing. In each case, the $E^2$ -terms can be viewed as a type of local cohomology relative to a map or an object in the diagram.
The extended persistence diagram introduced by Cohen-Steiner, Edelsbrunner, and Harer is an invariant of real-valued continuous functions, which are $\mathbb{F}$-tame in the sense that all open interlevel sets have degree-wise finite-dimensional cohomology with coefficients in a fixed field $\mathbb{F}$. We show that relative interlevel set cohomology (RISC), which is based on the Mayer--Vietoris pyramid by Carlsson, de Silva, and Morozov, categorifies this invariant. More specifically, we define an abelian Frobenius category $\mathrm{pres}(\mathcal{J})$ of presheaves, which are presentable in…
Ginzburg, Kapranov and Vasserot conjectured the existence of equivariant elliptic cohomology theories. In this paper, to give a description of equivariant spectra of the theories, we study an intermediate theory, quasi-elliptic cohomology. We formulate a new category of orthogonal G-spectra and construct explicitly an orthogonal G-spectrum of quasi-elliptic cohomology in it. The idea of the construction can be applied to a family of equivariant cohomology theories, including Tate K-theory and generalized Morava E-theories. Moreover, this construction provides a functor from the category of glo…
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen's iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long l…
In this paper we develop the definition of a global orthogonal spectrum and its unitary version. It relates $G-$equivariant spectra by equivariant weak equivalence in a coherent way. This category of global spectra has a model structure Quillen equivalent to the global model structure on orthogonal spectra. We also show that there is a large family of equivariant cohomology theories, including quasi-elliptic cohomology, that can be globalized in the new context. Starting from one global ring spectrum, we can construct infinitely many distinct global ring spectra. Moreover, in light of the resu…
Differential Characters on Orbifolds and String Connections I
本文引入了整体商orbifold的Cheeger-Simons上同调。我们证明了Orbifold的Cheeger-Simons上同调同构于它的Beilinson-Deligne上同调。此外，我们构造了一个字符串连接（à la Segal）从一个全球gerbe与连接的循环orbifold，完善相应的微分字符。
The author explains local and global model structures on higher orbifolds which are truncated étale differentiable higher stacks, and discuss the application of the model structures to quantum cohomology of higher and derived orbifolds.
Locally variational systems of differential equations on smooth manifolds, having certain de Rham cohomology group trivial, automatically possess a global Lagrangian. This important result due to Takens is, how-ever, of sheaf-theoretic nature. A new constructive method of finding a global Lagrangian for second-order ODEs on 2-manifolds is described on the basis of solvability of exactness equation for Lepage 2-forms, and the top-cohomology theorems. Examples from geometry and mechanics are discussed.
