相关论文: Cohomology for Anyone

We first introduce global arithmetic cohomology groups for quasi-coherent sheaves on arithmetic varieties, adopting an adelic approach. Then, we establish fundamental properties, such as topological duality and inductive long exact…

Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent…

This article explains basic constructions and results on group algebras and their cohomology, starting from the point of view of commutative algebra. It provides the background necessary for a novice in this subject to begin reading Dave…

The fundamental groupoid of a space becomes enriched over the category of topological spaces when the hom-sets are endowed with topologies intimately related to universal constructions of topological groups. This paper is devoted to a…

Homological algebra is often understood as the translator between the world of topology and algebra. However, this branch of mathematics is worth studying by itself, given that it provides fascinating perspectives about other disciplines,…

We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together…

The notion of meromorphic convexity is defined and studied on complex manifolds. Using this notion, in analogy with Stein manifolds, a new class of complex manifolds, called {\calligra M }-manifolds, is introduced. This is a class of…

Let f: M -> N be an even codimensional immersion between smooth manifolds. We derive an explicit formula for the Pontrjagin numbers and signature of the multiple point manifolds in terms of singular cohomology of M and N, the maps induced…

As a branch of algebraic and differential topology of manifolds, the theory of Morse functions and their higher dimensional versions or fold maps and its application to algebraic and differential topology of manifolds is fundamental,…

The polytopic definition introduced recently describing the topology of manifolds is used to formulate a generating function pertinent to its topological properties. In particular, a polynomial in terms of one variable and a tori underlying…

General relativity does not allow one to specify the topology of space, leaving the possibility that space is multiply rather than simply connected. We review the main mathematical properties of multiply connected spaces, and the different…

This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has…

Moser proved in 1965 in his seminal paper that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group…

A generalization of the triangle inequality is introduced by a mapping similar to a t-conorm mapping. This generalization leads us to a notion for which we use the $\star$-metric terminology. We are interested in the topological space…

Combining geometric group theory techniques with geometric topology tools, we show how primitive cohomologies provide useful insights towards unifying the mathematical formulation of Gromov-Witten invariants. In particular, we emphasise the…

The study of differential forms that are closed but not exact reveals important information about the global topology of a manifold, encoded in the de Rham cohomology groups $H^k(M)$, named after Georges de Rham (1903-1990). This expository…

This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures…

These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…

Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. For example, the automorphism group of a structure is, on top of being a group, a topological group when equipped with the…

There is a need in general relativity for a consistent and useful mathematical theory defining the multiplication of tensor distributions in a geometric (diffeomorphism invariant) way. Significant progress has been made through the concept…