Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry

The following sections are included:

• Preface
• Contents

One of the main problems, if not “the” problem of topology, is to understand when two spaces X and Y are similar or dissimilar. A related problem is to understand the connectivity structure of a space in terms of its holes and “higher-order” holes. Of course, one has to specify what “similar” means. Intuitively, two topological spaces X and Y are similar if there is a “good” bijection f : X → Y between them. More precisely, “good” means that f is a continuous bijection whose inverse f⁻¹ is also continuous; in other words, f is a homeomorphism. The notion of homeomorphism captures the notion proposed in the mid 1860s that X can be deformed into Y without tearing or overlapping. The problem then is to describe the equivalence classes of spaces under homeomorphism; it is a classification problem…

This chapter is an introduction to the crucial concepts and results of homological algebra needed to understand homology and cohomology in some depth. The two most fundamental concepts of homological algebra are:

• exact sequences.
• chain complexes

Differential forms offer a quick and rather easy approach to the cohomology groups (with real coefficients) of smooth manifolds. This approach was pioneered by Georges de Rham in the early 1930s. If M is a smooth manifold, then there is the de Rham complex

Historically, singular homology and cohomology were developed in the 1940s, starting with a seminal paper of Eilenberg published in 1944 (building up on work by Alexander and Lefschetz among others). It was not the first homology theory. Indeed, simplicial homology emerged in the early 1920s…

In Chapter 4 we introduced the singular homology groups and the singular cohomology groups and presented some of their properties. Historically, singular homology and cohomology was developed in the 1940s, starting with a seminal paper of Eilenberg published in 1944 (building up on work by Alexander and Lefschetz among others), but it was not the first homology theory. Simplicial homology emerged in the early 1920s, more than thirty years after the publication of Poincaré’s first seminal paper on “analysis situ” in 1892. Until the early 1930s, homology groups had not been defined and people worked with numerical invariants such as Betti numbers and torsion numbers. Emmy Noether played a significant role in introducing homology groups as the main objects of study…

Computing the singular homology (or cohomology) groups of a space X is generally very difficult. J.H.C. Whitehead invented a class of spaces called CW complexes for which the computation of the singular homology groups is much more tractable. Roughly speaking, a CW complex X is built up inductively starting with a collection of points, in such a way that if the space X^p has been obtained at stage p, then the space X^p+1 is obtained from X^p by gluing, or as it is customary to say attaching, a collection of closed balls whose boundaries are glued to X^p in a specific fashion. Each space X^p is called a p-skeleton of the space X. Every compact manifold is homotopy equivalent to a CW complex, so the class of CW complexes is quite rich. It also plays an important role in homotopy theory. In this short chapter we describe CW complexes and explain how their homology and cohomology can be computed…

Our goal is to state a version of the Poincaré duality for singular homology and cohomology, one of the most important results about the topology of manifolds. The basic version is that if M is a “nice” n-manifold, then there are isomorphisms

Presheaves and sheaves are two of the indispensable tools used in some of the more advanced parts of algebraic topology and algebraic geometry. Therefore it is important for the reader to be exposed to these concepts as soon as possible. Unfortunately, many presentations of these concepts quickly take a very abstract turn, especially when explaining the process of converting a presheaf into a sheaf…

Given a topological space X and a presehaf ℱ, there is a way of defining cohomology groups ${\stackrel{⌣}{H}}^p(\mathcal{X},\mathcal{F})$ as a limit process involving the definition of some cohomology groups ${\stackrel{⌣}{H}}^p(\mathcal{U},\mathcal{F})$ associated with open covers 𝒰 = (U_j)_j∈J of the space X. Given two open covers 𝒰 and 𝒱, we can define when 𝒱 is a refinement of 𝒰, and then we define the cohomology group ${\stackrel{⌣}{H}}^p(\mathcal{X},\mathcal{F})$ as the direct limit of the directed system of groups ${\stackrel{⌣}{H}}^p(\mathcal{U},\mathcal{F})$. When the presheaf ℱ has some special properties and when nice covers exist, the limit process can be bypassed…

One of the main goals of this chapter is to define the notion of exact sequence of sheaves

The main goal of this chapter is to define the notions of derived functors, δ-functors, and ∂-functors. This machinery plays a crucial role in the definition of sheaf cohomology, an indispensable tool in advanced algebraic geometry (based on schemes) and algebraic topology…

Suppose we have a homology chain complex

In this chapter we apply the results of Sections 11.4 and 11.8 to the case where C is the abelian category of sheaves of R-modules on a topological space X, D is the (abelian) category of abelian groups, and T is the left-exact global section functor Γ(X, –), with Γ(X, ℱ) = ℱ(X) for every sheaf ℱ on X. It turns out that the category of sheaves has enough injectives, thus the right derived functors R^pΓ(X, –) exist, and for every sheaf ℱ on X, the cohomology groups R^pΓ(X, –)(ℱ) are defined. These groups, denoted by H^p(X, ℱ), are called the cohomology groups of the sheaf ℱ (or the cohomology groups of X with values in ℱ)…

Our goal is to present various generalizations of Poincaré duality. These versions of duality involve taking direct limits of direct mapping families of singular cohomology groups which, in general, are not singular cohomology groups. However, such limits are isomorphic to Alexander–Spanier cohomology groups, and thus to Čech cohomology groups. These duality results also require relative versions of homology and cohomology. Thus, in preparation for Alexander–Lefschetz duality we need to define relative Alexander–Spanier cohomology and relative Čech cohomology…

A spectral sequence is a tool of homological algebra whose purpose is to approximate the cohomology (or homology) H(M) of a module M endowed with a family (F^pM)_p∈ℤ of submodules such that F^p+1M ⊆ F^pM for all p and

The following sections are included:

• Bibliography
• Index