Examples of derived functors (singular cohomology, De Rham cohomology)

Introduction

Derived functors are central to homological algebra, providing a systematic way to extend standard functors to the derived category. They arise when a functor fails to be exact, necessitating the construction of resolutions to compute derived values. In this article, we explore two key examples: singular cohomology and De Rham cohomology, illustrating their foundational roles in topology and differential geometry. These theories demonstrate how derived functors bridge abstract algebraic structures with geometric intuition, offering tools to study topological spaces and manifolds through algebraic methods.

Singular Cohomology

Singular cohomology is a fundamental tool in algebraic topology, defined as the derived functor of the global section functor on the category of singular chains. Given a topological space $ X $, the singular chain complex $ C_*(X) $ is constructed from continuous maps from standard simplices to $ X $. The homology groups $ H_n(X) $ capture the "holes" of dimension $ n $ in $ X $, while the cohomology groups $ H^n(X) $ are derived from the singular cochain complex $ C^n(X) $ via the alternating sum $ \sum (-1)^{n} \text{Hom}(C_n, \mathbb{Z}) $.

The derived functors of the global section functor yield the singular cohomology groups, which are computed via the cochain complex $ \mathbb{Z}[\text{Sing}(X)] $. This construction is exact in the category of cochain complexes, ensuring that the cohomology groups are well-defined. Singular cohomology is invariant under homotopy equivalences, making it a powerful invariant for topological spaces. Its connection to homotopy theory is profound, as the universal coefficient theorem relates cohomology groups to homology groups and torsion coefficients, providing a bridge between algebraic and geometric structures.

De Rham Cohomology

De Rham cohomology, on the other hand, is a differential geometric approach to cohomology, defined using differential forms on smooth manifolds. Given a smooth manifold $ M $, the De Rham complex $ \Omega^(M) $ consists of differential forms of all degrees, with the exterior derivative $ d $ as the differential operator. The De Rham cohomology groups $ H^n_{\text{dR}}(M) $ are the derived functors of the global section functor on the category of differential forms, computed via the cochain complex $ \Omega^(M) $.

This theory is closely tied to the Poincaré lemma, which states that every closed form is exact on a contractible manifold. However, for general manifolds, this fails, necessitating the use of de Rham cohomology to capture the topology of the space. The De Rham theorem establishes a profound connection between de Rham cohomology and singular cohomology, showing that they are isomorphic for smooth manifolds. This equivalence allows the use of differential geometric techniques to study topological invariants, such as the Euler characteristic, via the Hodge decomposition theorem.

Applications and Properties

Both singular cohomology and De Rham cohomology share properties such as being invariant under homotopy equivalences and being compatible with homotopy equivalences in their respective categories. Singular cohomology is particularly useful in algebraic topology, where it provides a way to classify topological spaces up to homotopy equivalence. De Rham cohomology, by contrast, is a tool in differential geometry, enabling the study of manifolds through the lens of differential forms and their topological invariants.

The interplay between these cohomology theories is further highlighted by the fact that singular cohomology is a special case of De Rham cohomology when the manifold is a CW complex. This relationship underscores the universality of derived functors in mathematics, as they provide a common framework for studying cohomology in both algebraic and geometric contexts.

Conclusion

Derived functors, exemplified by singular cohomology and De Rham cohomology, are pivotal in advancing our understanding of topological and geometric structures. Singular cohomology offers a robust algebraic framework for analyzing the homotopy properties of spaces, while De Rham cohomology provides a differential geometric perspective, linking topology to differential forms. Together, these theories illustrate the versatility of derived functors in mathematics, enabling the computation of cohomology groups through resolutions and the study of topological invariants. Their applications span fields from algebraic topology to differential geometry, demonstrating the enduring significance of derived functors in modern mathematics.