Definition of Cohomology and Methods of Computation

Introduction to Cohomology

Cohomology is a fundamental tool in algebraic topology that extends the concept of homology by assigning algebraic invariants to topological spaces. While homology measures the cycles and boundaries within a space, cohomology provides a way to study the global properties of a space through its cochains, which are functions defined on the space's cells or vertices. This extension allows cohomology to capture information about the space's structure that homology alone cannot detect, such as the existence of non-trivial maps or the behavior of higher-dimensional structures. Cohomology is particularly useful in distinguishing between spaces that are homotopy equivalent but not homeomorphic, and in computing topological invariants like Euler characteristics and Poincaré polynomials.

Fundamental Concepts and Definitions

Cohomology is built upon the concept of chain complexes, which are sequences of abelian groups connected by homomorphisms. For a topological space $ X $, a chain complex $ (C_*, \partial) $ is constructed by assigning a group $ C_n(X) $ to each dimension $ n $, where $ C_n(X) $ represents the free abelian group generated by the $ n $-dimensional simplices of $ X $. The boundary map $ \partial: C_n(X) \to C_{n-1}(X) $ is defined such that $ \partial(\sigma) = \sum_{i=0}^n (-1)^i \sigma_i $, where $ \sigma_i $ are the faces of the simplex $ \sigma $. The cohomology group $ H_n(X) $ is then defined as the quotient $ C_n(X) / \text{Im}(\partial_{n+1}) $, capturing the algebraic structure of the space.

To generalize homology, cohomology introduces cochain complexes, where the groups $ C^n(X) $ are dual to $ C_(X) $. A cochain map $ \phi: C^n(X) \to C^n(Y) $ induces a morphism $ \phi^: H^n(Y) \to H^n(X) $, preserving the cohomological invariants. The coboundary operator $ \delta: C^n(X) \to C^{n+1}(X) $ is defined analogously to the boundary operator but in reverse, ensuring that the cohomology groups satisfy the universal coefficient theorem.

Axioms of Cohomology

The cohomology theory is governed by a set of axioms that ensure consistency and compatibility with homology. One of the key axioms is the universal coefficient theorem, which states that for a cochain complex $ (C^*, \delta) $, the cohomology groups $ H^n(X) $ are isomorphic to the homology groups $ H^n(X) $ of the corresponding chain complex. This theorem provides a bridge between cohomology and homology, allowing the computation of cohomology groups from homology groups.

Another critical axiom is the Mayer-Vietoris sequence, which provides a way to compute cohomology groups of a space by decomposing it into two subspaces. If $ X = U \cup V $, then the Mayer-Vietoris sequence relates the cohomology groups of $ X $, $ U $, and $ V $, offering a powerful tool for analyzing complex spaces. This sequence is particularly useful in cases where $ U $ and $ V $ are compatible, allowing for the cancellation of overlapping cohomological information.

Methods of Computation

The computation of cohomology groups often relies on specific techniques tailored to the structure of the space. One of the most common methods is homology, which can be computed using the chain complex and the boundary operator. For a simply connected space, the homology groups are trivial, but for more complex spaces, the computation involves resolving the space into simpler components. The homology of a space can be calculated using the following steps:

1. Construct the chain complex $ (C_*, \partial) $ for the space.
2. Compute the homology groups $ H_n(X) $ by taking the quotient of the kernel of $ \partial $ by the image of $ \partial_{n+1} $.
3. Use the universal coefficient theorem to relate homology groups to cohomology groups.

In addition to homology, spectral sequences are a powerful tool for computing cohomology. Spectral sequences provide a way to approximate cohomology groups by breaking down the computation into manageable parts. The spectral sequence is constructed using the associated graded objects of the cochain complex and the filtration of the space. This method is particularly useful for spaces with high-dimensional structures, where direct computation is infeasible.

Another important method is the Künneth theorem, which provides a way to compute the cohomology of a product space $ X \times Y $ in terms of the cohomology of $ X $ and $ Y $. The theorem states that the cohomology groups of the product space are isomorphic to the tensor product of the cohomology groups of the individual spaces, provided certain conditions are met. This theorem is widely used in algebraic topology to simplify the computation of cohomology for complex spaces.

Applications and Examples

Cohomology has numerous applications in mathematics and physics, particularly in the study of topological spaces and their invariants. One of the most significant applications is in the classification of topological spaces, where cohomology groups provide a way to distinguish between spaces that are homotopy equivalent but not homeomorphic. For example, the cohomology of a sphere $ S^n $ is trivial for $ n \geq 1 $, while the cohomology of a projective space is non-trivial, reflecting its complex structure.

In physics, cohomology is used to study the topology of spacetime, particularly in the context of gauge theories. The co