Cohomology

Cohomology is a mathematical concept that plays an important role in algebraic topology, providing a robust way to study the properties of topological spaces. Its roots can be found in the notion of homology, but it has evolved to encompass a broader set of ideas that extend into other areas of mathematics as well. In this talk, we will dive deeper into what cohomology is, how it differs from homology, and look at some fascinating examples and applications.

Introduction to cohomology

Cohomology provides a way to associate algebraic structures, such as groups or rings, with topological spaces. These structures are derived from the idea of chains, cycles, and boundaries on a space. By understanding these associations, one can infer properties of spaces, such as connectivity, compactness, and dimensionality.

The basics: chains, cycles, and limits

Before venturing into cohomology, it is important to understand the building blocks of symmetry:

• Chains: These are formal sums of simplics, which are the basic building blocks of topological spaces. For example, in a triangle, the edges and vertices can serve as 1-chains and 0-chains, respectively.
• Cycle: A cycle is a series whose limit is zero. Think of it like a closed loop in space.
• Boundaries: Boundaries refer to the boundaries of a higher-dimensional series. For example, the boundary of a 2-dimensional triangle would be its 1-dimensional edges.

Homology vs. cohomology

While homology views cycles as boundaries, cohomology flips this view, focusing instead on cochains, cocycles, and coboundaries. The essence of cohomology is contained in these concepts:

• Cochains: maps that assign algebraic values (such as integers or real numbers) to series.
• Cocycles: Co-chains with no co-boundary, analogous to cycles in homology.
• Colimits: coseries that "fill in" coseries, much like limits fill in cycles in homology.

Construction of cohomology

To construct cohomology, we start with a sequence of cochain complexes. These sequences associate chains with groups, ultimately defining the cohomology group. Here's how it works step-by-step:

1. Start with a space X and define its chain complex, which is usually denoted as follows:

    ... → C_{n+1}(X) → C_n(X) → C_{n-1}(X) → ...

2. Construct the cochain complex as follows:

    ... ← Hom(C_{n+1}(X), G) ← Hom(C_n(X), G) ← Hom(C_{n-1}(X), G) ← ...

   Here, G is an abelian group (like the integers or real numbers).

3. Define the co-boundary operator, which maps one cochain to another.

4. The n-th cohomology group is defined as the kernel of a colimit operator modulo the image of the previous colimit operator:

 H^n(X; G) = Ker(d^n) / Im(d^{n-1})

Example: cohomology of a circle

Let's take the example of a familiar figure: the circle S^1.

The structure of the circle is simple. In terms of cohomology, consider:

• For n = 0, every point in S^1 can be continuously mapped to an integer, giving H^0(S^1; ℤ) ≈ ℤ.
• For n = 1, every loop around the circle corresponds to a cocycle. Since there are no higher-dimensional properties, we have that H^1(S^1; ℤ) ≈ ℤ.
• For n > 1, the higher cohomology groups are zero: H^n(S^1) = 0 for n > 1.

This highlights a key property of cohomology: its ability to capture and distinguish features of topological spaces across dimensions.

Applications of cohomology

Cohomology has abundant applications both inside and outside topology. Its usefulness ranges from solving equations in algebraic geometry to impressive applications in theoretical physics.

Algebraic geometry

In algebraic geometry, cohomology helps solve polynomial equations and investigate the geometry of solutions at higher dimensions. For example, the Lefschetz fixed-point theorem uses cohomology to calculate the fixed points of continuous maps.

Topological invariance

One of the main uses of cohomology is to prove topological invariance - showing that the cohomology groups of a space do not change under homeomorphisms. This explains why spaces that look different can still have the same underlying properties.

Theoretical physics

In physics, especially in string theory and quantum field theory, cohomology is used to calculate and understand the properties of fields and particles. De Rham cohomology, which connects differential forms to topology, is used to analyze differential equations and symmetries in these domains.

Example: Poincaré duality

An essential result in topology is the Poincaré duality theorem. It shows the relationship between symmetry and cohomology of dual manifolds. In the simplest terms, it states that there is an isomorphism between the k-th symmetry group and the (nk)-th cohomology group of an n-dimensional manifold. For example:

If M is a closed oriented manifold of dimension n, then:

H_k(M) ≈ H^{nk}(M)

This correspondence allows moving from a homology to a cohomology approach, thereby expanding the toolkit for understanding topological spaces.

Visual explanation of cohomology

Visualizing cohomology can greatly improve our understanding of its abstract concepts. Consider a simple finite complex, such as a triangle with vertices, edges, and one face. We can visualize cochain groups and coboundary operators as follows:

This visualization can help us think of cochains as maps that map onto vertices (C_0), edges (C_1), and faces (C_2), as well as show how coboundary operators move "up" these dimensions.

The future of cohomology

Cohomology remains a vibrant area of research, with many unresolved problems and potential applications. Efforts to extend cohomology theories, such as derived categories and deformed sheaves, continue to uncover new connections in mathematics and theoretical physics.

Thus, learning cohomology is not only a gateway to better understanding algebraic topology, but also an essential step towards discovering modern mathematical theories.

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