PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Topology → Algebraic Topology ↓


Homology Theory


In the world of mathematics, particularly in the field of topology, we explore the nature of spaces and the way they are interconnected. To understand complex structures, we use a powerful tool called homology theory. Homology theory provides a way to look at topological spaces through an algebraic lens, allowing us to understand various 'features' of such spaces, without needing to understand them as geometric objects.

Understanding topological spaces

Before we dive deeper into homology theory, we need to have a basic understanding of what a topological space is. A topological space is a set equipped with a topology, which is essentially a collection of open sets that includes the entire set and the empty set, as well as any unions or finite intersections of these open sets.

The two simplest examples of topological spaces are the line and the plane. More abstractly, topological spaces can include complex structures such as a doughnut shape (torus), a sphere, or even higher-dimensional spaces.

Simplicial complexes and series

One of the entry-level ideas in homology theory is to break down complex space into simple building blocks called simplices (plural of simplex). The simplex is a generalization of a triangle or tetrahedron to any number of dimensions. For example:

  • The 0-simplex is a single point.
  • A 1-simplex is a line segment.
  • The 2-simplex is a triangle.
  • The 3-simplex is a tetrahedron.

Think of any shape as being made by gluing these simple pieces together, similar to the way a Lego structure is assembled from individual bricks.

Chain complex

Once we understand a space in terms of simplices, we can form what we call a chain complex. A chain complex is a sequence of abelian groups connected by boundary operators. In terms of homology theory, this is a way to algebraically understand how these simplices are linked together.

For a given topological space, we deal with chain complexes by considering chains made up of simplices in each dimension and connecting them using a boundary map. This map tells us how a simplex in a given dimension 'bounds' or 'leads to' simplices in a lower dimension.

The boundary operator ∂_n takes an n-dimensional simplex and forms the formal sum of its (n-1)-dimensional faces. For example, the boundary of a line segment (1-simplex) would be a pair of points (0-simplices).

∂_1([v, w]) = [w] - [v]

This operation respects this formula, showing how the edges of the simplices fit together into a tessellated structure. To understand the role of homology theory, it is important that these boundary operations work in an order that converges to zero:

The composition of two limit operators from successive dimensions, say ∂_n o ∂_(n+1), will always give a zero result. This property is important because it allows us to study how these series relate and overlap within the space.

Visual example

A B C

This triangle ABC is a visual representation of the 2-simplex. Its boundary ∂, when applied, will result in edges AB, BC, CA as a formal sum. Each edge can have a boundary resulting in vertices A, B, C.

Homology group

The essence of homology theory lies in determining how to construct a space by looking at these limits and figuring out when these operations result in something 'non-trivial'. To get this algebraic information we use something called homology groups.

The n-th homology group, denoted H_n(X), for a space X finds equivalences: cycles that are not limits against all cycles. Essentially, it counts the n-dimensional 'holes' in a topological space.

These homology groups are invariant under homeomorphisms. This means that if two spaces are topologically identical (even if they look geometrically different), then they will have the same homology groups.

Calculating homology: an example

Let's consider a simple topological space: a loop or a circle. To calculate its homology groups, one should proceed as follows:

1. Simplicial complex

Break the circle into vertices and edges. Perhaps take four vertices and four edges connecting them.

2. Install the chain

Define series groups based on dimensions. Here, since the circle is one-dimensional, you focus on C_0 (vertices) and C_1 (edges).

3. Boundary map

In this case the boundary map identifies how the edges connect two vertices. However, when closed, the sum is zero, exposing a cycle that does not translate to bounding a 'higher' dimensional object because the circle encloses a void.

4. Identifying cycles and limits

The cycle group Z_1 contains loops, like moving around a circle. The limit group B_1 is zero because no enclosed region exists inside a 1-dimensional space.

5. Homology group computation

H_1(S^1) = Z_1 / B_1 = Z_1 / {0} = Z_1

This indicates that the first homology group of the circle is Z, indicating the presence of a one-dimensional 'hole' in the loop.

Beyond basic symmetries: higher dimensions

Once you master the basics, homology theory allows moving into higher dimensions. While the basic idea remains the same, investigating spaces like the torus (donut shape) becomes complex and fascinating.

For the torus, we determine the symmetry by decomposing it into a grid-like surface of square spaces, which reveals both 'horizontal' and 'vertical' loops, identifiable by independent cycles.

Homology of more complex spaces provides practical insights into their organization and also helps in areas such as data analysis (permanent homology) and theoretical physics (exploring behavior in space-time).

Applications of homology theory

Although homology theory originates in pure mathematics, its applications are varied:

  • Biological data analysis: This method provides important information about patterns in genetic data when used as a permanent homology.
  • Robotics and path planning: Whether a robot can pass around obstacles without getting stuck can be modeled using homology groups.
  • Computer graphics: Understanding and rendering complex shapes copied from the real world, incorporating underlying topographical considerations.

Despite its abstract nature, homology theory forms a bridge between topology and algebra, opening up myriad possibilities and further contemplation, making it an enduring area of study for mathematicians globally.

Armed with an understanding of simplices, chain complexes, and homology groups, one can rigorously seek solutions to understand the amazing mathematical symmetry language that underlies the structure of reality.


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Homology Theory

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