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 ↓


Fundamental Groups


In the field of algebraic topology, the concept of fundamental group emerges as an important and informative tool. It provides information about the shape, structure, and nature of spaces using algebraic tools. A fundamental group reveals intuitive properties of loops within a space, revealing important information about that space.

The fundamental group is centered around the idea of continuous deformation of paths. Instead of focusing on the minutiae of a space, it abstracts the fundamental features by looking at how these paths can be transformed. In essence, it is the search for homotopy classes of loops within a space that retain their topological essence despite possible bending or stretching, but do not break or stick.

Basic concepts

Before delving deeper into the structure of the fundamental group, it is necessary to understand some key concepts like paths, loops, and homotopy:

Paths and loops

A path in a topological space (X) is a continuous function (f) from the unit interval ([0, 1]) to (X), i.e., (f: [0, 1] to X). The points (f(0)) and (f(1)) are known as the initial and terminal points of the path, respectively. A loop is a special type of path where the initial and terminal points coincide, i.e., (f(0) = f(1)).

f: [0, 1] to X, quad text{such that} quad f(0) = f(1)

Homotopy

The concept of homotopy captures the idea of continuously deforming one path into another without breaking it. Formally, two paths (f_0) and (f_1) are homotopic if there exists a continuous function (H: [0, 1] times [0, 1] to X) such that:

H(t, 0) = f_0(t), quad H(t, 1) = f_1(t), quad forall t in [0, 1]

Moreover, for loop homotopy, the initial and terminal points are preserved throughout the deformation:

H(0, s) = f_0(0) = f_1(0), quad H(1, s) = f_0(1) = f_1(1), quad forall s in [0, 1]

Checking that two loops (f_0) and (f_1) are isotopic involves observing that one loop transforms into the other smoothly and without any jumps.

h(t, s)

In the above illustration, the blue line represents a loop, and the red dashed line represents the homotopy, which smoothly transforms this loop into a trivial point.

Definition of fundamental group

The fundamental group, denoted by (pi_1(X, x_0)), is the group of homotopy classes of loops starting and ending at (x_0) over a point (x_0 in X) for a space (X). The group operation is concatenation, where two loops are connected end-to-end.

pi_1(X, x_0) = {[f] : f text{ is a loop at } x_0}

Let (f) and (g) be two loops based on (x_0). Their combination (f * g) is defined as:

(f * g)(t) = begin{cases} f(2t) & text{if } 0 leq t leq 0.5 \ g(2t - 1) & text{if } 0.5 < t leq 1 end{cases}

This combination operation follows the group properties: associativity, identity (stationary loop at (x_0)), and inverse (moving the loop in the opposite direction).

Visual example: circle

Consider the fundamental group of the circle (S^1). Intuitively, the loop around the circle can be wrapped any number of times, including zero or negative, which corresponds to a rotation in the opposite direction.

pi_1(S^1) cong mathbb{Z}

Here, (mathbb{Z}) denotes the set of integers, which highlights that each loop around the circle is determined by the number of rotations around (S^1).

S1

The green outline of the circle represents the location (S^1) and the purple path represents a possible loop, rotating once around (S^1).

Additional text examples

Let us consider some more text examples to strengthen our understanding:

  • Simply connected spaces: Consider (mathbb{R}^n) for (n geq 2). The fundamental group (pi_1(mathbb{R}^n, x_0)) is trivial (consisting only of the identity element) since any loop can be compactified to a point.
  • Torus: The fundamental group of a torus, which can be thought of as a rectangle with opposite edges identified, is (mathbb{Z} times mathbb{Z}). This is due to the ability to rotate around the torus in two independent directions.
  • Path-connectedness: For path-connected spaces, the choice of base point does not matter, that is, the fundamental group does not change fundamentally with a different choice of base point.

Properties and applications of fundamental groups

Fundamental groups serve as a bridge between topology and group theory, unlocking tools for analyzing complex spaces through algebraic methods. They tell whether a space is simply connected and provide insight into possible coverings or symmetric actions on spaces.

A notable property is that continuous functions between spaces induce isomorphisms between their fundamental groups. A function (f: X to Y) induces an isomorphism:

f_*: pi_1(X, x_0) to pi_1(Y, f(x_0))

It connects functions between spaces and mappings between corresponding algebraic structures, and demonstrates the interaction between geometry and algebra.

Closing thoughts

The journey into the concept of fundamental groups highlights the beauty of algebraic topology, as they navigate through the abstract anatomy of space, understanding specific features through homotopy and path analysis. As we have explored, fundamental groups have the potential to unlock greater understanding, uniting seemingly disparate mathematical domains.

Whether through primitive spaces such as the circle or complex spaces such as high-dimensional manifolds, fundamental groups consistently reveal essential properties, and serve as the cornerstone of topological exploration.


PHD → 3.2.1


U
username
0%
completed in PHD

← Prev (3.2)
Algebraic Topology
Next (3.2.2) →
Homotopy Theory

Comments 



Fundamental Groups

https://www.buddymath.com/phd/3.2.1?bmal=en