Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Topology → Algebraic Topology ↓


Fundamental Groups


Algebraic topology is a field of mathematics that uses tools from abstract algebra to study topological spaces. One of the central concepts in this field is the fundamental group. To understand fundamental groups, we first need to familiarize ourselves with some essential ideas in topology.

Basic concepts in topology

Topology is the study of spaces, shapes, and their properties, through continuous deformations such as stretching or bending, but not through tearing or sticking. Imagine that a space is like a rubber sheet; you can bend, stretch, and fold it, but you cannot tear or stick parts off.

Fundamental concepts for the study of topology are open and closed sets, continuity, and homeomorphism (which is an equivalence relation that shows that two spaces are topologically the same).

Paths and loops

Before diving into the fundamental group, it is important to understand what paths and loops are. A path in a topological space (X) is a continuous map:

    (gamma: [0, 1] rightarrow X)
    (gamma: [0, 1] rightarrow X)

Here, ([0, 1]) is the closed interval of real numbers from 0 to 1, and (gamma(t)) gives the position at any point (t) along the path from the start ((gamma(0))) to the end ((gamma(1))).

A loop is a special type of path where the start and end points are the same: (gamma(0) = gamma(1)). The common point is often called the "base point".

Symmetry of paths

Two paths (gamma_1) and (gamma_2) are called isotopic if one can be continuously deformed into the other while remaining within the space. Formally, they are isotopic if there exists a continuous function:

    H: [0, 1] times [0, 1] rightarrow X
    H: [0, 1] times [0, 1] rightarrow X

Such that for every (t in [0, 1]):

  • (h(s, 0) = gamma_1(s))
  • (h(s, 1) = gamma_2(s))
  • (H(0, t) = gamma_1(0) = gamma_2(0)) and (H(1, t) = gamma_1(1) = gamma_2(1))

Then, the homotopy (H) defines a continuous deformation from the path (gamma_1) to the path (gamma_2).

Fundamental group definition

The fundamental group, denoted by (pi_1(X, x_0)), is a topological invariant that encodes information about the fundamental shapes or holes of a topological space (X). Each element of the fundamental group is an equivalence class of loops based on a point (x_0) in the space.

The main multiplication operation in the fundamental group is 'combination of loops'. Given two loops (alpha) and (beta), their product is a loop consisting of passing through (alpha) followed by (beta).

The formal definition of a fundamental group is as follows:

    (pi_1(X, x_0) = { gamma | text{loops at} x_0 }/sim)
    (pi_1(X, x_0) = { gamma | text{loops at} x_0 }/sim)

where (sim) denotes homotopy equivalence.

Example: fundamental group of a circle

One of the most classic examples is the circle, denoted by (S^1). Consider a loop on a circle based on a point. Intuitively, these loops can rotate any number of times around the circle, either clockwise or counterclockwise.

Each loop can be denoted by an integer indicating how many times it goes around the circle. This integer forms the fundamental group of the circle:

    (pi_1(S^1) cong mathbb{Z})
    (pi_1(S^1) cong mathbb{Z})

Here, (mathbb{Z}) denotes the set of integers under addition.

For example, a loop that rotates once clockwise corresponds to (1), and a loop that rotates twice counterclockwise corresponds to (-2).

Visual example: path on a torus

The torus is another interesting example for understanding fundamental groups. Unlike a circle, the torus has two different types of loops: one goes around the hole of the donut, and the other loops through the hole.

A B

In the above illustration:

  • The path marked with (a) represents looping around the hole.
  • The path marked with (b) represents looping through the hole.

In algebraic terms, the fundamental group of a torus can be represented as:

    (pi_1(T^2) = mathbb{Z} times mathbb{Z})
    (pi_1(T^2) = mathbb{Z} times mathbb{Z})

This means that the group is generated by two independent loops (such as (a) and (b)).

Properties of fundamental groups

When studying fundamental groups, it is important to consider several properties:

  • Invariance under homeomorphisms: If two spaces (X) and (Y) are homeomorphic, then their fundamental groups at corresponding points are isomorphic.
  • Dependence on the base point: although the choice of the base point may change the real elements of the fundamental group, the group structure is independent of this, provided the space is path-connected.
  • Functionality: continuous maps between spaces yield group isomorphisms between their fundamental groups.

Applications and significance

Fundamental groups are a powerful tool in topology and mathematics in general because they provide information about the shape and structure of spaces. They allow mathematicians to distinguish between different types of spaces based on their structure.

For example, although a sphere and a disk are both topological spaces, their fundamental groups are different, reflecting their different nature. This is essential in areas such as manifold theory and complex analysis.

Closing thoughts

The study of fundamental groups forms the backbone for many areas of mathematics, especially those dealing with shapes, spaces, and transformations. The ability to classify spaces through algebraic tools provides a powerful lens through which topologists can understand complexity in a structured way.

Understanding fundamental groups lays the groundwork for more advanced topics in algebraic topology, such as higher homotopy groups and homology, which further expand our understanding of spatial and geometric structure.


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Fundamental Groups

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