🔢Algebraic Topology Unit 4 – Homotopy and the Fundamental Group

Key Concepts and Definitions

• Homotopy describes a continuous deformation of one function into another, capturing the idea of topological equivalence
• Path homotopy is a homotopy between two paths in a topological space with fixed endpoints
• Loop is a path that starts and ends at the same point (basepoint) in a topological space
• Fundamental group $\pi_1(X,x_0)$ is the set of all homotopy equivalence classes of loops based at a point $x_0$ in a topological space $X$
  • The group operation is concatenation of loops, and the identity element is the constant loop at $x_0$
• Simply connected space is a path-connected space whose fundamental group is trivial (consists only of the identity element)
• Covering space is a topological space that locally looks like the base space but may have a different global structure
  • The fundamental group of the base space acts on the fibers of the covering space

Historical Context and Motivation

• The concept of homotopy was introduced by Poincaré in the early 20th century as a tool to study topological spaces
• Homotopy theory emerged as a way to classify topological spaces by considering continuous deformations between them
• The fundamental group was one of the first homotopy invariants, capturing the notion of "holes" in a space
• Homotopy groups, including the fundamental group, became central objects of study in algebraic topology
  • They provide a bridge between topology and algebra, allowing the use of algebraic tools to study topological problems
• The development of homotopy theory led to significant advances in understanding the structure and properties of topological spaces
• Homotopy ideas have found applications in various areas of mathematics, including complex analysis, differential equations, and physics

Homotopy Basics

• A homotopy between two continuous functions $f,g:X\to Y$ is a continuous map $H:X\times [0,1]\to Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for all $x\in X$
  • The parameter $t\in [0,1]$ can be thought of as "time" during the deformation process
• Homotopic functions are topologically equivalent and share many properties (homotopy invariance)
• Homotopy equivalence is an equivalence relation on topological spaces, leading to the notion of homotopy type
  • Spaces of the same homotopy type have isomorphic homotopy groups and share many topological properties
• Contractible space is a space that is homotopy equivalent to a point (has trivial homotopy groups)
• Homotopy lifting property characterizes the relationship between homotopies in the base space and their lifts to a covering space
• Homotopy extension property allows extending homotopies defined on a subspace to the entire space under certain conditions

The Fundamental Group

• The fundamental group $\pi_1(X,x_0)$ captures the essential loop structure of a topological space $X$ based at a point $x_0$
• Elements of the fundamental group are homotopy equivalence classes of loops, denoted by $[\gamma]$ for a loop $\gamma$
• The group operation is defined by concatenation of loops: $[\gamma]*[\eta]=[\gamma\cdot\eta]$, where $\gamma\cdot\eta$ is the loop obtained by first traversing $\gamma$ and then $\eta$
  • The operation is well-defined on homotopy classes and satisfies the group axioms
• The identity element is the homotopy class of the constant loop at $x_0$, and the inverse of $[\gamma]$ is the class of the loop traversed in the opposite direction
• Isomorphic fundamental groups imply homotopy equivalent spaces, but the converse is not always true
• The fundamental group is a homotopy invariant and can be used to distinguish non-homotopy equivalent spaces
  • For example, $\pi_1(S^1)\cong\mathbb{Z}$, while $\pi_1(\mathbb{R}^2)$ is trivial, showing that $S^1$ and $\mathbb{R}^2$ are not homotopy equivalent

Computation Techniques

• Fundamental groups can be computed using various techniques depending on the nature of the topological space
• For CW complexes, the fundamental group can be computed using the cellular approximation theorem and the attaching maps of cells
  • The presentation of the fundamental group can be derived from the 1-skeleton and the relations imposed by the attaching maps of 2-cells
• Van Kampen's theorem allows computing the fundamental group of a space by decomposing it into simpler subspaces and considering their intersection
  • The fundamental group of the whole space is expressed as a free product with amalgamation of the fundamental groups of the subspaces
• For covering spaces, the fundamental group of the total space is related to the fundamental group of the base space by the lifting correspondence
  • The fundamental group of the total space is a subgroup of the fundamental group of the base space, determined by the covering map
• Seifert-van Kampen theorem is a generalization of van Kampen's theorem for arbitrary open covers of a space
• Homotopy classes of maps from a circle to a space $X$ correspond to conjugacy classes in the fundamental group $\pi_1(X)$

Applications and Examples

• The fundamental group is a crucial tool in classifying and studying topological spaces
• Contractible spaces (simply connected) have trivial fundamental groups, while spaces with non-trivial fundamental groups have "holes" or "loops" that cannot be contracted
  • For example, the fundamental group of a circle $S^1$ is isomorphic to $\mathbb{Z}$, reflecting the winding number of loops around the circle
• The fundamental group can be used to study covering spaces and their properties
  • The universal cover of a space has a trivial fundamental group and is simply connected
• Knot theory utilizes the fundamental group of the complement of a knot to distinguish and classify knots
  • The knot group (fundamental group of the knot complement) is a powerful invariant in knot theory
• The fundamental group plays a role in the classification of surfaces and the study of 3-manifolds
• Homotopy groups, including the fundamental group, are essential in the study of obstruction theory and the existence of certain continuous maps between spaces

Connections to Other Areas

• The fundamental group is related to other homotopy groups, such as higher homotopy groups $\pi_n(X)$ (homotopy classes of maps from $n$-spheres to $X$)
• Homology groups, another set of topological invariants, are abelianizations of homotopy groups and capture different aspects of the structure of a space
  • The first homology group $H_1(X)$ is the abelianization of the fundamental group $\pi_1(X)$
• The fundamental group is connected to the theory of covering spaces and the classification of covering spaces over a given base space
• Group theory, especially free groups and group presentations, is closely tied to the study of fundamental groups
• The fundamental group appears in the study of vector bundles and principal bundles over topological spaces
• Homotopy theory has connections to other areas of mathematics, such as algebraic geometry (étale fundamental group) and mathematical physics (topological quantum field theories)

Advanced Topics and Extensions

• Higher homotopy groups $\pi_n(X)$ generalize the fundamental group and capture higher-dimensional hole structure of a space
  • These groups are abelian for $n\geq 2$ and can be challenging to compute explicitly
• Homotopy groups of spheres $\pi_n(S^k)$ are a central object of study in algebraic topology and have a rich and complex structure
• The Hurewicz theorem relates homotopy groups and homology groups, providing a bridge between these two invariants
• Homotopy excision and the Blakers-Massey theorem are powerful tools for computing homotopy groups of certain spaces
• Spectral sequences, such as the Serre spectral sequence and the Atiyah-Hirzebruch spectral sequence, are advanced computational tools in homotopy theory
• Homotopy limits and colimits, as well as homotopy pullbacks and pushouts, provide a framework for studying homotopy-theoretic constructions
• Simplicial sets and simplicial homotopy theory offer a combinatorial approach to homotopy theory, allowing for explicit computations and constructions
• Model categories and $\infty$-categories provide a general framework for studying homotopy theories and higher categorical structures