🔢Algebraic Topology Unit 1 – Introduction to Algebraic Topology

Key Concepts and Definitions

• Algebraic topology studies topological spaces using algebraic structures such as groups, rings, and modules
• Topological spaces are mathematical structures that allow for the formal definition of concepts such as convergence, connectedness, and continuity
• Homeomorphisms are continuous functions between topological spaces with continuous inverses, capturing the notion of topological equivalence
• Homotopy is a way to continuously deform one continuous function into another, providing a notion of equivalence between functions
• The fundamental group of a topological space $X$, denoted $\pi_1(X)$, captures information about the loops in $X$ and their deformations
• Homology groups are algebraic invariants that measure the "holes" in a topological space at different dimensions (e.g., connected components, loops, voids)
• Simplicial complexes are combinatorial structures used to represent topological spaces, built from simplices (points, edges, triangles, tetrahedra, etc.)
• The Euler characteristic is a topological invariant that can be computed from a simplicial complex, given by the alternating sum of the number of simplices in each dimension

Fundamental Groups and Homotopy

• The fundamental group $\pi_1(X,x_0)$ of a topological space $X$ at a basepoint $x_0$ consists of homotopy equivalence classes of loops based at $x_0$
  • Loops are continuous functions $f: [0,1] \to X$ with $f(0) = f(1) = x_0$
  • Two loops are homotopic if one can be continuously deformed into the other while keeping the endpoints fixed
• The group operation in $\pi_1(X,x_0)$ is given by concatenation of loops, with the constant loop serving as the identity element
• Homotopy equivalence is a weaker notion than homeomorphism, allowing for spaces to be "equivalent" even if they are not homeomorphic
• The fundamental group is a topological invariant, meaning that homeomorphic spaces have isomorphic fundamental groups
• The fundamental group can be used to distinguish between spaces that are not homeomorphic, such as the circle $S^1$ and the disk $D^2$
  • $\pi_1(S^1) \cong \mathbb{Z}$, while $\pi_1(D^2)$ is trivial
• Higher homotopy groups $\pi_n(X,x_0)$ can be defined using homotopy classes of maps from the $n$-dimensional sphere $S^n$ to $X$, but they are more difficult to compute and interpret than the fundamental group

Covering Spaces

• A covering space of a topological space $X$ is another space $\tilde{X}$ together with a continuous surjective map $p: \tilde{X} \to X$ that is a local homeomorphism
  • Each point in $X$ has an open neighborhood $U$ such that $p^{-1}(U)$ is a disjoint union of open sets in $\tilde{X}$, each mapped homeomorphically onto $U$ by $p$
• Covering spaces can be thought of as "unwrapping" the original space, allowing for a simpler and more tractable representation
• The fiber of a point $x \in X$ is the preimage $p^{-1}(x)$, which consists of all the points in $\tilde{X}$ that map to $x$ under the covering map
• A covering space is called regular (or normal) if the deck transformation group (the group of automorphisms of $\tilde{X}$ that preserve the covering map) acts transitively on each fiber
• The universal cover of a space $X$ is a simply connected covering space $\tilde{X}$, which is unique up to homeomorphism
• The fundamental group of a space $X$ acts on the universal cover $\tilde{X}$ by deck transformations, and the quotient space of this action is homeomorphic to $X$
• Covering spaces are closely related to the fundamental group, as there is a bijective correspondence between connected covering spaces of $X$ and conjugacy classes of subgroups of $\pi_1(X)$

Simplicial Complexes

• A simplicial complex is a collection of simplices (points, edges, triangles, tetrahedra, etc.) that fit together in a combinatorial way to form a topological space
• An $n$-simplex is the convex hull of $n+1$ affinely independent points in a real vector space
  • A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron
• A simplicial complex $K$ is a set of simplices that satisfies two conditions:
  1. Every face of a simplex in $K$ is also in $K$
  2. The intersection of any two simplices in $K$ is either empty or a common face of both simplices
• The dimension of a simplicial complex is the maximum dimension of its simplices
• A simplicial map between simplicial complexes is a function that maps simplices to simplices, preserving the face relations
• Simplicial homology is a method for computing the homology groups of a topological space using a simplicial complex representation
  • The boundary operator $\partial_n$ maps $n$-chains (formal sums of $n$-simplices) to $(n-1)$-chains, encoding the face relations
  • The $n$-th homology group is defined as the quotient of the kernel of $\partial_n$ (the $n$-cycles) by the image of $\partial_{n+1}$ (the $n$-boundaries)
• The Euler characteristic of a simplicial complex $K$ is given by $\chi(K) = \sum_{i=0}^{\dim K} (-1)^i f_i$, where $f_i$ is the number of $i$-dimensional simplices in $K$

Homology Groups

• Homology groups are algebraic invariants that measure the "holes" in a topological space at different dimensions
  • The 0-th homology group $H_0(X)$ measures the connected components of $X$
  • The 1-st homology group $H_1(X)$ measures the 1-dimensional holes (loops) in $X$
  • The 2-nd homology group $H_2(X)$ measures the 2-dimensional holes (voids) in $X$, and so on
• Homology groups are defined using chain complexes, which are sequences of abelian groups (or modules) connected by boundary operators
  • The boundary operators satisfy the property that the composition of any two consecutive boundary operators is zero
• The $n$-th homology group $H_n(X)$ is defined as the quotient of the kernel of the $n$-th boundary operator (the $n$-cycles) by the image of the $(n+1)$-st boundary operator (the $n$-boundaries)
• Homology groups are topological invariants, meaning that homeomorphic spaces have isomorphic homology groups
• Different homology theories can be constructed by using different coefficient groups, such as integer coefficients (yielding $H_n(X; \mathbb{Z})$) or field coefficients (yielding $H_n(X; \mathbb{F})$ for a field $\mathbb{F}$)
• The Betti numbers of a space $X$ are the ranks of its homology groups, denoted $\beta_n(X) = \rank H_n(X)$
  • The Betti numbers provide a coarser invariant than the homology groups themselves but are easier to compute and interpret
• The Euler characteristic can be expressed in terms of the Betti numbers as $\chi(X) = \sum_{n=0}^{\infty} (-1)^n \beta_n(X)$, connecting homology with other topological invariants

Applications in Mathematics

• Algebraic topology has numerous applications within mathematics, providing tools and insights for studying problems in geometry, algebra, and analysis
• In differential geometry, the homology and cohomology of manifolds are used to define and study characteristic classes, which provide obstructions to the existence of certain geometric structures
  • The Euler class and Chern classes are examples of characteristic classes that arise from the topology of vector bundles
• In complex analysis, the homology and cohomology of complex manifolds are used to study the relationships between analytic and topological properties
  • The Dolbeault cohomology groups capture information about the complex structure of a manifold and are related to the theory of sheaves and holomorphic functions
• In algebraic geometry, the étale fundamental group and étale cohomology provide a way to study the topology of algebraic varieties over arbitrary fields
  • The étale fundamental group is an analog of the classical fundamental group that captures arithmetic information about the variety
• In representation theory, the cohomology of groups and Lie algebras plays a key role in understanding the structure and properties of representations
  • Group cohomology can be used to classify extensions of groups and to define important invariants such as the Euler class and the Chern-Simons invariant
• Algebraic topology also has connections to mathematical physics, particularly in the areas of gauge theory, string theory, and topological quantum field theory
  • The path integral formulation of quantum mechanics can be understood in terms of the topology of the configuration space and the fundamental group
  • Topological invariants such as the Jones polynomial and the Witten-Reshetikhin-Turaev invariants arise from the study of knots and 3-manifolds in the context of quantum field theory

Computational Techniques

• Computational algebraic topology is concerned with the development and implementation of algorithms for computing topological invariants and solving problems in algebraic topology
• Simplicial complexes provide a natural framework for computational topology, as they can be represented using data structures such as incidence matrices or adjacency lists
• The boundary operator on a simplicial complex can be represented as a sparse matrix, allowing for efficient computation of homology groups using linear algebra techniques
  • The Smith normal form of the boundary matrix can be used to compute the homology groups and their generators
• Persistent homology is a computational technique that studies the evolution of homology groups across a filtration of a simplicial complex
  • A filtration is a sequence of nested subcomplexes, often obtained by varying a parameter such as a distance threshold or a scalar function value
  • Persistent homology tracks the birth and death of homology classes across the filtration, providing a multi-scale description of the topological features in the data
• The Vietoris-Rips complex is a commonly used construction in computational topology, obtained by taking the clique complex of a proximity graph on a set of points
  • The Vietoris-Rips complex can be used to approximate the shape of a point cloud and to compute its persistent homology
• Discrete Morse theory is a combinatorial analog of classical Morse theory that can be used to simplify the computation of homology groups
  • A discrete Morse function assigns a value to each simplex in a complex, satisfying certain conditions that allow for the collapsing of simplices without changing the homotopy type
  • The critical simplices of a discrete Morse function generate the homology of the complex, often providing a much smaller and more tractable set of generators than the original complex
• Software packages such as GUDHI, Dionysus, and PHAT provide implementations of various algorithms in computational topology, including persistent homology, discrete Morse theory, and simplicial homology

Advanced Topics and Further Reading

• Cohomology is a dual notion to homology, assigning abelian groups (or modules) to a topological space in a contravariant functor
  • Cohomology groups are defined using cochains, which are dual to chains, and coboundary operators, which are dual to boundary operators
  • Cup product and cap product provide additional structure on cohomology and homology, allowing for the definition of ring structures and Poincaré duality
• Spectral sequences are a powerful tool in algebraic topology for computing homology and cohomology groups in situations where direct computation is difficult
  • The Serre spectral sequence relates the homology of a fiber bundle to the homology of its base and fiber
  • The Eilenberg-Moore spectral sequence computes the homology of a pullback or a pushout in terms of the homology of its components
• K-theory is a generalization of linear algebra that studies vector bundles, projective modules, and other categorical constructions
  • The K-groups of a space or a ring provide important invariants that capture information about the stable structure of vector bundles or projective modules
  • Algebraic K-theory extends these ideas to the study of rings and schemes, providing a deep connection between topology, algebra, and geometry
• Homotopy theory is a vast generalization of the ideas of homotopy and homotopy equivalence, encompassing topics such as model categories, $\infty$-categories, and spectra
  • Model categories provide a framework for studying homotopy theories in a general setting, allowing for the development of powerful tools such as the homotopy category and the derived functors
  • $\infty$-categories (or quasi-categories) are a higher categorical analog of topological spaces, where the morphisms form a topological space rather than a set
  • Spectra are a stable homotopy theoretic analog of abelian groups, providing a natural setting for the study of generalized cohomology theories such as K-theory and cobordism
• Further reading:
  • "Algebraic Topology" by Allen Hatcher, a comprehensive introduction to the subject, covering fundamental groups, homology, cohomology, and homotopy theory
  • "Elements of Homotopy Theory" by George W. Whitehead, a classic text on homotopy theory, covering topics such as CW complexes, fibrations, and spectral sequences
  • "Computational Topology: An Introduction" by Herbert Edelsbrunner and John L. Harer, an introduction to the algorithms and data structures used in computational topology, with a focus on persistent homology
  • "Categories and Sheaves" by Masaki Kashiwara and Pierre Schapira, a modern treatment of categorical and homological methods in geometry and analysis, including sheaf theory and derived categories