🔢Algebraic Topology Unit 6 – Homology and Cohomology Theories

Key Concepts and Definitions

• Homology studies the properties of topological spaces that are preserved under continuous deformations (homotopies)
  • Captures "holes" or "voids" in a space
  • Provides a way to distinguish between spaces that cannot be continuously deformed into one another
• Cohomology is the dual notion to homology, assigning algebraic objects (abelian groups) to a topological space
  • Cohomology groups measure the "twistedness" or "obstructions" in a space
• Chain complexes are sequences of abelian groups connected by boundary operators satisfying $\partial_{n+1} \circ \partial_n = 0$
  • Homology groups are defined as the quotient of kernel and image of boundary operators
• Exact sequences are sequences of homomorphisms between abelian groups where the image of each homomorphism equals the kernel of the next
  • Short exact sequences ($0 \to A \to B \to C \to 0$) and long exact sequences are particularly useful in computations
• Functoriality expresses the idea that homology and cohomology are not just assignments of groups to spaces, but also induce maps between these groups when there is a map between spaces

Historical Context and Development

• Homology theory originated in the work of Henri Poincaré in the late 19th century
  • Poincaré introduced the concept of Betti numbers, which count the number of "holes" of each dimension in a space
• Emmy Noether's work in the early 20th century laid the foundation for the modern algebraic approach to homology
  • Noether introduced the concept of homology groups and the use of chain complexes
• In the 1920s and 1930s, topologists such as Solomon Lefschetz and Heinz Hopf further developed homology theory
  • Lefschetz introduced the concept of relative homology and the Lefschetz fixed-point theorem
  • Hopf introduced cohomology as the dual notion to homology
• The Eilenberg-Steenrod axioms, introduced in the 1940s, provided a unified framework for homology and cohomology theories
  • These axioms characterize homology and cohomology as functors satisfying certain properties (homotopy invariance, exactness, excision)
• The development of category theory in the mid-20th century had a significant impact on homology and cohomology
  • Homology and cohomology are now understood as functors between categories of topological spaces and categories of algebraic objects (abelian groups, modules, etc.)

Homology Groups and Their Properties

• For a topological space $X$, the $n$-th homology group $H_n(X)$ is an abelian group that captures information about the $n$-dimensional "holes" in $X$
  • $H_0(X)$ measures the connectedness of $X$
  • $H_1(X)$ detects loops or "1-dimensional holes" in $X$
  • Higher homology groups ($H_n(X)$ for $n \geq 2$) detect higher-dimensional holes
• Homology groups are homotopy invariant: if $X$ and $Y$ are homotopy equivalent, then $H_n(X) \cong H_n(Y)$ for all $n$
• The rank of the $n$-th homology group, called the $n$-th Betti number $\beta_n(X)$, provides a numerical invariant of the space $X$
  • $\beta_0(X)$ counts the number of connected components of $X$
  • $\beta_1(X)$ counts the number of independent loops in $X$
• Homology groups satisfy the Eilenberg-Steenrod axioms, which include homotopy invariance, exactness, and excision
• The Mayer-Vietoris sequence relates the homology of a space $X$ to the homology of subspaces $A$ and $B$ such that $X = A \cup B$
  • Provides a powerful tool for computing homology groups of spaces that can be decomposed into simpler pieces

Cohomology Groups and Their Properties

• Cohomology groups $H^n(X)$ are the dual notion to homology groups, capturing information about the "twistedness" or "obstructions" in a space
  • Can be defined using cochains (maps from chains to a coefficient group) and coboundary operators
  • For a field $k$, the cohomology group $H^n(X; k)$ is the dual vector space to the homology group $H_n(X; k)$
• Like homology, cohomology is homotopy invariant and satisfies the Eilenberg-Steenrod axioms
• The cup product gives cohomology the structure of a graded ring
  • Allows for the definition of cohomology operations, such as Steenrod squares in mod 2 cohomology
• Poincaré duality relates the homology and cohomology of a compact, oriented manifold $M$
  • $H^k(M) \cong H_{n-k}(M)$, where $n$ is the dimension of $M$
• Cohomology has important applications in obstruction theory and the study of vector bundles
  • The Euler class of a vector bundle is a cohomology class that measures the "twistedness" of the bundle

Computational Techniques and Examples

• The homology groups of a simplicial complex can be computed using the simplicial chain complex and its boundary operators
  • Reduces the computation to linear algebra (kernels and images of matrices)
• The Mayer-Vietoris sequence is a powerful tool for computing homology groups of spaces that can be decomposed into simpler pieces
  • Example: the homology groups of the torus can be computed by decomposing it into two cylinders
• The homology groups of a CW complex can be computed using the cellular chain complex
  • Each cell of dimension $n$ contributes a generator to the $n$-th chain group
  • Boundary operators are determined by the attaching maps of the cells
• Cohomology groups can be computed using the dual cochain complex or by applying the Universal Coefficient Theorem
  • The Universal Coefficient Theorem relates homology and cohomology with coefficients in a field
• Spectral sequences provide a systematic way to compute homology and cohomology in certain situations
  • The Serre spectral sequence relates the homology of a fiber bundle to the homology of its base and fiber
  • The Leray-Serre spectral sequence is used to compute the cohomology of a fibration

Applications in Topology and Beyond

• Homology and cohomology are essential tools for classifying topological spaces
  • Spaces with different homology or cohomology groups cannot be homeomorphic
  • Example: the torus and the Klein bottle have different homology groups, so they are not homeomorphic
• The Brouwer fixed point theorem states that any continuous map from a ball to itself has a fixed point
  • Can be proved using homology (the Lefschetz fixed-point theorem is a generalization)
• The Borsuk-Ulam theorem states that any continuous map from the n-sphere to Euclidean n-space maps some pair of antipodal points to the same point
  • Has applications in combinatorics and graph theory (e.g., the Lovász-Kneser theorem)
• Cohomology is used to define and study characteristic classes of vector bundles
  • The Stiefel-Whitney classes (in mod 2 cohomology) and Chern classes (in integer cohomology) are important invariants
  • Characteristic classes have applications in differential geometry and physics (e.g., the Chern-Weil theory of connections on principal bundles)
• Persistent homology is a recent development that applies homology to data analysis and machine learning
  • Studies the evolution of homology groups as a parameter (such as a distance threshold) varies
  • Has applications in image analysis, sensor networks, and computational biology

Connections to Other Mathematical Theories

• Homology and cohomology are deeply connected to homotopy theory
  • The Hurewicz theorem relates the first non-trivial homotopy group of a space to its first homology group
  • The cohomology ring of a space is a homotopy invariant that captures more information than homology groups alone
• K-theory, which studies vector bundles and projective modules, is closely related to cohomology
  • The Chern character is a ring homomorphism from K-theory to rational cohomology
• De Rham cohomology, which is defined using differential forms on a smooth manifold, is isomorphic to singular cohomology with real coefficients
  • This isomorphism is a key result in the development of differential topology
• Sheaf theory provides a general framework for studying cohomology theories
  • Cohomology groups can be defined as the derived functors of the global sections functor on the category of sheaves
• Homological algebra abstracts the ideas of homology and cohomology to the study of chain complexes and their homology in arbitrary abelian categories
  • Provides a unified framework for studying various cohomology theories in algebra, topology, and geometry

Advanced Topics and Current Research

• Spectral sequences are a powerful computational tool in homology and cohomology theory
  • The Adams spectral sequence is used to compute stable homotopy groups of spheres
  • The Atiyah-Hirzebruch spectral sequence relates the generalized cohomology theories of a space (such as K-theory) to its ordinary cohomology
• Equivariant homology and cohomology study spaces with group actions
  • The Borel construction and the Serre spectral sequence are key tools in this setting
  • Has applications in the study of transformation groups and the topology of manifolds
• Intersection homology, introduced by Mark Goresky and Robert MacPherson, is a homology theory for singular spaces
  • Assigns homology groups to a space based on the intersection properties of chains with the singular strata
  • Satisfies a generalized form of Poincaré duality for singular spaces
• Floer homology is a family of homology theories defined using the solutions to certain partial differential equations
  • Includes Lagrangian Floer homology, Heegaard Floer homology, and instanton Floer homology
  • Has applications in symplectic topology, low-dimensional topology, and the study of knots and 3-manifolds
• Topological data analysis applies ideas from algebraic topology, including homology and persistence, to the study of large datasets
  • Has led to the development of new computational methods for studying the "shape" of data
  • Has applications in various fields, such as biology, neuroscience, and materials science