🧬Cohomology Theory Unit 11 – Manifolds and Algebraic Topology Applications

Key Concepts and Definitions

• Manifold: A topological space that locally resembles Euclidean space near each point
  • Locally Euclidean: Every point has a neighborhood homeomorphic to an open subset of Euclidean space
  • Examples include circles, spheres, tori, and projective spaces
• Algebraic topology: The study of topological spaces using algebraic structures such as groups, rings, and modules
  • Assigns algebraic invariants to topological spaces to distinguish them up to homeomorphism or homotopy equivalence
• Cohomology: A contravariant functor from the category of topological spaces to the category of abelian groups or rings
  • Measures the "holes" in a topological space
  • Cohomology groups are denoted as $H^n(X; R)$, where $X$ is the space and $R$ is the coefficient ring
• Exact sequence: A sequence of homomorphisms between algebraic objects (e.g., groups, modules) such that the image of each homomorphism is equal to the kernel of the next
  • Short exact sequences: $0 \to A \to B \to C \to 0$, where $A$ injects into $B$, and $C$ is the quotient of $B$ by the image of $A$
• Cup product: A bilinear operation on cohomology classes that gives the cohomology ring structure
  • For classes $\alpha \in H^p(X; R)$ and $\beta \in H^q(X; R)$, their cup product is $\alpha \smile \beta \in H^{p+q}(X; R)$
• Poincaré duality: A fundamental relationship between the homology and cohomology of orientable closed manifolds
  • For an orientable closed $n$-manifold $M$, $H^k(M; R) \cong H_{n-k}(M; R)$

Manifold Fundamentals

• Smooth manifolds: Manifolds with a differentiable structure, allowing for calculus on the space
  • Smooth functions: Maps between smooth manifolds that are infinitely differentiable
  • Tangent spaces: Vector spaces attached to each point of a smooth manifold, representing the space of directional derivatives
• Riemannian manifolds: Smooth manifolds equipped with a Riemannian metric, which assigns an inner product to each tangent space
  • Allows for the definition of lengths, angles, and curvature on the manifold
• Orientability: A property of manifolds related to the existence of a consistent choice of orientation for all tangent spaces
  • Orientable manifolds: Manifolds that admit a global orientation (e.g., spheres, tori)
  • Non-orientable manifolds: Manifolds that do not admit a global orientation (e.g., Möbius strip, projective plane)
• Submanifolds: A subset of a manifold that is itself a manifold with the subspace topology
  • Examples include circles as submanifolds of the plane, and spheres as submanifolds of Euclidean space
• Immersions and embeddings: Different ways of mapping one manifold into another
  • Immersion: A differentiable map with everywhere injective derivative
  • Embedding: An injective immersion that is a homeomorphism onto its image

Algebraic Topology Basics

• Homotopy: A continuous deformation of one map into another
  • Homotopy equivalence: Two spaces are homotopy equivalent if there exist maps between them that are homotopy inverses of each other
• Fundamental group: The group of homotopy classes of loops based at a point in a topological space
  • Measures the "1-dimensional holes" in a space
  • Denoted as $\pi_1(X, x_0)$, where $X$ is the space and $x_0$ is the basepoint
• Homology: A covariant functor from the category of topological spaces to the category of abelian groups or modules
  • Measures the "holes" in a topological space
  • Homology groups are denoted as $H_n(X; R)$, where $X$ is the space and $R$ is the coefficient ring
• Simplicial complexes: A combinatorial structure used to represent topological spaces
  • Consists of vertices, edges, triangles, and higher-dimensional simplices
  • Allows for the computation of homology and cohomology using linear algebra
• CW complexes: A topological space constructed by attaching cells of increasing dimension
  • Provides a convenient way to build spaces with desired homotopy or homology properties
• Eilenberg-Steenrod axioms: A set of axioms characterizing homology and cohomology theories
  • Includes homotopy invariance, exactness, and excision

Cohomology Theory Overview

• Cochain complexes: A sequence of abelian groups or modules connected by coboundary maps
  • The cohomology groups are the quotients of the kernel of one coboundary map by the image of the previous
• Relative cohomology: The cohomology of a pair $(X, A)$, where $A$ is a subspace of $X$
  • Measures the cohomology of $X$ "relative to" the cohomology of $A$
  • Fits into a long exact sequence with the absolute cohomology groups of $X$ and $A$
• Mayer-Vietoris sequence: A long exact sequence relating the cohomology of a space to the cohomology of two overlapping subspaces
  • Useful for computing the cohomology of spaces that can be decomposed into simpler pieces
• Cohomology operations: Maps between cohomology groups that are natural with respect to continuous maps
  • Examples include the cup product, Steenrod squares, and Bockstein homomorphisms
• Characteristic classes: Cohomology classes associated with vector bundles over a space
  • Measure the twisting and non-triviality of the bundle
  • Examples include Stiefel-Whitney classes, Chern classes, and Pontryagin classes
• Spectral sequences: Algebraic tools for computing homology or cohomology groups in stages
  • Arise from filtered complexes or double complexes
  • Examples include the Serre spectral sequence and the Atiyah-Hirzebruch spectral sequence

Applications to Manifolds

• De Rham cohomology: A cohomology theory for smooth manifolds based on differential forms
  • Isomorphic to singular cohomology with real coefficients for smooth manifolds
  • Allows for the use of differential geometry techniques in cohomology computations
• Hodge theory: The study of harmonic forms on Riemannian manifolds
  • Hodge decomposition: Every differential form can be uniquely written as the sum of a harmonic form, an exact form, and a co-exact form
  • Hodge theorem: On a compact oriented Riemannian manifold, the de Rham cohomology is isomorphic to the space of harmonic forms
• Intersection theory: The study of how submanifolds intersect in a larger manifold
  • Intersection product: A bilinear operation on homology classes dual to the cup product on cohomology
  • Poincaré duality: Relates the intersection product of submanifolds to the cup product of their Poincaré dual cohomology classes
• Characteristic classes of manifolds: Cohomology classes associated with the tangent bundle or other natural vector bundles of a manifold
  • Stiefel-Whitney classes: Measure the orientability and parallelizability of a manifold
  • Pontryagin classes: Measure the non-triviality of the tangent bundle and are related to the curvature of Riemannian metrics
• Obstruction theory: The study of when continuous maps between manifolds can be extended or lifted
  • Obstruction cocycles: Cohomology classes that measure the obstruction to extending or lifting a map
  • Whitehead tower: A sequence of spaces approximating a given space, with successive obstructions lying in higher homotopy groups

Advanced Techniques and Methods

• Sheaf cohomology: A generalization of cohomology theory to the setting of sheaves on topological spaces
  • Sheaves: A way of assigning algebraic data (e.g., abelian groups, rings) to open sets of a space in a compatible way
  • Čech cohomology: A sheaf cohomology theory based on open covers of a space
• Morse theory: The study of the topology of manifolds using critical points of smooth functions
  • Morse functions: Smooth functions on a manifold with non-degenerate critical points
  • Morse inequalities: Relate the number of critical points of a Morse function to the Betti numbers (ranks of homology groups) of the manifold
• Floer homology: A family of homology theories for manifolds based on the Morse theory of the action functional on the loop space
  • Variants include Hamiltonian Floer homology, Lagrangian Floer homology, and Heegaard Floer homology
  • Applications to symplectic geometry, low-dimensional topology, and knot theory
• Equivariant cohomology: The cohomology of spaces with group actions, taking into account the symmetries of the space
  • Borel construction: A way to construct a space with a free group action from a space with any group action
  • Equivariant characteristic classes: Characteristic classes for equivariant vector bundles
• K-theory: A generalized cohomology theory based on vector bundles or projective modules
  • Algebraic K-theory: The K-theory of rings or schemes, related to the study of projective modules
  • Topological K-theory: The K-theory of topological spaces, related to the study of vector bundles

Problem-Solving Strategies

• Identify the type of space: Determine if the space is a manifold, CW complex, or has other special properties
  • Use the properties of the space to guide the choice of cohomology theory or computational techniques
• Compute cohomology groups: Use the tools and techniques of algebraic topology to compute the cohomology groups of the space
  • Simplicial or cellular cohomology: Compute cohomology using a simplicial complex or CW complex structure
  • Mayer-Vietoris sequence: Decompose the space into simpler pieces and use the Mayer-Vietoris sequence to compute cohomology
  • Spectral sequences: Use spectral sequences to compute cohomology in stages, especially for fibrations or filtered spaces
• Interpret the results: Understand what the cohomology groups tell you about the space
  • Betti numbers: The ranks of the cohomology groups give information about the number of "holes" in each dimension
  • Cup product structure: The cup product gives information about the ring structure of the cohomology and the intersection properties of submanifolds
  • Characteristic classes: The presence of non-trivial characteristic classes indicates non-trivial vector bundles or obstructions to certain structures
• Apply duality and functoriality: Use the relationships between different cohomology theories and the functorial properties of cohomology to gain further insight
  • Poincaré duality: Relate the cohomology of a manifold to its homology
  • De Rham theorem: Relate the de Rham cohomology of a smooth manifold to its singular cohomology
  • Functoriality: Use the behavior of cohomology under maps between spaces to compare cohomology groups or transfer information between spaces
• Seek connections to other areas: Look for ways to apply cohomological techniques to problems in other areas of mathematics
  • Algebraic geometry: Sheaf cohomology and intersection theory are important tools in the study of algebraic varieties
  • Differential geometry: De Rham cohomology and characteristic classes are closely related to curvature and the geometry of manifolds
  • Mathematical physics: Cohomological methods appear in gauge theory, string theory, and other areas of theoretical physics

Real-World Applications

• Sensor networks: Algebraic topology can be used to analyze the coverage and connectivity of sensor networks
  • Homology groups can detect coverage holes and redundancies in the network
  • Persistent homology can be used to study the evolution of the network over time
• Image analysis: Cohomological methods can be applied to problems in image processing and computer vision
  • Persistent homology can be used to identify and classify features in images
  • Sheaf cohomology can be used to study the global structure of images and detect inconsistencies or anomalies
• Data analysis: Topological data analysis (TDA) uses techniques from algebraic topology to study the shape and structure of high-dimensional datasets
  • Mapper algorithm: A method for visualizing and summarizing the topological structure of a dataset
  • Persistent homology: A way to quantify the multi-scale topological features of a dataset and identify significant features
• Robotics and motion planning: Cohomological methods can be used to study the configuration spaces of robots and plan motions
  • Homology groups can detect obstacles and holes in the configuration space
  • Cohomology classes can represent potential motion plans or strategies
• Network science: Algebraic topology can be used to study the structure and dynamics of complex networks
  • Persistent homology can identify cycles and higher-order structures in networks
  • Sheaf cohomology can model the flow of information or resources through a network
• Computational biology: Topological methods are increasingly being applied to problems in biology and biomedicine
  • Persistent homology can be used to study the structure of biomolecules and identify binding sites or functional regions
  • Sheaf cohomology can model the organization and interactions of biological systems at different scales