🧬Cohomology Theory Unit 1 – Simplicial and Singular Homology

Key Concepts and Definitions

• Homology studies topological spaces by associating algebraic objects (homology groups) to them
• Simplicial complexes combinatorial objects built from simplices (points, edges, triangles, tetrahedra, etc.) glued together in a specific way
• Singular simplices continuous maps from the standard n-simplex to a topological space
• Chain complexes algebraic structures consisting of abelian groups (or modules) connected by boundary operators satisfying $\partial_{n-1} \circ \partial_n = 0$
• Homology groups measure the "holes" in a topological space by considering cycles (elements with zero boundary) modulo boundaries (elements that are the boundary of a higher-dimensional object)
• Betti numbers rank of the homology groups, providing a numerical invariant of the space
• Exact sequences algebraic tools for studying the relationship between homology groups of related spaces, such as long exact sequences and Mayer-Vietoris sequences

Simplicial Complexes and Simplicial Homology

• Simplicial complexes built from simplices (vertices, edges, triangles, tetrahedra, etc.) glued together according to certain rules
  • Each face of a simplex in the complex must also be in the complex
  • The intersection of any two simplices must be a face of both simplices or empty
• Oriented simplicial complex assigns a consistent orientation to each simplex, allowing for the definition of boundary operators
• Simplicial k-chain formal sum of k-dimensional oriented simplices with coefficients in an abelian group (usually integers or a field)
• Boundary operator $\partial_k$ maps k-chains to (k-1)-chains by taking the alternating sum of the faces of each simplex
  • Satisfies $\partial_{k-1} \circ \partial_k = 0$, forming a chain complex
• Simplicial homology groups $H_k(X)$ defined as the quotient of k-cycles (ker $\partial_k$) by k-boundaries (im $\partial_{k+1}$)
• Simplicial homology computable using linear algebra techniques (Smith normal form, reduction algorithms) due to the finite nature of simplicial complexes

Singular Homology: Basics and Construction

• Singular n-simplex continuous map $\sigma: \Delta^n \to X$ from the standard n-simplex to a topological space $X$
• Singular n-chain formal sum of singular n-simplices with coefficients in an abelian group
• Boundary operator $\partial_n$ maps singular n-chains to singular (n-1)-chains by taking the alternating sum of the faces of each singular simplex
  • Satisfies $\partial_{n-1} \circ \partial_n = 0$, forming a chain complex
• Singular homology groups $H_n(X)$ defined as the quotient of n-cycles (ker $\partial_n$) by n-boundaries (im $\partial_{n+1}$)
• Singular homology functorial associates homomorphisms between homology groups to continuous maps between spaces
• Homotopy invariance singular homology invariant under homotopy equivalence of spaces
• Excision theorem relates the homology of a space to the homology of a subspace and its complement

Homology Groups and Their Computation

• Homology groups $H_n(X)$ measure the "n-dimensional holes" in a topological space $X$
  • $H_0(X)$ measures the connected components of $X$
  • $H_1(X)$ measures the 1-dimensional holes or "loops" in $X$
  • $H_2(X)$ measures the 2-dimensional holes or "voids" in $X$
• Betti numbers $\beta_n$ rank of the homology group $H_n(X)$, providing a numerical invariant of the space
• Euler characteristic $\chi(X) = \sum_{n=0}^{\infty} (-1)^n \beta_n$, a topological invariant that can be computed combinatorially for simplicial complexes
• Homology with coefficients homology groups can be defined with coefficients in any abelian group, leading to different information about the space
  • Homology with integer coefficients most common, captures torsion information
  • Homology with field coefficients (e.g., $\mathbb{Q}$, $\mathbb{F}_p$) simpler computationally, torsion information lost
• Computational tools for homology include the Smith normal form algorithm for integer coefficients and reduction algorithms for field coefficients

Comparison of Simplicial and Singular Homology

• Simplicial homology computable for simplicial complexes, singular homology defined for arbitrary topological spaces
• Simplicial approximation theorem every continuous map between simplicial complexes is homotopic to a simplicial map
  • Allows for the comparison of simplicial and singular homology
• Subdivision operators refine simplicial complexes, leading to a direct system of simplicial homology groups
• Singular homology isomorphic to the direct limit of simplicial homology groups under subdivision
  • Justifies the use of simplicial homology as a computationally tractable approximation to singular homology
• Eilenberg-Steenrod axioms characterize singular homology as the unique functor satisfying certain properties (homotopy invariance, excision, dimension axiom)
  • Simplicial homology also satisfies these axioms for simplicial complexes

Applications and Examples

• Homology of surfaces classify surfaces by their genus (number of handles) using homology groups
  • Sphere: $H_0 = \mathbb{Z}$, $H_1 = 0$, $H_2 = \mathbb{Z}$
  • Torus: $H_0 = \mathbb{Z}$, $H_1 = \mathbb{Z}^2$, $H_2 = \mathbb{Z}$
• Homology of cell complexes CW complexes provide a flexible way to construct spaces with prescribed homology groups
  • Real projective plane: $H_0 = \mathbb{Z}$, $H_1 = \mathbb{Z}_2$, $H_2 = 0$
  • Klein bottle: $H_0 = \mathbb{Z}$, $H_1 = \mathbb{Z} \oplus \mathbb{Z}_2$, $H_2 = 0$
• Homology of simplicial complexes Vietoris-Rips and Čech complexes used in topological data analysis to study the shape of point cloud data
  • Persistent homology studies how homology groups change as the scale parameter of these complexes varies
• Homology of chain complexes algebraic topology provides tools for studying the homology of chain complexes arising in other areas of mathematics (e.g., Khovanov homology in knot theory)

Theoretical Foundations and Proofs

• Proof of the Eilenberg-Steenrod axioms for singular homology relies on the homotopy invariance and excision properties
  • Homotopy invariance proved using the prism operator, which relates cycles in homotopic spaces
  • Excision proved using the barycentric subdivision operator and the Lebesgue number lemma
• Proof of the simplicial approximation theorem uses the barycentric subdivision operator to refine simplicial maps
  • Homotopy between the original map and the simplicial approximation constructed using a partition of unity argument
• Proof of the isomorphism between simplicial and singular homology uses the direct limit construction and the simplicial approximation theorem
  • Chain maps between simplicial and singular chain complexes induced by the inclusion of simplicial chains into singular chains and the simplicial approximation of the identity map
• Künneth theorem relates the homology of a product space to the homology of its factors via a tensor product of homology groups
  • Proof uses the Eilenberg-Zilber theorem, which establishes a chain homotopy equivalence between the singular chain complex of the product and the tensor product of the singular chain complexes of the factors

Advanced Topics and Extensions

• Relative homology studies pairs of spaces $(X,A)$, where $A$ is a subspace of $X$, leading to long exact sequences relating the homology of $X$, $A$, and the quotient $X/A$
• Cohomology dual theory to homology, obtained by dualizing the chain complex and boundary operators
  • Cohomology groups $H^n(X)$ are contravariant functors, leading to the study of cohomological operations and ring structures
• Cup product gives cohomology the structure of a graded ring, with applications to intersection theory and the study of manifolds
• Cap product relates homology and cohomology, allowing for the definition of Poincaré duality for manifolds
• Sheaf cohomology generalizes singular cohomology to the setting of sheaves on topological spaces, leading to powerful tools in algebraic geometry and complex analysis
• Spectral sequences algebraic tools for computing homology and cohomology groups, arising from filtered chain complexes or double complexes
  • Serre spectral sequence relates the homology of a fiber bundle to the homology of its base and fiber
  • Atiyah-Hirzebruch spectral sequence computes generalized cohomology theories (e.g., K-theory, cobordism) using singular cohomology