and are powerful tools in algebraic topology. They help us understand the structure of by studying their "holes" and other geometric features. These concepts build on the ideas of chain complexes and .
Homology groups capture information about and in a space, while provide a dual perspective. Together, they offer insights into the topology of spaces and allow us to compute important invariants.
Singular Homology
Simplicial Complexes and Chain Complexes
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Singular simplex maps standard n-simplex into a topological space X
Collection of singular simplices forms a
constructed from simplicial complex by taking free abelian groups generated by n-simplices (Cn(X))
∂n:Cn(X)→Cn−1(X) defined by mapping each n-simplex to its oriented boundary (alternating sum of its (n-1)-dimensional faces)
Boundary operator satisfies ∂n−1∘∂n=0, forming a chain complex
Homology Groups and Their Properties
n-th homology group Hn(X) defined as kernel of ∂n modulo image of ∂n+1
Elements of Hn(X) are equivalence classes of n-cycles (elements of kernel of ∂n) modulo boundaries (elements of image of ∂n+1)
Homology groups are topological invariants, independent of the choice of simplicial complex
Functorial properties allow for the study of induced homomorphisms between homology groups
Homology groups capture "holes" in a topological space (connected components for H0, loops for H1, voids for H2, etc.)
Singular Cohomology
Cohomology Groups and Cochains
Cohomology groups Hn(X) defined as the dual of homology groups Hn(X)
Cochain complex formed by taking Hom(Cn(X),G) for each n, where G is an abelian group (coefficients)
δn:Hom(Cn(X),G)→Hom(Cn+1(X),G) defined as the dual of the boundary operator
n-th cohomology group Hn(X;G) defined as kernel of δn modulo image of δn−1
Elements of Hn(X;G) are equivalence classes of n- modulo n-coboundaries
Universal Coefficient Theorem and Künneth Formula
relates homology and cohomology groups via a short involving Ext and Tor functors
Allows for the computation of cohomology groups from homology groups and vice versa
Künneth Formula expresses the homology (or cohomology) of a product space in terms of the homology (or cohomology) of its factors and their tensor products
Useful for computing homology and cohomology of product spaces (torus, product of spheres, etc.)
Computational Tools
Mayer-Vietoris Sequence
is a long exact sequence relating homology groups of a space X to homology groups of subspaces A and B, where X = A ∪ B
Sequence involves homology groups of A, B, A ∩ B, and X, connected by boundary operators and inclusion-induced homomorphisms
Useful for computing homology groups of spaces that can be decomposed into simpler subspaces (CW complexes, simplicial complexes, etc.)
Provides a way to break down the computation of homology groups into smaller, more manageable pieces
Can be applied iteratively to compute homology groups of spaces with multiple decompositions