10.1 Singular homology and cohomology

Singular homology and cohomology are powerful tools in algebraic topology. They help us understand the structure of topological spaces by studying their "holes" and other geometric features. These concepts build on the ideas of chain complexes and homology groups.

Homology groups capture information about cycles and boundaries in a space, while cohomology groups 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

• Singular simplex maps standard n-simplex into a topological space X
• Collection of singular simplices forms a simplicial complex
• Chain complex constructed from simplicial complex by taking free abelian groups generated by n-simplices ($C_n(X)$)
• Boundary operator $\partial_n: C_n(X) \to C_{n-1}(X)$ defined by mapping each n-simplex to its oriented boundary (alternating sum of its (n-1)-dimensional faces)
• Boundary operator satisfies $\partial_{n-1} \circ \partial_n = 0$, forming a chain complex

Homology Groups and Their Properties

• n-th homology group $H_n(X)$ defined as kernel of $\partial_n$ modulo image of $\partial_{n+1}$
• Elements of $H_n(X)$ are equivalence classes of n-cycles (elements of kernel of $\partial_n$) modulo boundaries (elements of image of $\partial_{n+1}$)
• 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 $H_0$, loops for $H_1$, voids for $H_2$, etc.)

Singular Cohomology

Cohomology Groups and Cochains

• Cohomology groups $H^n(X)$ defined as the dual of homology groups $H_n(X)$
• Cochain complex formed by taking $Hom(C_n(X), G)$ for each $n$, where $G$ is an abelian group (coefficients)
• Coboundary operator $\delta^n: Hom(C_n(X), G) \to Hom(C_{n+1}(X), G)$ defined as the dual of the boundary operator
• n-th cohomology group $H^n(X; G)$ defined as kernel of $\delta^n$ modulo image of $\delta^{n-1}$
• Elements of $H^n(X; G)$ are equivalence classes of n-cocycles modulo n-coboundaries

Universal Coefficient Theorem and Künneth Formula

• Universal Coefficient Theorem relates homology and cohomology groups via a short exact sequence 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

• 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