Singular homology and cohomology

Singular homology and cohomology
Name	Singular homology and cohomology
Field	Algebraic topology
Introduced	1930s
Contributors	Henri Poincaré, Emmy Noether, Samuel Eilenberg, Norman Steenrod

Singular homology and cohomology Singular homology and cohomology are foundations of modern algebraic topology that assign algebraic invariants to topological spaces and continuous maps. These theories, developed by figures such as Henri Poincaré, Emmy Noether, Samuel Eilenberg, and Norman Steenrod, connect geometric intuition with algebraic structures and have influenced work across mathematics, physics, computer science, engineering, and economics.

Introduction

Singular homology and cohomology produce sequences of abelian groups or modules that are invariant under homeomorphism, homotopy equivalence, and other central notions in topology. Originating in the work of Poincaré and formalized using algebraic methods by Eilenberg and Steenrod, the theory interacts with theorems and structures introduced by Noether, Emmy Noether, David Hilbert, Élie Cartan, Hermann Weyl, and Kurt Gödel in the broader development of 20th-century mathematics. Theories like homotopy theory, cohomology theory, and homological algebra provide frameworks linking singular homology to concepts from category theory, functoriality, exact sequences, and chain complexes.

Singular Homology: Definitions and Computation

Singular homology begins with singular simplices: continuous maps from the standard simplex Δ^n into a topological space X, as in constructions used by Poincaré and later by Eilenberg and Steenrod; these maps form free abelian groups called chain groups, with boundary operators defining chain complexes central to homological algebra and formalized by Emmy Noether and Samuel Eilenberg. Computing singular homology uses tools such as the Mayer–Vietoris sequence, simplicial homology, and cellular methods akin to techniques developed by J. H. C. Whitehead, Jean Leray, André Weil, and Henri Cartan. Examples include calculations for spheres via the Hurewicz theorem, tori related to work by Johann Benedict Listing and William Rowan Hamilton, and surfaces classified using the classification theorem for compact surfaces, which connects to matrices and bilinear forms studied by Carl Friedrich Gauss, Bernhard Riemann, and Augustin-Louis Cauchy.

Fundamental Properties and Theorems

Singular homology satisfies axioms analogous to those in the Eilenberg–Steenrod axioms established by Eilenberg and Steenrod, including homotopy invariance, excision, and long exact sequences of pairs; these properties mirror structural results in category theory influenced by Saunders Mac Lane and Samuel Eilenberg. The long exact sequence of a pair, the Mayer–Vietoris sequence, and excision facilitate computations and comparisons with algebraic invariants found in the work of Alexander Grothendieck and Jean-Pierre Serre. Key theorems include the Hurewicz theorem, which relates homotopy groups studied by Henri Poincaré and George Whitehead to homology, and the Universal Coefficient Theorem connecting homology and cohomology analogous to duality principles developed by Alexander Grothendieck and Jean-Pierre Serre. Poincaré duality, originally formulated by Poincaré and extended in modern forms by Hermann Weyl and André Weil, links homology and cohomology on manifolds and underpins deep results in differential topology and algebraic geometry pursued by René Thom, John Milnor, and Michael Atiyah.

Singular Cohomology and Cup Product

Singular cohomology arises by applying Hom functors to chain complexes, producing cochain complexes and cohomology groups that encapsulate additional algebraic structure; foundational contributors include Eilenberg, Steenrod, and Cartan. The cup product endows singular cohomology with a graded-commutative ring structure, enabling intersection-theoretic calculations and connections to characteristic classes elaborated by Shiing-Shen Chern, Atiyah–Singer contexts studied by Michael Atiyah and Isadore Singer, and Chern–Weil theory developed with André Weil. Cohomology operations, notably the Steenrod squares introduced by Norman Steenrod, and the structure of the Steenrod algebra, link to stable homotopy theory advanced by J. F. Adams and Daniel Quillen. Cup and cap products interact with Poincaré duality and with spectral sequences like the Leray–Serre spectral sequence, tools used in analyses by Jean Leray and Jean-Pierre Serre.

Relationships with Other Homology Theories

Singular homology relates to simplicial homology, cellular homology, de Rham cohomology, and sheaf cohomology, tying to the work of Henri Cartan, André Weil, Alexander Grothendieck, and Jean-Pierre Serre. The de Rham theorem, connecting singular cohomology with real differential forms, builds on ideas from Élie Cartan, Georges de Rham, and applications in differential geometry explored by Shiing-Shen Chern and John Milnor. Comparisons with extraordinary homology theories, such as K-theory developed by Atiyah and Michael Atiyah, and bordism theory advanced by René Thom, occur via generalized cohomology frameworks introduced by Brown–Pontryagin and formalized in stable homotopy theory by J. F. Adams and Daniel Quillen. Sheaf-theoretic perspectives from Alexander Grothendieck and derived functor methods by Grothendieck and Henri Cartan connect singular cohomology to Čech cohomology and étale cohomology used in algebraic geometry by Grothendieck and Jean-Pierre Serre.

Applications and Examples

Applications span classification problems in topology, invariants in knot theory studied by James Waddell Alexander II and Vaughan Jones, fixed-point theorems such as Brouwer fixed-point theorem and results related to Lefschetz fixed-point theorem developed by Solomon Lefschetz, and problems in differential topology and surgery theory influenced by William Browder and C. T. C. Wall. In geometry, singular cohomology and cup product computations inform characteristic classes in the work of Shiing-Shen Chern, Raoul Bott, and Michael Atiyah, and have consequences in physical theories including gauge theory and aspects of quantum field theory explored by Edward Witten and Michael Atiyah. Computational topology and persistent homology in data analysis draw on singular homology concepts adapted for algorithmic work by researchers associated with Stanford University, MIT, and Princeton University. Classical examples include homology groups of spheres, tori, projective spaces like Real projective space and Complex projective space, and configuration spaces relevant to robotics and motion planning studied by Michael Farber.

Category:Algebraic topology