Singular cohomology

Singular cohomology
Name	Singular cohomology
Field	Algebraic topology
Introduced	20th century
Key concepts	Cohomology group, cup product, long exact sequence

Singular cohomology is a fundamental tool in Algebraic topology used to assign algebraic invariants to topological spaces via cochains on singular simplices, providing contravariant functors from spaces to graded rings that detect global properties such as connectivity, orientability, and obstructions to maps. Developed in the context of mid-20th century advances in Topology, Homological algebra, and Category theory, it connects with classical results like the Poincaré duality and modern frameworks including Sheaf cohomology and K-theory. Singular cohomology plays a central role in work by figures and institutions associated with Eilenberg–Steenrod axioms, Henri Cartan, Jean Leray, and research programs at places such as École Normale Supérieure and Institute for Advanced Study.

Definition

Singular cohomology is defined using singular simplices which are continuous maps from the standard simplex Δ^n into a topological space, assembling chain complexes analogous to those in Simplicial homology and formalized with techniques from Homological algebra and Category theory. Given a coefficient group such as Z or a module over a ring like Z/pZ or Q one forms the cochain complex Hom(C_*(X), G) whose coboundary maps yield the cohomology groups H^n(X; G), respecting axioms originating in the work of Samuel Eilenberg and Norman Steenrod. The construction is natural under maps induced by continuous maps of spaces studied in contexts including Homotopy theory and results influenced by research at institutions like Princeton University.

Cohomology groups and functoriality

Cohomology assigns to each topological space X a graded group H^*(X; G) functorially in maps f: X → Y, producing pullback homomorphisms f^*: H^*(Y; G) → H^*(X; G) that interact with homotopy classes and monoidal structures studied in Homotopy category and Model categories. The groups H^n(X; G) recover dual information to homology groups H_n(X; G) and are computed via chain complexes whose exactness properties are illuminated by the Eilenberg–Steenrod axioms and examples from Manifold theory and CW complex theory as in texts by authors associated with Princeton University Press and Cambridge University Press.

Cup product and ring structure

The cup product endows H^*(X; R) with a graded-commutative ring structure where α ∪ β ∈ H^{p+q}(X; R) is defined using the Alexander–Whitney map and properties explored by Norman Steenrod and Samuel Eilenberg. This multiplicative structure interacts with dualities such as Poincaré duality for Orientable manifolds and appears in signature theorems and invariants studied by Hirzebruch and Atiyah–Singer. The ring structure is central in obstruction theory as developed by researchers affiliated with Harvard University and University of Cambridge.

Universal coefficient theorem and coefficients

The universal coefficient theorem relates H^n(X; G) to Hom(H_n(X; Z), G) and Ext^1(H_{n-1}(X; Z), G), connecting singular cohomology to Ext functors and Hom functors from Homological algebra. Choice of coefficients such as Z, Q, R, Z/pZ or modules over a principal ideal domain influences torsion detection, as encountered in calculations by mathematicians at ETH Zurich and investigations tied to the Künneth theorem and computational frameworks used at institutions like Max Planck Institute for Mathematics.

Relative cohomology and long exact sequence

Relative cohomology H^*(X, A; G) for a pair (X, A) fits into long exact sequences connecting H^*(A; G), H^*(X; G), and H^*(X, A; G) analogously to relative homology, enabling excision-based arguments and connecting to spectral sequence techniques developed in works from Princeton and University of Chicago. These long exact sequences underpin computations involving pairs arising in studies at places such as University of California, Berkeley and are instrumental in proofs of duality theorems used by scholars including those at Columbia University.

Mayer–Vietoris sequence and excision

The Mayer–Vietoris sequence provides a tool to compute H^*(X; G) from an open cover U ∪ V = X, giving a long exact sequence built from the cohomology of U, V, and U ∩ V and used in calculations for spaces appearing in research at Massachusetts Institute of Technology and Imperial College London. Excision properties justified by the Eilenberg–Steenrod axioms ensure local-to-global arguments and feed into computational spectral sequences like the Leray–Serre spectral sequence applied in studies at University of Oxford.

Computations and examples

Computations of singular cohomology for spheres, tori, projective spaces, and CW complexes illustrate phenomena such as torsion, ring generators, and cup products; classical results include H^*(S^n; Z), H^*(T^n; Z), and H^*(RP^n; Z/2Z), appearing in foundational texts by authors affiliated with Cambridge University Press, Springer-Verlag, and researchers like Jean-Pierre Serre and Raoul Bott. Calculations leverage cellular cohomology on CW complexs and relate to invariants from Characteristic classes studied by Chern, Pontryagin, and Stiefel–Whitney who are central figures in the development of modern topology at institutions such as Stanford University and University of Chicago.

Applications and relations to other theories

Singular cohomology interfaces with sheaf cohomology, de Rham cohomology for smooth manifolds, Alexander duality, and generalized cohomology theories like K-theory and cobordism studied by Michael Atiyah, Bott, and groups at Institute for Advanced Study. It informs obstruction theory, fixed-point theorems like Lefschetz fixed-point theorem, and classification results in Differential topology and Algebraic geometry where comparisons to étale cohomology and Hodge theory appear in work at Université Paris-Saclay and Institut des Hautes Études Scientifiques.

Category:Algebraic topology