Singular CohomologyEdit
Singular cohomology is a central construction in algebraic topology that assigns to every topological space X and every abelian group A a sequence of abelian groups H^n(X; A). Built from the contravariant functor of cochains on the space, it provides algebraic invariants that detect holes of various dimensions and behave well with respect to continuous maps. It is defined for all spaces, not only well-behaved ones, and has a rich algebraic structure in addition to its groups, including a cup product that makes H^*(X; A) into a graded ring. For readers who want to situate it in the broader landscape, singular cohomology sits alongside other cohomology theories as part of the foundational toolkit of cohomology and homology theories.
Background and construction
The construction begins with the notion of a topological space X and the set of singular n-simplices, i.e., continuous maps σ: Δ^n → X from the standard n-simplex. The free abelian group on the set of all such maps is denoted C_n(X). The boundary operator ∂: C_n(X) → C_{n-1}(X) encodes how an n-simplex is glued to its faces, yielding the singular chain complex (C_(X), ∂). Dually, the cochain complex is formed by taking C^n(X; A) = Hom(C_n(X), A) with coboundary δ: C^n(X; A) → C^{n+1}(X; A) defined by δϕ(σ) = ϕ(∂σ). The n-th singular cohomology group is the quotient H^n(X; A) = ker(δ: C^n → C^{n+1}) / im(δ: C^{n-1} → C^n). This construction is functorial in X: a continuous map f: X → Y induces a pullback f^: H^n(Y; A) → H^n(X; A).
A defining feature of singular cohomology is its cup product, a graded operation ⌣: H^p(X; A) × H^q(X; B) → H^{p+q}(X; A ⊗ B) that endows the direct sum of cohomology groups with a ring structure. With appropriate coefficients, one obtains a natural, graded-commutative ring H^*(X; A) that reflects the interaction of cohomology classes in different degrees. The existence of the cup product and, more generally, Steenrod operations in mod p cohomology, are among the key algebraic features that enable computations and structural insight.
Basic properties and axioms
Functoriality and homotopy invariance: If f, g: X → Y are homotopic, then the induced maps f^, g^: H^n(Y; A) → H^n(X; A) coincide. This makes H^*(−; A) a homotopy-invariant functor of spaces.
Excision and Mayer–Vietoris: The theory satisfies excision, allowing computation from local data, and the Mayer–Vietoris sequence provides long exact sequences for unions of open sets, enabling inductive or patching arguments for computations.
Universal coefficient theorem: For each space X and coefficient group A, there is a short exact sequence 0 → Ext^1_Z(H_{n-1}(X), A) → H^n(X; A) → Hom_Z(H_n(X), A) → 0. This relates cohomology with arbitrary coefficients to the (often simpler) homology of X and explains phenomena like torsion detection in cohomology.
Representability and universal properties: For a fixed group G, the cohomology functor H^n(−; G) is representable by an Eilenberg–MacLane space K(G, n) in the sense that H^n(X; G) ≅ [X, K(G, n)], the set of homotopy classes of maps from X to K(G, n). This bridging of algebra and topology is a cornerstone of modern homotopy theory.
Computations and examples
Point: H^0(point; A) ≅ A and H^n(point; A) = 0 for n > 0. This base case anchors many computations.
Spheres: For a sphere S^k, H^n(S^k; A) ≅ A when n = 0 or n = k and is zero otherwise. This reflects the basic “hole” in dimension k.
Products and the Künneth formula: For spaces X and Y and suitable coefficient rings, there is a Künneth-type formula describing H^n(X × Y; A) in terms of the cohomology of X and Y. This enables the systematic analysis of products of spaces.
Classic rings: The cohomology ring of complex projective space with integer coefficients is H^*(CP^n; Z) ≅ Z[x]/(x^{n+1}) with deg x = 2. Here the cup product completely determines the ring structure in low dimensions, illustrating how cohomology encodes geometric information about complex line bundles over CP^n.
Universal coefficient phenomena: When X has torsion in its homology, the universal coefficient theorem shows how torsion contributes to H^n(X; A) via Ext terms, producing new information not visible in ordinary homology alone.
Mod p cohomology and operations: When working with coefficients Z/p, the cohomology groups carry an action of the Steenrod algebra, giving rise to cohomology operations like Steenrod squares. These operations provide finer invariants than the cohomology groups alone and are central to many computations and classifications in topology.
Foundations, comparisons, and debates
Singular cohomology sits in a network of closely related theories. De Rham cohomology, defined for smooth manifolds in terms of differential forms, agrees with singular cohomology with real coefficients via de Rham’s theorem. Čech cohomology approaches, which can be more geometric in nature for certain spaces, agree with singular cohomology for a broad class of spaces such as paracompact spaces. These relationships illuminate when different constructions yield the same invariants and when they may diverge, deepening the understanding of spaces with pathological features.
From a foundational perspective, singular cohomology is part of the Eilenberg–Steenrod framework of axioms that characterize ordinary cohomology theories. While these axioms codify the essential features one expects of a “good” cohomology theory, they also leave room for alternative theories (e.g., generalized cohomology theories like K-theory) that extend or refine the information captured by singular cohomology in various contexts. The representability by Eilenberg–MacLane spaces is a particularly powerful vantage point, translating topological problems into homotopy-theoretic questions about maps into fixed classifying spaces.
Contemporary computations and applications
Singular cohomology remains a practical workhorse in topology and geometry. It underpins the classification of vector bundles via classifying spaces BU(n) and characteristic classes such as Chern classes and Stiefel–Whitney classes, which live in cohomology rings and govern bundle geometry. The theory also informs fixed-point results, obstruction theory, and the study of smooth and topological manifolds through Poincaré duality, which relates cohomology in complementary degrees on closed oriented manifolds.
In a computational setting, tools like spectral sequences (notably the Serre spectral sequence) reduce complex calculations to successive, more manageable pages. The interplay between algebra and topology—through rings, modules, and homological algebra—makes singular cohomology a crossroads of computation and conceptual insight.
See also