Cyclic CohomologyEdit
Cyclic cohomology is a cohomology theory for associative algebras that extends geometric intuition from smooth manifolds to the broader realm of noncommutative spaces. Conceived in the framework of noncommutative geometry, it provides natural invariants that interact with K-theory through the Chern character, enabling index-theoretic and topological statements for algebras that arise in analysis, topology, and mathematical physics. For a smooth manifold M, the algebra of smooth functions C∞(M) serves as a bridge between classical differential geometry and noncommutative constructions, because the cyclic cohomology of C∞(M) recovers ordinary de Rham cohomology in a precise sense while suggesting how to generalize those ideas to noncommutative settings.
In the broader landscape, cyclic cohomology sits beside Hochschild cohomology as part of a family of invariants derived from the algebraic structure of A. Its periodic version, periodic cyclic cohomology HP^•, exhibits a 2-periodic behavior that makes it especially suited to pairing with K-theory. The theory rests on a careful algebraic apparatus—the cyclic category, cyclic modules, and the (b,B) bicomplex—that encodes how cyclic symmetry interacts with coboundary operators. This machinery yields a robust framework for computing invariants of algebras that do not come from ordinary geometric spaces, as well as for formulating index-type results in a noncommutative context.
History
Cyclic cohomology emerged in the 1980s within the program of noncommutative geometry, a project associated with Alain Connes. It was developed to capture geometric and topological information of spaces that are not well described by classical manifolds. The development built on and extended earlier ideas from Hochschild cohomology, which already encoded algebraic deformations and extensions. A key feature of the theory is its compatibility with K-theory, which supplies a natural source of topological data. Connes and others showed how cyclic cohomology pairs with K-theory through a Chern character map, enabling index-type theorems to be formulated in the noncommutative setting. The SBI long exact sequence, introduced by Connes, relates Hochschild cohomology, cyclic cohomology, and related theories, and it yields the 2-periodicity that characterizes HP^•. For a classical anchor, the case A = C∞(M) ties cyclic cohomology directly to topological invariants of M via de Rham cohomology.
Key contributors to the development of cyclic cohomology include Alain Connes, Jean-Pierre Serre in spirit through the broader algebraic topology milieu, and later researchers who clarified the relationships among cyclic cohomology, Hochschild cohomology, and various flavors of cyclic theories such as entire cyclic cohomology. The field has matured as part of Noncommutative geometry, with diverse applications across analysis, topology, and mathematical physics.
Mathematical framework
Cyclic cohomology is built from the notion of cyclic modules and the cyclic category. Given an associative algebra A over a field, one constructs a cochain complex whose cochains are multilinear functionals on repeated tensor powers of A, subject to a cyclic symmetry. The two primary differentials, the Hochschild coboundary b and Connes’ boundary B, interact to form a bicomplex. The total cohomology of this (b,B) bicomplex yields HC^•(A), the cyclic cohomology groups of A. A closely related invariant is periodic cyclic cohomology HP^•(A), which arises by stabilizing HC^•(A) under a degree shift, yielding a 2-periodic theory: HP^n(A) ≅ HP^{n+2}(A).
The cyclic category and cyclic modules: The cyclic structure is organized by the cyclic category Λ, and cyclic modules provide a natural language for encoding invariants that respect cyclic permutations of entries in A⊗(n+1). See Cyclic category for the foundational setup.
The (b,B) bicomplex: The Hochschild coboundary b captures the standard cohomological relations of multilinear functionals, while B encodes cyclicity, producing a bi-graded picture from which HC^•(A) and HP^•(A) are extracted.
The SBI sequence: Connes’ long exact sequence, often referred to as the SBI sequence, links Hochschild cohomology with cyclic cohomology and relates various flavors of cyclic theories. This exact sequence is a central computational and conceptual tool in the theory and underpins the periodicity phenomenon in HP^•.
The Chern character and pairings with K-theory: Cyclic cohomology admits natural pairings with K-theory, producing numerical invariants of projections and unitary elements. The Chern character ch: K_•(A) → HP_•(A) translates K-theoretic data into cyclic cohomology, enabling index-type formulas in noncommutative settings. See K-theory and Chern character for the bridging concepts.
Examples and computation strategies: For the commutative case A = C∞(M) with M a compact smooth manifold, HP^•(A) recovers the de Rham cohomology of M with a 2-periodic grading. This serves as a guiding example showing how noncommutative invariants extend classical topology. See De Rham cohomology for the classical antecedent.
The main objects and their properties
Hochschild cohomology: A precursor to cyclic cohomology, Hochschild cohomology HH^•(A) encodes infinitesimal deformations of the algebra A and relates to extensions and derivations. See Hochschild cohomology for the broader theory and computations.
Cyclic cohomology: HC^•(A) captures invariants that are sensitive to cyclic symmetry in multilinear functionals on A. It generalizes de Rham cohomology in the noncommutative setting and furnishes a natural recipient for index-theoretic data.
Periodic cyclic cohomology: HP^•(A) is the stabilized, 2-periodic version of cyclic cohomology. HP^• is especially well suited to pairing with K-theory and to formulating index theorems in noncommutative geometry.
The cyclic category and cyclic modules: The organizing framework that underpins the cyclic symmetry in the cohomology theory. See Cyclic category.
The Chern character and index pairing: The Chern character ch links K-theory to HP^•, enabling the computation of indices of elliptic operators in both classical and noncommutative contexts. See Index theory and Chern character.
Applications and perspectives
Connections to index theory: Cyclic cohomology supplies tools to formulate and prove index theorems in noncommutative geometry, where the “space” of interest is encoded by an algebra rather than a geometric manifold. The Connes–Moscovici framework for foliations and the broader noncommutative index theory illustrate how analytic data (like Dirac-type operators) interact with algebraic invariants.
Noncommutative geometry and physics: The theory provides a language for modeling spaces that arise in quantum physics and gauge theories, where the algebra of observables may be noncommutative. The pairing of cyclic cohomology with K-theory helps formulate topological invariants in these settings and has influenced perspectives on the geometry of space-time at small scales.
Examples that illuminate the theory: For a smooth manifold M, HP^•(C∞(M)) recovering the de Rham cohomology illustrates how classical geometry sits inside the noncommutative framework. For group algebras or algebras associated with foliations, cyclic cohomology yields invariants that reflect global geometric and analytic structure of the underlying objects.
Computational challenges and alternatives: While the theory is powerful, explicit computations can be intricate for general noncommutative algebras. In analytic contexts, entire cyclic cohomology and other refinements sometimes provide more flexible analytic tools, and researchers compare cyclic, entire cyclic, and other variants to suit specific problems. See Entire cyclic cohomology for one such refinement and Group algebra for a primary source of noncommutative examples.
Controversies and debates (mathematical perspectives)
Within the mathematical community, debates about cyclic cohomology often center on the most effective framework for particular problems. Some points of discussion include:
The right invariant for noncommutative geometry: While HP^• provides a clean 2-periodic theory well-suited for pairing with K-theory, other variants (such as entire cyclic cohomology) can be more apt for analytic questions or for categories beyond purely algebraic settings. Researchers weigh the trade-offs between algebraic generality and analytic control.
Computational tractability: For many natural algebras arising in geometry and physics, direct computations in cyclic cohomology are challenging. This has led to the development of computational tools, spectral sequences, and comparison theorems that relate cyclic cohomology to more familiar invariants in special cases.
Interpretational scope: The noncommutative geometry viewpoint emphasizes that many phenomena traditionally described by spaces and differential forms have algebraic counterparts captured by cyclic cohomology. Some questions concern how broadly these invariants should be taken to reflect geometric content, and how they relate to classical invariants in limiting cases.
Interplay with physics: The application of cyclic cohomology to models inspired by quantum physics has generated dialogue about the physical meaning of noncommutative invariants. Critics sometimes urge caution about over-interpreting purely mathematical invariants in physical theories, while proponents view the noncommutative framework as a natural generalization of geometric thinking.