K-homology

K-homology
Name	K-homology
Field	Algebraic topology; Operator algebras
Introduced	1960s
Key people	Michael Atiyah; Isadore Singer; Gennadi Kasparov; Nigel Higson; John Roe

K-homology is a generalized homology theory dual to K-theory that arises in the interface of index theory, operator algebras, and topology. It connects constructions of Atiyah–Singer index theorem with analytic cycles built from Fredholm operators, and it plays a central role in the work of Michael Atiyah, Isadore Singer, and Gennadi Kasparov. K-homology informs classification problems studied by Elliott classification program, Connes–Moscovici index theorem, and research around the Baum–Connes conjecture.

Definition and basic properties

K-homology is the homology theory representing classes of elliptic-type operators on a space or of extensions of C*-algebras; it is the dual theory to topological K-theory developed by Atiyah and Bott and later formalized in analytic terms by Kasparov. As a homology theory it satisfies versions of the Eilenberg–Steenrod axioms adapted to generalized homology, and it exhibits long exact sequences associated to pairs and Mayer–Vietoris sequences used in computations by researchers such as Rosenberg and Schochet. Functoriality under proper maps connects it to transformations studied in works by Gromov, Thurston, and applications to manifold invariants considered by Hirzebruch.

Cycles and analytic formulation

Analytic K-homology classes are represented by Fredholm-module cycles built from representations of C*-algebras on Hilbert spaces together with an operator whose commutators are controlled; this approach was axiomatized by Kasparov and refined by Baaj and Julg. Typical cycles involve graded or ungraded Hilbert spaces, bounded operators, and compactness conditions reminiscent of constructions by Brown and Douglas in extension theory. The analytic picture relates to geometric cycles like elliptic pseudodifferential operators on manifolds studied by Atiyah, Singer, and Seeley; analytical techniques from Voiculescu and Connes influence regularity and summability conditions.

Relation to K-theory and index pairing

K-homology pairs naturally with K-theory via an index pairing that generalizes the classical index of elliptic operators appearing in the Atiyah–Singer index theorem. The pairing is expressed analytically using Kasparov product in KK-theory and topologically via pushforward maps studied by Grothendieck and Hirzebruch. This duality underpins conjectures such as the Baum–Connes conjecture and ties into classification programs by Elliott; computations of pairings have been central in work by Higson, Kasparov, and Skandalis.

Examples and computations

Basic examples include K-homology of spheres, tori, and projective spaces; explicit generators arise from Dirac operators on spin manifolds studied by Lawson and Michelsohn and from signature operators investigated by Novikov. Calculations for noncommutative tori and crossed product algebras feature prominently in the literature of Connes, Rieffel, and Pimsner–Voiculescu; combinatorial tools due to Atiyah–Bott–Shapiro assist in computations for Clifford module constructions. Concrete computations for group C*-algebras of lattices in Lie groups and for Roe algebras were developed by Yu, Roe, and Lafforgue.

Kasparov KK-theory and duality

Kasparov's bivariant KK-theory provides a powerful framework in which K-homology appears as KK(A, C) while K-theory appears as KK(C, A); the Kasparov product furnishes composition laws and dualities exploited in proofs by Kasparov and in the formulation of the Brown–Douglas–Fillmore theory. KK-theory has been used to establish Poincaré duality results for manifolds and noncommutative spaces in the work of Connes, Skandalis, and Higson–Roe and to approach assembly maps central to the Novikov conjecture and Baum–Connes conjecture.

Applications in geometry and topology

K-homology classes capture geometric invariants such as higher indices of elliptic operators, obstructions to positive scalar curvature via the Gromov–Lawson and Rosenberg index frameworks, and signature-type invariants related to the Novikov conjecture. Applications extend to classification of manifolds and rigidity theorems pursued by Sullivan and Weinberger, and to noncommutative geometry programs by Connes and Moscovici linking cyclic cohomology, characteristic classes, and index theory. The theory also informs analysis on metric spaces studied by Cheeger, coarse geometry investigations by Roe, and dynamics on foliations treated by Alain Connes and Moore.

Historical development and key results

The conceptual roots trace to early work on topological K-theory by Atiyah and Bott and the analytic index by Atiyah–Singer; the analytic formulation of K-homology and the bivariant framework were introduced and developed by Kasparov in the 1970s. Subsequent milestones include the Brown–Douglas–Fillmore classification program, Connes's noncommutative geometry reformulations, and advances in the Baum–Connes conjecture by Kasparov, Higson, Lafforgue, and Yu. Important results include index theorems for families by Atiyah–Singer collaborators, duality theorems in KK-theory proven by Kasparov and extended by Skandalis, and rigidity and positive scalar curvature obstructions advanced by Gromov, Lawson, and Rosenberg.

Category:Algebraic topology