Novikov conjecture

Novikov conjecture
Name	Novikov conjecture
Field	Topology
Proposer	Sergei Novikov
Proposed	1965

Novikov conjecture is a conjecture in Topology concerning the homotopy invariance of higher signatures of closed smooth manifolds and the rational injectivity of certain assembly maps in K-theory and L-theory. It connects problems in Algebraic topology with questions in Operator algebras, Differential geometry, Geometric group theory, and Global analysis, and it has driven interactions among researchers at institutions such as the Steklov Institute of Mathematics, Harvard University, Princeton University, Massachusetts Institute of Technology, and the Institute for Advanced Study.

Statement of the conjecture

The conjecture asserts that the higher signature invariants defined using cohomology classes pulled back from the classifying space of the fundamental group of a closed smooth oriented manifold are invariant under oriented homotopy equivalences; equivalently, the rational injectivity of the assembly map in topological L-theory or the Baum–Connes-style assembly map in K-theory implies homotopy invariance of these higher signatures. This statement ties together constructions from Hirzebruch signature theorem, the Pontryagin classs, the Atiyah–Singer index theorem, and the algebraic machinery developed in Surgery theory by figures such as William Browder, C. T. C. Wall, and Andrew Ranicki.

Historical background and motivation

The conjecture originates with work of Sergei Novikov in the 1960s on characteristic classes and homotopy invariants, building on earlier results by Friedrich Hirzebruch and the development of the signature in cobordism theory and Characteristic class theory. Its formulation responded to questions posed by the International Congress of Mathematicians era developments and relates to the classification efforts of topologists like John Milnor, Raoul Bott, and René Thom. Motivating examples came from investigations in high-dimensional topology and connections with index-theoretic ideas of Michael Atiyah and Isadore Singer as well as later operator-theoretic perspectives advocated by Alain Connes, Gennadi Kasparov, and Nigel Higson.

Known results and partial cases

The conjecture has been proved for many classes of groups and spaces: for groups with the homotopy type of finite complexes, for amenable groups via work related to Yu Guoliang's property A and coarse geometry techniques, and for groups acting on nonpositively curved spaces such as CAT(0) spaces and Gromov hyperbolic groups. Positive results were established using methods associated with Kasparov's KK-theory, the Baum–Connes conjecture implications, and controlled topology techniques developed by Frank Quinn and Shmuel Weinberger. Notable milestones include proofs for word-hyperbolic groups by researchers influenced by Gromov (mathematician), for groups with finite asymptotic dimension by Guoliang Yu, and for groups admitting suitable foliations treated by Alain Connes and Ralf Meyer.

Methods and approaches

Approaches to the conjecture exploit tools from Surgery theory, assembly map formulations in algebraic K-theory and L-theory, index theory from the Atiyah–Singer index theorem tradition, and operator-algebraic techniques originating in the Baum–Connes conjecture program of Paul Baum and Alain Connes. Controlled topology strategies stem from work by Andrew Ranicki and Jerome Levine, while coarse geometric and analytic methods derive from contributions by Guoliang Yu, Gennadi Kasparov, Nigel Higson, and John Roe. Additional inputs include equivariant homotopy theory developed by G. W. Whitehead and assembly map analysis by Wolfgang Lück.

Related conjectures and consequences

The Novikov conjecture is closely related to the Borel conjecture on topological rigidity, the Baum–Connes conjecture in operator K-theory, and the Farrell–Jones conjecture in algebraic K-theory and L-theory. Consequences of the Novikov conjecture include rigidity phenomena for manifolds conjectured by Armand Borel and applications to classification results pursued by F. T. Farrell and L. E. Jones as well as implications for the structure of group C*-algebras studied by G. A. Elliott and David R. Roe. Progress on Novikov has influenced work on conjectures posed at venues such as the International Congress of Mathematicians and in programs at the Mathematical Sciences Research Institute.

Examples and counterexamples in special settings

Positive verifications include closed manifolds with fundamental groups in classes such as virtually cyclic groups, word-hyperbolic groups, amenable groups of certain types, and groups acting properly on Euclidean buildings or Hadamard manifolds; these examples draw on constructions by Gromov (mathematician), Jean-Pierre Serre, Michael Davis, and Werner Ballmann. No counterexamples are known in the smooth closed manifold category; however, related pathologies and failures arise in contexts involving wild group actions, exotic PL manifold phenomena studied by C. T. C. Wall and exotic smooth structures explored by John Milnor, or when analytic assembly map hypotheses fail as in certain exotic C*-algebra constructions considered by Alain Connes and Gennadi Kasparov.

Category:Conjectures in topology