Andrew Ranicki
A survey of the applications of the noncommutative Cohn localization of rings to the topology of manifolds with infinite fundamental group, with particular emphasis on the algebraic K- and L-theory of generalized free products.
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Given a noncommutative (Cohn) localization $A \to σ^{-1}A$ which is injective and stably flat we obtain a lifting theorem for induced f.g. projective $σ^{-1}A$-module chain complexes and localization exact sequences in algebraic $L$-theory, matching the algebraic $K$-theory localization exact sequence of Neeman and Ranicki.
The noncommutative (Cohn) localization S^{-1}R of a ring R is defined for any collection S of morphisms of f.g. projective left R-modules. We exhibit S^{-1}R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if S^{-1}R is "stably flat over R" (meaning that Tor^R_i(S^{-1}R,S^{-1}R)=0 for i>0) then every bounded f.g. projective S^{-1}R-module chain complex D with [D] \in im(K_0(R)-->K_0(S^{-1}R)) is chain equivalent to S^{-1}C for a bounded f.g. projective R-module chain complex C, and that there is a localization exact sequence in higher…
In this paper we discuss how the question about the rationality of L^2-Betti numbers is related to the Isomorphism Conjecture in algebraic K-theory and why in this context noncommutative localization appears as an important tool.
Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomo…
We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a homotopy class, generalizing the Novikov inequalities.
Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum $Mξ$ in terms of characteristic numbers (indexed by quasi-symmetric functions) for complex-oriented quasitoric manifolds, and show that automorphisms or cohomology operations on this representation are defined by a `renormalization' Hopf algebra of formal diffeomorphisms at the origin of the noncommutative line, previously considered (over $Q$) in quantum electrodynamics. The resulting structure can be presented in purely algebraic terms, as a grou…
We use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a "classical" T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be "non-classical," that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted $K$-theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the non-classical case by an isomorphism of twisted cyclic homology. An important part…
WeintroducethenotionofderivedformalnoncommutativeZariski immersion for differential graded algebra with examples from topology. We il- lustrate the importance of such notion by reformulating the Friedlander-Milnor conjecture in terms of formal noncommutative Zariski immersions. This paper is based on the language developed by Dwyer, Greenless and Iyendar.
In this survey, written for the proceedings of the conference K-theory in algebra, analysis and topology, Buenos-Aires, Argentina (satellite event of the ICM 2018), we give a rigorous overview of the noncommutative counterparts of some celebrated conjectures of Grothendieck, Voevodsky, Beilinson, Weil, Tate, Parshin, Kimura, Schur, and others.
In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which may have closed 1-forms within a given cohomology class. The Main Theorem of the paper generalizes the result of a joint paper with A. Ranicki, which treats the special case of closed 1-forms having integral cohomology classes. The present paper also describes a number of new ine…
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