John Lott

Notes on Perelman’s papers

e Conjecture, and more generally Thurston’s Geometrization Conjecture, using the Ricci flow approach of Hamilton. Perelman’s proofs are concise and, at times, sketchy. The purpose of these notes is to provide the details that are missing in [46] and [47], which contain Perelman’s arguments for the Geometrization Conjecture. Among other things, we cover the construction of the Ricci flow with surgery of [47]. We also discuss the long-time behavior of the Ricci flow with surgery, which is needed for the full Geometrization Conjecture. The papers of Colding and Minicozzi [23; 24] and Perelman [48], which are not covered in these notes, each provide a shortcut in the case of the Poincar´ e Conjecture. Namely, these papers show that if the initial manifold is simply-connected then the Ricci flow with surgery becomes extinct in a finite time, thereby removing the issue of the long-time behavior. Combining this claim with the proof of existence of Ricci flow with surgery gives the shortened proof in the simply-connected case. These notes are intended for readers with a solid background in geometric analysis. Good sources for background material on Ricci flow are Chow and Knopf [21], Chow, Lu and Ni [22], Hamilton [30] and Topping [60]. The notes are self-contained but are designed to be read along with [46; 47]. For the most part we follow the format of [46; 47] and use the section numbers of [46; 47] to label our sections. We have done this in

Particle models and noncommutative geometry

Abstract We write three particle models in terms of noncommutative gauge theory: the Glashow-Weinberg-Salam model, the Peccei-Quinn model and the standard model.

Flat vector bundles, direct images and higher real analytic torsion

We prove a Riemann-Roch-Grothendieck-type theorem concerning the direct image of a flat vector bundle under a submersion of smooth manifolds. We refine this theorem to the level of differential forms. We construct associated secondary invariants, the analytic torsion forms, which coincide in degree 0 with the Ray-Singer real analytic torsion. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

The Metric Aspect of Noncommutative Geometry

Most of the previous work on “noncommutative geometry” could more accurately be labeled as noncommutative differential topology, in that it deals with the homology of differential forms on noncommutative spaces (cyclic homology) and vector bundles on noncommutative spaces (K-theory) [Col]. However, the essence of geometry has to do with the metric properties of spaces.

L2-Topological invariants of 3-manifolds.

SummaryWe give results on theL2-Betti numbers and Novikov-Shubin invariants of compact manifolds, especially 3-manifolds. We first study the Betti numbers and Novikov-Shubin invariants of a chain complex of Hilbert modules over a finite von Neumann algebra. We establish inequalities among the Novikov-Shubin invariants of the terms in a short exact sequence of chain complexes. Our algebraic results, along with some analytic results on geometric 3-manifolds, are used to compute theL2-Betti numbers of compact 3-manifolds which satisfy a weak form of the geometrization conjecture, and to compute or estimate their Novikov-Shubin invariants.

Superconnections and higher index theory

LetM be a smooth closed spin manifold. The higher index theorem, as given for example in Proposition 6.3 of [CM], computes the pairing between the group cohomology of π1(M) and the Chern character of the “higher” index of a Dirac-type operator on M. Using superconnections, we give a heat equation proof of this theorem on the level of differential forms on a noncommutative base space. As a consequence, we obtain a new proof of the Novikov conjecture for hyperbolic groups.

Higher Eta-Invariants

AMtraet. We define the higher eta-invariant of a Dirac-type operator on a nonsimply-connected closed manifold. We discuss its variational properties and how it would fit into a higher index theorem for compact manifolds with boundary. We give applications to questions of positive scalar curvature for manifolds with boundary, and to a Novikov conjecture for manifolds with boundary.

An index theorem in differential K-theory

Let X --> B be a proper submersion with a Riemannian structure. Given a differential K-theory class on X, we define its analytic and topological indices as differential K-theory classes on B. We prove that the two indices are the same.

Dimensional reduction and the long-time behavior of Ricci flow

If g.t/ is a three-dimensional Ricci flow solution, with sectional curvatures that are O.t / and diameter that is O.t/, then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton. Mathematics Subject Classification (2010). 53C44, 57M50.

Higher-degree analogs of the determinant line bundle

Abstract: In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-type operators whose index lies in the subspace of the reduced K-theory of the parametrizing space, we construct a set of Deligne cohomology classes of degree i whose curvatures are the i-form component of the Atiyah-Singer families index theorem.