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Aspects of topology: On dimension theory

Background from Riemannian geometry and Ricci flow: Preliminaries from Riemannian geometry Manifolds of non-negative curvature Basics of Ricci flow The maximum principle Convergence results for Ricci flow Perelman's length function and its applications: A comparison geometry approach to the Ricci flow Complete Ricci flows of bounded curvature Non-collapsed results $\kappa$-non-collapsed ancient solutions Bounded curvature at bounded distance Geometric limits of generalized Ricci flows The standard solution Ricci flow with surgery: Surgery on a $\delta$-neck Ricci flow with surgery: The definition Controlled Ricci flows with surgery Proof of non-collapsing Completion of the proof of Theorem 15.9 Completion of the proof of the Poincare conjecture: Finite-time extinction Completion of the proof of Proposition 18.24 3-manifolds covered by canonical neighborhoods Bibliography Index.

Ricci Flow and the Poincare Conjecture

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

/pdf/notes-on-perelman-s-papers-4m0k9pxzx3.pdf

Notes on Perelman's papers

These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds".

/pdf/notes-on-perelman-s-papers-xlp640e452.pdf

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Metric Spaces Of Non Positive Curvature

We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curv > k Gromov-Hausdorff converging to a compact Alexandrov space X , Xi is homeomorphic to X for all large i .

/pdf/perelman-s-stability-theorem-56ktrsio7q.pdf

Perelman???s Stability Theorem

Verifying a conjecture of Gromov we establish a generalized Margulis Lemma for manifolds with lower Ricci curvature bound Among the various applications are finiteness results for fundamental groups of compact $n$-manifolds with upper diameter and lower Ricci curvature bound modulo nilpotent normal subgroups

/pdf/structure-of-fundamental-groups-of-manifolds-with-ricci-4zviynahzd.pdf

Structure of fundamental groups of manifolds with Ricci curvature bounded below

We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound are topologically spheres. As an application we show that for any finite dimensional Alexandrov space X

 n
 with 
$ n \ge 5 $
 there exists an Alexandrov space Y homeomorphic to X which cannot be obtained as such a limit.

/pdf/regularity-of-limits-of-noncollapsing-sequences-of-manifolds-38fovze47r.pdf

Regularity of limits of noncollapsing sequences of manifolds

We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.

https://www.math.toronto.edu/vtk/nilpotency.pdf

Nilpotency, almost nonnegative curvature and the gradient flow

We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound are topologically spheres. As an application we show that for any finite dimensional Alexandrov space 
$X^n$ with $n\ge 5$ there exists an Alexandrov space $Y$ homeomorphic to $X$ which can not be obtained as such a limit.

Vitali Kapovitch

Papers

Perelman???s Stability Theorem

Structure of fundamental groups of manifolds with Ricci curvature bounded below

Regularity of limits of noncollapsing sequences of manifolds

Nilpotency, almost nonnegative curvature and the gradient flow

Regularity of limits of noncollapsing sequences of manifolds