Searcharxiv⌕ Search

arXiv subjects

Vitali Kapovitch

Publications and source records attributed to Vitali Kapovitch.

35 records · Page 2Linked to original sources
arXiv2017

On the torsion in the center conjecture

We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower. Our result is motivated by the conjecture that every almost nonnegatively curved closed m-dimensional manifold M admits a finite cover M' for which the number of leafs is bounded in terms of m such that the torsion of the fundamental group of M' lies in its center.

math.AT↗
arXiv2017

Biquotients with singly generated rational cohomology

We classify all biquotients whose rational cohomology rings are generated by one element. As a consequence we show that the Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient.

math.DG↗
arXiv2017

Space of nonnegatively curved metrics and pseudoisotopies

Let V be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on V are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many V the space of complete nonnegatively curved metrics has infinite higher homotopy groups.

math.DG↗
arXiv2015

On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds

We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with Ricci curvature bounded below and $γ$ is a limit geodesic in $X$ then along the interior of $γ$ same scale measure metric tangent cones $T_{γ(t)}X$ are Hölder continuous with respect to measured Gromov-Hausdorff topology and have the same dimension in the sense of Colding-Naber.

math.DG↗
arXiv2012

On fibrations with formal elliptic fibers

We prove that for a fibration of simply-connected spaces of finite type $F\hookrightarrow E\to B$ with $F$ being positively elliptic and $H^*(F,\qq)$ not possessing non-trivial derivations of negative degree, the base $B$ is formal if and only if the total space $E$ is formal. Moreover, in this case the fibration map is a formal map. As a geometric application we show that positive quaternion Kähler manifolds are formal and so are their associated twistor fibration maps.

math.AT↗
arXiv2010

Pinching estimates for negatively curved manifolds with nilpotent fundamental groups

Let $M$ be a complete Riemannian metric of sectional curvature within $[-a^2,-1]$ whose fundamental group contains a $k$-step nilpotent subgroup of finite index. We prove that $a\ge k$ answering a question of M. Gromov. Furthermore, we show that for any $ε>0$, the manifold $M$ admits a complete Riemannian metric of sectional curvature within $[-(k+ε)^2,-1]$.

math.DG↗
arXiv2007

Perelman's Stability Theorem

We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence $X_i$ of Alexandrov spaces with curvature bounded below Gromov-Hausdorff converging to a compact Alexandrov space $X$, $X_i$ is homeomorphic to $X$ for all large $i$.

math.DG↗
arXiv2004

Nonnegative pinching, moduli spaces and bundles with infinitely many souls

We show that in each dimension $n\ge 10$ there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian $n$-manifolds with $0\le \sec\le 1$, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of noncompact manifolds whose moduli spaces of complete metrics with $\sec\ge 0$ have infinitely many connected components.

math.DG↗
arXiv2004

Curvature bounds via Ricci smoothing

We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow.

math.DG↗
arXiv2001

Restrictions on collapsing with a lower sectional curvature bound

We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.

math.DG↗
arXiv2001

Regularity of limits of noncollapsing sequences of manifolds

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.

math.DG↗
arXiv2000

Finiteness theorems for nonnegatively curved vector bundles

We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of ``bounded geometry''.

math.DG↗
arXiv2000

Topological obstructions to nonnegative curvature

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric.

math.DG↗