# Quantitative null-cobordism

@article{Chambers2018QuantitativeN, title={Quantitative null-cobordism}, author={Gregory R. Chambers and Dominic Dotterrer and Fedor Manin and Shmuel Weinberger}, journal={Journal of the American Mathematical Society}, year={2018} }

<p>For a given null-cobordant Riemannian <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear…

## 16 Citations

Quantitative nullhomotopy and rational homotopy type

- Mathematics
- 2016

In a 2014 survey, Gromov asks the following question: given a nullhomotopic map $${f:S^{m} \to S^{n}}$$f:Sm→Sn of Lipschitz constant L, how does the Lipschitz constant of an optimal nullhomotopy of f…

Embedding the Heisenberg group into a bounded-dimensional Euclidean space with optimal distortion

- Computer Science
- 2018

This paper shows that one can in fact retain the optimal distortion bound $O( \varepsilon^{-1/2} )$ and still embed into a bounded dimensional space ${\mathbf R}^D$, answering a question of Naor and Neiman.

Mathematisches Forschungsinstitut Oberwolfach

- Mathematics
- 2020

The workshop concentrated on the interplay of advances in the understanding of manifolds and geometric group theory. In particular, we discussed mapping class groups and moduli spaces of manifolds…

Algorithmic aspects of immersibility and embeddability

- Mathematics, Computer ScienceArXiv
- 2018

It is shown that the smooth embeddability of an $m$-manifold with boundary in $\mathbb{R}^n$ is undecidable when $n-m$ is even and $11m \geq 10n+1$.

Plato’s cave and differential forms

- Mathematics
- 2017

In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences…

A zoo of growth functions of mapping class sets

- MathematicsJournal of Topology and Analysis
- 2018

Suppose [Formula: see text] and [Formula: see text] are finite complexes, with [Formula: see text] simply connected. Gromov conjectured that the number of mapping classes in [Formula: see text] which…

Yang–Mills Measure on the Two-Dimensional Torus as a Random Distribution

- MathematicsCommunications in Mathematical Physics
- 2019

We introduce a space of distributional 1-forms $$\Omega ^1_\alpha $$Ωα1 on the torus $$\mathbf {T}^2$$T2 for which holonomies along axis paths are well-defined and induce Hölder continuous functions…

Recent progress in quantitative topology

- Mathematics
- 2017

We discuss recent work of Chambers, Dotterrer, Ferry, Manin, and Weinberger, which resolved a fundamental question in quantitative topology: if f : Sm ! Sn is a contractible map with Lipschitz…

Quantitative nullhomotopy and the Hopf Invariant

- Mathematics
- 2021

Let G : S → S be a map with nonzero Hopf Invariant. Using the generalized Hopf invariant introduced by Haj lasz, Schikorra and Tyson, we show that any null-homotopy F : B → B of G with small (2n +…

Enhanced Bounds for rho-invariants for both general and spherical 3-manifolds

- MathematicsJournal of Topology and Analysis
- 2022

We establish enhanced bounds on Cheeger–Gromov [Formula: see text]-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain…

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