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Urysohn Width and Surgeries

Aleksandr Berdnikov, Brendan Isley

TL;DR

This work analyzes how Urysohn width $\mathrm{UW}_k$ behaves under surgeries along a boundary piece $A$ by studying the connected sum $M_1\#_A M_2$. It develops a spectrum of bounds: a basic no-covers approach giving $\mathrm{UW}_k(M_i) \le 2\mathrm{UW}_k(M_1\#_A M_2)+\mathrm{diam}_{M_i}(A)$, a refined collar-type method yielding $\mathrm{UW}_k(M_i) \le \mathrm{UW}_k(M_1\#_A M_2)+\mathrm{diam}_{M_i}(A)$ under suitable hypotheses, and a covers version for universal covers with a constant $C'$: $\mathrm{UW}_k(\tilde{M}\#_A M') \le C'\max_i\mathrm{UW}_k(\tilde{M}_i)+2\operatorname{diam}(\partial A)$. Extensions to multiple attachments and to universal covers are developed, and counterexamples demonstrate the optimality of the constants involved. The results connect width bounds to macroscopic dimension and provide tools that can yield alternative proofs of scalar-curvature-related constraints in Gromov-type settings. Overall, the paper clarifies how width and macroscopic dimension behave under surgeries and highlights intrinsic limits via explicit counterexamples.

Abstract

We analyze the behavior of Urysohn width of manifolds under a connected sum operation, specifically, bounding widths of summands in terms of widths of the sum and vice versa. Our methods also apply to the universal covers of these spaces, and to more general type of surgeries. Lastly, we provide examples that show the optimality of constants in our estimates.

Urysohn Width and Surgeries

TL;DR

This work analyzes how Urysohn width behaves under surgeries along a boundary piece by studying the connected sum . It develops a spectrum of bounds: a basic no-covers approach giving , a refined collar-type method yielding under suitable hypotheses, and a covers version for universal covers with a constant : . Extensions to multiple attachments and to universal covers are developed, and counterexamples demonstrate the optimality of the constants involved. The results connect width bounds to macroscopic dimension and provide tools that can yield alternative proofs of scalar-curvature-related constraints in Gromov-type settings. Overall, the paper clarifies how width and macroscopic dimension behave under surgeries and highlights intrinsic limits via explicit counterexamples.

Abstract

We analyze the behavior of Urysohn width of manifolds under a connected sum operation, specifically, bounding widths of summands in terms of widths of the sum and vice versa. Our methods also apply to the universal covers of these spaces, and to more general type of surgeries. Lastly, we provide examples that show the optimality of constants in our estimates.
Paper Structure (8 sections, 14 theorems, 38 equations)

This paper contains 8 sections, 14 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Simple "No Covers" Case: collapsing $A$
  3. Refined "No Covers" Case: contracting $\partial A$
  4. Covers
  5. Counterexamples
  6. Constant 2 in theorem \ref{['nocover']}
  7. Constant $C$ theorem \ref{['cover']}, case $k=1$
  8. Constant $C$ theorem \ref{['cover']}, case $k=n-1$

Key Result

Theorem 1

Let $M_1$ and $M_2$ be complete $n$-manifolds, each containing an embedded copy of a compact $n$-manifold $A$ with boundary. Then for $i\in \{1,2\}$ and $k\in \mathop{\mathrm{\mathbb{N}}}\nolimits$ we have In particular, if $M_1 \#_A M_2$ has macroscopic dimension bounded above by $k$, then so does every $M_i$.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • proof : Proof of theorem \ref{['nocover']}
  • proof
  • Lemma 2
  • ...and 23 more