• DOI:10.1090/jams/903
• Corpus ID: 54040187

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}
}

• Gregory R. Chambers, Dominic Dotterrer, +1 author S. Weinberger
• Published 16 October 2016
• Computer Science
• Journal of the American Mathematical Society

<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. We show that it is at most a polynomial whose degree depends on <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>. In the appendix the bound is improved to one that is <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis upper L Superscript 1 plus epsilon Baseline right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>O</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mn>1</mml:mn>
<mml:mo>+</mml:mo>
<mml:mi>ε<!-- ε --></mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">O(L^{1+\varepsilon })</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> for every <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0">
<mml:semantics>
<mml:mrow>
<mml:mi>ε<!-- ε --></mml:mi>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\varepsilon >0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p>

<p>This construction relies on another of independent interest. Take <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X">
<mml:semantics>
<mml:mi>X</mml:mi>
<mml:annotation encoding="application/x-tex">X</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y">
<mml:semantics>
<mml:mi>Y</mml:mi>
<mml:annotation encoding="application/x-tex">Y</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y">
<mml:semantics>
<mml:mi>Y</mml:mi>
<mml:annotation encoding="application/x-tex">Y</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
<mml:semantics>
<mml:mi>L</mml:mi>
<mml:annotation encoding="application/x-tex">L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-Lipschitz maps <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f comma g colon upper X right-arrow upper Y">
<mml:semantics>
<mml:mrow>
<mml:mi>f</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>g</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi>X</mml:mi>
<mml:mo stretchy="false">→<!-- → --></mml:mo>
<mml:mi>Y</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">f,g:X \to Y</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> are homotopic via a <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper L">
<mml:semantics>
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mi>L</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">CL</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y">
<mml:semantics>
<mml:mi>Y</mml:mi>
<mml:annotation encoding="application/x-tex">Y</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p>