<p>We construct infinitely many examples of macroscopically large manifolds of dimension <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_723_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>m</mi> <mo>≥</mo> <mn>4</mn> </math></EquationSource> <EquationSource Format="TEX">$m \geq 4$</EquationSource> </InlineEquation> equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_723_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">⌈</mo> <mi>m</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">⌉</mo> <mo>+</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$\lceil m/2 \rceil + 1$</EquationSource> </InlineEquation>. In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.</p>

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Circle Bundles with PSC over Large Manifolds

  • Aditya Kumar,
  • Balarka Sen

摘要

We construct infinitely many examples of macroscopically large manifolds of dimension m 4 $m \geq 4$ equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most m / 2 + 1 $\lceil m/2 \rceil + 1$ . In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.