Circle Bundles with PSC over Large Manifolds

Abstract We construct infinitely many examples of macroscopically large manifolds of dimension $m \geq 4$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:math> equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most $\lceil m/2 \rceil + 1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⌈</mml:mo> <mml:mi>m</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>⌉</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> . 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.