The Yamabe invariant for non-simply connected manifolds

Details The Yamabe invariant for non-simply connected manifolds The Yamabe invariant for non-simply connected manifolds

2001-04-18

arxiv.org/abs/math/0104186v1

J. Differential Geom. 62 (2002), no. 2, 175-208

Mathematics

Differential Geometry

25 pages

Scientific paper

The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is non-negative for all closed simply connected manifolds of dimension $\ge 5$. We extend this to show that Yamabe invariant is non-negative for all closed manifolds of dimension $\ge 5$ with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed manifolds of dimension $\ge 5$ with fundamental group of odd order having all Sylow subgroups elementary abelian, with non-spin universal cover, admit metrics of positive scalar curvature, once one restricts to the ``complement'' of manifolds whose homology classes are ``toral.'' The exceptional toral homology classes only exist in dimensions not exceeding the ``rank'' of the fundamental group, so this proves important cases of the Gromov-Lawson-Rosenberg Conjecture once the dimension is sufficiently large.

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