Positive scalar curvature and local actions of nonabelian Lie groups (Q5966207)

\textit{H. B. Lawson jun.} and \textit{S. T. Yau} proved in [(*) Comment. Math. Helv. 49, 232-244 (1974; Zbl 0297.57016)] that if a compact, connected, nonabelian Lie group G acts smoothly and effectively on a compact manifold M, then M admits a riemannian metric of positive scalar curvature. In Theorem A below we show that the same conclusion holds under somewhat weaker assumptions described by the following definition: A local action of nonabelian Lie groups (or \({\mathcal N}\)-structure) on a smooth manifold M consists of a finite cover \((U_ i)_{i\in I}\) of M by open, connected sets \(U_ i\) and a family \(F_ i: G_ i\times U_ i\to U_ i\quad (i\in I)\) of smooth, effective actions of compact, connected, nonabelian Lie groups \(G_ i\) such that the following compatibility condition holds: for i,j\(\in I\) the set \(U_{ij}=U_ i\cap U_ j\) (if nonempty) is both \(G_ i\)- and \(G_ j\)-invariant and one of the two groups contains the other if we treat them as subgroups of \(Homeo(U_{ij}).\) Theorem A. If a compact manifold M admits a local action by non-abelian Lie groups, then it admits a riemannian metric of positive scalar curvature. {\S}{\S} 4 and 5 contain the main conceptual body of the proof of Theorem A and explain its relation to (*). The technical core of the proof is deferred to {\S}{\S} 9 and 10. Theorem B (see {\S} 2) states that if M and N are two manifolds with \({\mathcal N}\)-structures and \(\dim (M)=\dim (N)\geq 6,\) then the connected sum M{\#}N also has an \({\mathcal N}\)- structure. This theorem thus provides a method of constructing local actions from global ones and illustrates some flexibility of \({\mathcal N}\)- structures, which is not shared by global actions. Theorem C (see {\S} 3) supplies examples of manifolds (with the family \((T^ n\times S^ 2)\#(T^ n\times S^ 2),\) \(n\geq 3\), among them) which admit local actions but no global action by a nonabelian group. As those manifolds have metrics with positive scalar curvature, they also prove that the converse of the theorem of Lawson and Yau does not hold. Our interest in local actions of Lie groups comes from the work of \textit{M. Gromov} and \textit{J. Cheeger} [J. Differ. Geom. 23, 309-346 (1986; Zbl 0606.53028) and \textit{M. Gromov}, Publ. Math., Inst. Hautes Etud. Sci. 56, 5-99 (1982; Zbl 0516.53046)] who introduced the notion (using different terminology though) and explored the case of abelian groups (tori). Quite naturally the geometric features in the two (abelian and nonabelian) cases differ substantially. Finally let us mention that some related results were obtained by Phillipe Fullsack.