Torus manifolds with non-abelian symmetries (Q2880687)

Let \(G\) be a connected compact non-abelian Lie group and let \(T\) be a maximal torus of \(G\). In this paper the author studies torus manifolds with \(G\)-action, i.e, smooth connected closed oriented manifolds \(M\) on which \(G\) acts almost effectively, with \(\dim M=2\dim T\) and \(M^T\not=\emptyset\). He shows that the action of a finite covering group of \(G\) on \(M\) factors through the group NEWLINE\[NEWLINE\tilde{G}=\prod { SU}(l_i+1)\times \prod{ SO}(2l_i+1) \times\prod{ SO}(2l_i)\times T^{l_0}.NEWLINE\]NEWLINE The action of \(\tilde{G}\) on \(M\) restricts to an action of NEWLINE\[NEWLINE\tilde{G}'=\prod { SU}(l_i+1)\times \prod{ SO}(2l_i+1) \times\prod{ U}(l_i)\times T^{l_0}NEWLINE\]NEWLINE and the \(\tilde{G}'\)-action has the same orbits as the \(\tilde{G}\)-action. He defines invariants called admissible \(5\)-tuples and uses these invariants to determine the \(\tilde{G}'\)-equivariant diffeomorphism type of torus manifolds \(M\) with \(G\)-action. If \(M\) is simply connected then its admissible \(5\)-tuple determines its \(\tilde{G}\)-equivariant diffeomorphism type.NEWLINENEWLINEThe author then applies the classification to obtain more explicit results in several special cases. For example, he shows that if \(\tilde{G}\) is semi-simple and has two simple factors and if \(M\) is simply connected, then \(M\) is one of the following: NEWLINE\[NEWLINE {\mathbb{C}}P^{l_1}\times {\mathbb{C}}P^{l_2},\;{\mathbb{C}}P^{l_1}\times S^{2l_2},\;\#_i( S^{2l_1}\times S^{2l_2})_i,\;S^{2l_1+2l_2}, NEWLINE\]NEWLINE The \(\tilde{G}\)-action on each of these spaces is unique up to equivariant diffeomorphism.