Semisimple degree of symmetry and maps of non-zero degree into a product of 1-spheres and 2-spheres (Q1060473)

This paper generalizes the author's result in [J. Math. Soc. Japan 35, 683-692 (1983; Zbl 0528.57029)]. Let M be a closed topological manifold which admits a map of nonzero degree into \((S^ 1)^ r\times (S^ 2)^ s\), the product of 1- and 2-spheres. A connected sum L{\#}((S\({}^ 1)^ r\times (S^ 2)^ s)\) is an example of such a manifold. It is shown that SU(2) or SO(3) is the only compact connected simple Lie group G that acts on M almost effectively. If G is not simple, G is locally isomorphic to \(T^ u\times SU(2)^ v\) with \(v\leq s\). Moreover, if the Euler characteristic of M is nonzero, then \(u+v\leq s\). It is also shown that SU(2) cannot act almost effectively on the connected sum M{\#}N, where M is as above and N is orientable and not a rational homology sphere. If a torus \(T^ n\) acts on \(M=L\#((S^ 1)^ r)\times (S^ 2)^ s)\) almost effectively and L is orientable, then \(n\leq r+s\) is shown.