Semisimple degree of symmetry of manifold with the homotopy type of product \((S^ 1)^ r\times (S^ 2)^ s\times (S^ 3)^ t\) (Q1060475)

Let M be a closed topological manifold with the same integral cohomology ring as \(S^ 3\times...\times S^ 3\), the product of 3-spheres, or let M be with the same homotopy type as \((S^ 1)^ r\times (S^ 2)^ s\times (S^ 3)^ t\), the product of 1-, 2- and 3-spheres. 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 M is in the latter case and G is not simple, it is also shown that G is locally isomorphic to \(T^ u\times SU(2)^ v\) with \(u+v\leq r+2(s+t)\). In the proof of these results the authors use the Leray spectral sequence of the orbit map \(M\to M/G\) and observe the cohomological structure of the spaces in question.