Degree-1 maps into lens spaces and free cyclic actions on homology 3- spheres (Q749926)

The paper deals with the existence of degree-1 maps f: \(M\to L_{n,m}\) from a closed orientable 3-manifold M to some lens space \(L_{n,m}\). The main result of the paper asserts that if \(H_ 1(M)={\mathbb{Z}}_ n\), then such a map f exists and the lens space \(L_{n,m}\) is determined uniquely (up to homotopy equivalence) by the mod n intersection number \(<\beta (x),x>=\pm mk^ 2\), where x is a generator of \(H_ 2(M,{\mathbb{Z}}_ n)={\mathbb{Z}}_ n\) and \(\beta: H_ 2(M,{\mathbb{Z}}_ n)\to H_ 1(M,{\mathbb{Z}}_ n)\) is the Bockstein homomorphism defined by the exact sequence \(0\to {\mathbb{Z}}_ n\to {\mathbb{Z}}_{n^ 2}\to {\mathbb{Z}}_ n\to 0\). As a corollary, the authors show that if \(\tilde M\) is a homology 3- sphere and p: \(\tilde M\to M\) is a regular covering with \({\mathbb{Z}}_ n\) as its group of covering transformations, then p is induced from the universal covering \(S^ 3\to L_{n,m}\) by a degree-1 map f: \(M\to L_{n,m}\).