Splittings of free groups, normal forms and partitions of ends (Q601191)

The authors analyse embedded essential spheres in \(M\), the connected sum of \(k\) copies of \(S^2\times S^1\), and in its universal cover \(\tilde M\). They start by recalling some standard results, namely that there is a one-to-one correspondence between embedded oriented spheres in \(\tilde M\) up to homology and partitions of the space of ends of \(\tilde M\) into two non-empty and closed subsets, and the fact that two spheres intersects translates into properties of the corresponding partitions (i.e. they ``cross''). Using the data of crossings, the authors show that for each partition of ends into two non-empty closed subsets there is a construction that produces directly a normal sphere (in the sense of Hatcher) in \(\tilde M\) which induces it. In the last part of the paper, using graphs of trees, the authors give a constructive proof of a result of Gadgil characterising the classes of \(\pi_2(M)\) which can be represented by embedded spheres.