Simply connected 4-manifolds of second Betti number 1 bounded by homology lens spaces (Q1263131)

The lens space L(p,1) is the boundary of a disk bundle V(p) which is a homotopy 2-sphere. This paper studies when a homology lens space M bounds a smooth 4-manifold V which is a homology 2-sphere. It gives examples of homology lens spaces which bound topological homotopy 2-spheres but don't bound smooth ones. \textit{S. Fukuhara} had given similar examples in J. Math. Soc. Japan 36, 259-277 (1984; Zbl 0527.57005), which were detected by an invariant \({\tilde \mu}\)(M). Examples are constructed where Fukuhara's invariant vanishes. The proofs hinge on constructing 4- manifolds with vanishing first homology and second homology of rank 2 or 3. These manifolds contain smoothly embedded 2-spheres and the results follow from applying results of \textit{K. Kuga} [Topology 23, 133-137 (1984; Zbl 0551.57019)] and the reviewer [Mich. Math. J. 34, 85-91 (1987; Zbl 0624.57019)] which restrict which homology classes can be represented by an embedded 2-sphere. As an application, it is shown that if K is a slice knot and \(p>2\) an integer which is either even or has a prime factor p' with p'\(\equiv 3 mod 4\), then the Dehn surgered 3-manifold M(K,p/1) bounds a smooth homotopy 2-sphere V iff V is homeomorphic to the handlebody V(K,p). An example when \(p=5\) shows the necessity of some condition on p and is tied into the question of which homology classes in a homology \({\mathbb{C}}{\mathbb{P}}^ 2\#{\mathbb{C}}{\mathbb{P}}^ 2\) can be represented by a smoothly embedded 2-sphere. The hypothesis on p allows the author to restrict to the characteristic case, where the reviewer showed that the only characteristic homology classes represented are the sum of two generators. It is conjectured that for any primitive class \(ax+by\) which is represented one must have \(| a|\), \(| b| \leq 2\). If this holds one can change the hypothesis to \(p\neq 5\). As a second application it is shown that there are infinitely many homology lens spaces M which are Dehn surgeries M(K,p/q) on knots which satisfy: (1) M bounds a topological homotopy \(S^ 2\); (2) Fukuhara's invariant is defined for M and it vanishes; (3) M can't bound any smooth homology \(S^ 2\). In particular, M can't be obtained from \(S^ 3\) by integral Dehn surgery on a knot.