Poincare Complexes: I

Abstract

Summary:In this note we give examples in every dimension $m \ge 9$ of piecewise linearly homeomorphic, closed, connected, smooth $m$-manifolds which admit two smoothness structures with differing spans, stable spans, and immersion co-dimensions. In dimension $15$ the examples include the total spaces of certain $7$-sphere bundles over $S^8$. The construction of such manifolds is based on the topological variance of the second Pontrjagin class: a fact which goes back to Milnor and which was used by Roitberg to give examples of span variation in dimensions $m \ge 18$. We also show that span does not vary for piecewise linearly homeomorphic smooth manifolds in dimensions less than or equal to $8$, or under connected sum with a smooth homotopy sphere in any dimension. Finally, we use results of Morita to show that in all dimensions $m \ge 19$ there are topological manifolds admitting two piecewise linear structures having different $PL$-spans