On the non-invariance of span and immersion co-dimension for manifolds

Abstract

In this note we give examples in every dimension m ≥ 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 S8. 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 ≥ 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 ..