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

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...