Geodesic
On the non-invariance of span and immersion co-dimension for manifolds
Archivum mathematicum, Tome 44 (2008) no. 5, pp. 353-365
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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.
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.
Classification : 57Q25, 57R20, 57R25, 57R55
Keywords: span; stable span; manifolds; non-invariance 
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Crowley, Diarmuid J.; Zvengrowski, Peter D. On the non-invariance of span and immersion co-dimension for manifolds. Archivum mathematicum, Tome 44 (2008) no. 5, pp. 353-365. http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a2/

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