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.
Classification : 57Q25, 57R20, 57R25, 57R55
Keywords: span; stable span; manifolds; non-invariance 
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     author = {Crowley, Diarmuid J. and Zvengrowski, Peter D.},
     title = {On the non-invariance of span and immersion co-dimension for manifolds},
     journal = {Archivum mathematicum},
     pages = {353--365},
     publisher = {mathdoc},
     volume = {44},
     number = {5},
     year = {2008},
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     zbl = {1212.57009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2008__44_5_a2/}
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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/