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.
@article{127122,
author = {Diarmuid J. Crowley and Peter D. Zvengrowski},
title = {On the non-invariance of span and immersion co-dimension for manifolds},
journal = {Archivum Mathematicum},
volume = {044},
year = {2008},
pages = {353-365},
zbl = {1212.57009},
mrnumber = {2501571},
language = {en},
url = {http://dml.mathdoc.fr/item/127122}
}
Crowley, Diarmuid J.; Zvengrowski, Peter D. On the non-invariance of span and immersion co-dimension for manifolds. Archivum Mathematicum, Tome 044 (2008) pp. 353-365. http://gdmltest.u-ga.fr/item/127122/
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