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

Diarmuid J. Crowley; Peter D. Zvengrowski

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 5, page 353-365
  • ISSN: 0044-8753

Abstract

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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 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 ≥ 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 ≥ 19 there are topological manifolds admitting two piecewise linear structures having different P L -spans.

How to cite

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Crowley, Diarmuid J., and Zvengrowski, Peter D.. "On the non-invariance of span and immersion co-dimension for manifolds." Archivum Mathematicum 044.5 (2008): 353-365. <http://eudml.org/doc/250500>.

@article{Crowley2008,
abstract = {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.},
author = {Crowley, Diarmuid J., Zvengrowski, Peter D.},
journal = {Archivum Mathematicum},
keywords = {span; stable span; manifolds; non-invariance; span; stable span; manifold; non-invariance},
language = {eng},
number = {5},
pages = {353-365},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the non-invariance of span and immersion co-dimension for manifolds},
url = {http://eudml.org/doc/250500},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Crowley, Diarmuid J.
AU - Zvengrowski, Peter D.
TI - On the non-invariance of span and immersion co-dimension for manifolds
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 5
SP - 353
EP - 365
AB - 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.
LA - eng
KW - span; stable span; manifolds; non-invariance; span; stable span; manifold; non-invariance
UR - http://eudml.org/doc/250500
ER -

References

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