| Title:
|
On the non-invariance of span and immersion co-dimension for manifolds (English) |
| Author:
|
Crowley, Diarmuid J. |
| Author:
|
Zvengrowski, Peter D. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
44 |
| Issue:
|
5 |
| Year:
|
2008 |
| Pages:
|
353-365 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
span |
| Keyword:
|
stable span |
| Keyword:
|
manifolds |
| Keyword:
|
non-invariance |
| MSC:
|
57Q25 |
| MSC:
|
57R20 |
| MSC:
|
57R25 |
| MSC:
|
57R55 |
| idZBL:
|
Zbl 1212.57009 |
| idMR:
|
MR2501571 |
| . |
| Date available:
|
2009-01-29T09:15:51Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127122 |
| . |
| Reference:
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