# Article

 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: [1] Atiyah, M.: Thom complexes.Proc. London Math. Soc. 11 (3) (1961), 291–310. Zbl 0124.16301, MR 0131880 Reference: [2] Benlian, R., Wagoner, J.: Type d’homotopie et réduction structurale des fibrés vectoriels.C. R. Acad. Sci. Paris Sér. A-B 207-209. 265 (1967), 207–209. MR 0221524 Reference: [3] Bredon, G. 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