Hostname: page-component-5c6d5d7d68-ckgrl Total loading time: 0 Render date: 2024-08-08T05:26:21.936Z Has data issue: false hasContentIssue false
  • English
  • Français

Stratified path spaces and fibrations

Published online by Cambridge University Press:  14 November 2011

Bruce Hughes
Bruce Hughes
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240, USA, (hughes@math.vanderbilt.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

The main objects of study are the homotopically stratified metric spaces introduced by Quinn. Closed unions of strata are shown to be stratified forward tame. Stratified fibrations between spaces with stratifications are introduced. Paths that lie in a single stratum, except possibly at their initial points, form a space with a natural stratification, and the evaluation map from that space of paths is shown to be a stratified fibration. Applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 2 , 1999 , pp. 351 - 384
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Akbulut, S. and King, H.. Topology of real algebraic sets. Mathematical Sciences Reseach Institute Publishers, vol. 25 (New York: Springer, 1992).CrossRefGoogle Scholar
2Beshears, A.. G-isovariant structure sets and stratified structure sets (Vanderbilt University, preprint, 1997).Google Scholar
3Cappell, S. and Shaneson, J.. The mapping cone and cylinder of a stratified map. In Prospects in topology; proceedings of a conference in honor of William Browder (ed. Quinn, F.). Ann. Math. Stud. 138, 58–66 (Princeton, NJ: Princeton University Press, 1995).Google Scholar
4Dovermann, K. H. and Schultz, R.. Equivariant surgery theories and their periodicity properties. Lecture Notes in Mathematics, vol. 1443 (New York: Springer, 1990).CrossRefGoogle Scholar
5Goresky, M. and MacPherson, R.. Stratified Morse theory. In Ergeb. Math. Grenzgeb (3) vol. 14 (New York: Springer, 1988).CrossRefGoogle Scholar
6Hughes, B.. Geometric topology of stratified spaces. Electron. Res. Announc. Am. Math. Soc. 2 1996, 73–81.CrossRefGoogle Scholar
7Hughes, B.. The geometric topology of stratified spaces. (In preparation.)Google Scholar
8Hughes, B. and Ranicki, A.. Ends of complexes. In Cambridge Tracts in Mathematics vol. 123 (Cambridge University Press, 1996).Google Scholar
9Hughes, B., Taylor, L., Weinberger, S. and Williams, B.. Neighborhoods in stratified spaces with two strata. (Preprint.)Google Scholar
10Hughes, B., Taylor, L. and Williams, B.. Bundle theories for topological manifolds. Trans. Am. Math. Soc. 319 1990, 1–65.CrossRefGoogle Scholar
11Mather, J.. Notes on topological stability (Cambridge, MA: Harvard University Press, 1970). (Photocopied.)Google Scholar
12Mather, J.. Stratifications and mappings. In Dynamical systems, Proc. Symp., Univ. Bahia, Salvador, Brazil, 1971 (ed. Peixoto, M. M.), pp. 195–232 (New York: Academic, 1973).Google Scholar
13Quinn, F.. Ends of maps. II. Invent. Math. 68 1982, 353–424.CrossRefGoogle Scholar
14Quinn, F.. Homotopically stratified sets. J. Am. Math. Soc. 1 1988, 441–499.CrossRefGoogle Scholar
15Spanier, E.. Algebraic topology (New York: McGraw-Hill, 1966).Google Scholar
16Talbert, R.. Stratified and equivariant homology via homotopy colimits. PhD thesis, Vanderbilt University (1997).Google Scholar
17Verona, A.. Stratified mappings—structure and triangulability. Lecture Notes in Mathematics, no. 1102 (New York: Springer, 1984).CrossRefGoogle Scholar
18Whitehead, G. W.. Elements of homotopy theory. Graduate Texts in Mathematics, vol. 61 (New YorK: Springer, 1978).CrossRefGoogle Scholar