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Geometric topology of stratified spaces
Author:
Bruce Hughes
Journal:
Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 73-81
MSC (1991):
Primary 57N80, 57N37; Secondary 55R65, 57N40
MathSciNet review:
1412945
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Abstract: A theory of tubular neighborhoods for strata in manifold stratified spaces is developed. In these topologically stratified spaces, manifold stratified approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include a multiparameter isotopy extension theorem, neighborhood germ classification and a topological version of Thom's First Isotopy Theorem.
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Thom, Ensembles et morphismes
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0239613 (39 #970), http://dx.doi.org/10.1090/S0002-9904-1969-12138-5
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Andrei
Verona, Stratified mappings—structure and
triangulability, Lecture Notes in Mathematics, vol. 1102,
Springer-Verlag, Berlin, 1984. MR 771120
(86k:58010)
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Shmuel
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developments, Group actions on manifolds (Boulder, Colo., 1983)
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Shmuel
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spaces, Chicago Lectures in Mathematics, University of Chicago Press,
Chicago, IL, 1994. MR 1308714
(96b:57024)
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and Combinatorial Topology (A Symposium in Honor of Marston Morse),
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- 29.
Min
Yan, The periodicity in stable equivariant surgery, Comm. Pure
Appl. Math. 46 (1993), no. 7, 1013–1040. MR 1223661
(95d:57017), http://dx.doi.org/10.1002/cpa.3160460704
- 1.
- D. R. Anderson and W. C. Hsiang, The functors
 and pseudo-isotopies of polyhedra, Ann. of Math. (2) 105 (1977), 201--203. MR 55:13447
- 2.
- N. P. Buchdahl, S. Kwasik, and R. Schultz, One fixed point actions on low-dimensional spheres, Invent. Math. 102 (1990), 633--662. MR 92b:57047
- 3.
- A. V. Cernavskii, Local contractibility of the group of homeomorphisms of a manifold, Mat. Sb. 8 (1969), 287--233. MR 41:4554
- 4.
- T. A. Chapman, Approximation results in topological manifolds, Mem. Amer. Math. Soc. 34 (1981), no. 251. MR 83i:57005
- 5.
- F. Connolly and B. Vajiac, An end theorem for stratified spaces, (preprint).
- 6.
- K. H. Dovermann and R. Schultz, Equivariant surgery theories and their periodicity properties, Lecture Notes in Math. 1443, Springer-Verlag, New York, 1990. MR 92c:57032
- 7.
- R. D. Edwards, TOP regular neighborhoods, handwritten manuscript, 1973.
- 8.
- R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63--88. MR 44:1032
- 9.
- M. Goresky and R. MacPherson, Stratified Morse theory, Ergeb. Math. Grenzgeb. (3) 14, Springer-Verlag, New York, 1988. MR 90d:57039
- 10.
- B. Hughes, Approximate fibrations on topological manifolds, Michigan Math. J. 32 (1985), 167--183. MR 87e:57025
- 11.
- ------, The geometric topology of stratified spaces, (in preparation).
- 12.
- B. Hughes and A. Ranicki, Ends of complexes, Cambridge Tracts in Math. 123, Cambridge Univ. Press, Cambridge, 1996.
- 13.
- B. Hughes, L. Taylor, and B. Williams, Bundle theories for topological manifolds, Trans. Amer. Math. Soc. 319 (1990), 1--65. MR 91e:57035
- 14.
- ------, Manifold approximate fibrations are approximately bundles, Forum Math. 3 (1991), 309--325. MR 92k:57040
- 15.
- B. Hughes, L. Taylor, S. Weinberger, and B. Williams, Neighborhoods in stratified spaces with two strata, (preprint).
- 16.
- J. Mather, Notes on topological stability, Harvard Univ., Cambridge, 1970, (photocopied).
- 17.
- F. Quinn, Ends of maps. II, Invent. Math. 68 (1982), 353--424. MR 84j:57011
- 18.
- ------, Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988), 441--499. MR 89g:57050
- 19.
- ------, Speculations on Gromov convergence of stratified sets, and Riemannian volume collapse, Prospects in topology; Proceddings of a conference in honor of William Browder (F. Quinn, ed.), Ann. of Math. Studies, vol. 138, Princeton Univ. Press, Princeton, 1995. CMP 96:06
- 20.
- C. Rourke and B. Sanderson, An embedding without a normal microbundle, Invent. Math. 3 (1967), 293--299. MR 36:5954
- 21.
- L. Siebenmann, Deformations of homeomorphisms on stratified sets, Comment. Math. Helv. 47 (1971), 123--165. MR 47:7752
- 22.
- M. Steinberger, The equivariant topological
 -cobordism theorem, Invent. Math. 91 (1988), 61--104. MR 89e:57015
- 23.
- M. Steinberger and J. West, Equivariant
 -cobordisms and finiteness obstructions, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 217--220. MR 86j:57019
- 24.
- R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240--282. MR 39:970
- 25.
- A. Verona, Stratified mappings -- structure and triangulability, Lecture Notes in Math., vol. 1102, Springer-Verlag, New York, 1984. MR 86k:58010
- 26.
- S. Weinberger, Construction of group actions: a survey of some recent developments, Contemp. Math. 36 (1985), 269--298. MR 87k:57029
- 27.
- ------, The topological classification of stratified spaces, Chicago Lectures in Math., Univ. Chicago Press, Chicago, 1994. MR 96b:57024
- 28.
- H. Whitney, Local properties of analytic varieties, Differentiable and combinatorial topology (S. Cairns, ed.), Princeton Univ. Press, Princeton, 1965, pp. 205--244. MR 32:5924
- 29.
- M. Yan, The periodicity in stable equivariant surgery, Comm. Pure Appl. Math. 46 (1993), 1013--1040. MR 95d:57017
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Additional Information
Bruce Hughes
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email:
hughescb@math.vanderbilt.edu
DOI:
http://dx.doi.org/10.1090/S1079-6762-96-00010-8
PII:
S 1079-6762(96)00010-8
Keywords:
Stratified space,
approximate fibration,
teardrop,
locally conelike,
isotopy extension,
strata,
homotopy link,
neighborhood germ
Received by editor(s):
May 20, 1996
Additional Notes:
Supported in part by NSF Grant DMS–9504759.
Communicated by:
Walter Neumann
Article copyright:
© Copyright 1996 American Mathematical Society
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