Geometric topology of stratified spaces

Hughes, Bruce

doi:10.1090/S1079-6762-96-00010-8

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
DOI: https://doi.org/10.1090/S1079-6762-96-00010-8
MathSciNet review: 1412945
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

Douglas R. Anderson and W. C. Hsiang, The functors $K_{-i}$ and pseudo-isotopies of polyhedra, Ann. of Math. (2) 105 (1977), no. 2, 201–223. MR 440573, DOI https://doi.org/10.2307/1970997
N. P. Buchdahl, Sławomir Kwasik, and Reinhard Schultz, One fixed point actions on low-dimensional spheres, Invent. Math. 102 (1990), no. 3, 633–662. MR 1074489, DOI https://doi.org/10.1007/BF01233442
A. V. Černavskiĭ, Local contractibility of the group of homeomorphisms of a manifold., Mat. Sb. (N.S.) 79 (121) (1969), 307–356 (Russian). MR 0259925
T. A. Chapman, Approximation results in topological manifolds, Mem. Amer. Math. Soc. 34 (1981), no. 251, iii+64. MR 634341, DOI https://doi.org/10.1090/memo/0251

F. Connolly and B. Vajiac, An end theorem for stratified spaces, (preprint).

Karl Heinz Dovermann and Reinhard Schultz, Equivariant surgery theories and their periodicity properties, Lecture Notes in Mathematics, vol. 1443, Springer-Verlag, Berlin, 1990. MR 1077825

R. D. Edwards, TOP regular neighborhoods, handwritten manuscript, 1973.

Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88. MR 283802, DOI https://doi.org/10.2307/1970753

Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR 932724

C. Bruce Hughes, Approximate fibrations on topological manifolds, Michigan Math. J. 32 (1985), no. 2, 167–183. MR 783571, DOI https://doi.org/10.1307/mmj/1029003184

---, The geometric topology of stratified spaces, (in preparation).

B. Hughes and A. Ranicki, Ends of complexes, Cambridge Tracts in Math. 123, Cambridge Univ. Press, Cambridge, 1996.

C. B. Hughes, L. R. Taylor, and E. B. Williams, Bundle theories for topological manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 1, 1–65. MR 1010410, DOI https://doi.org/10.1090/S0002-9947-1990-1010410-8

C. Bruce Hughes, Laurence R. Taylor, and E. Bruce Williams, Manifold approximate fibrations are approximately bundles, Forum Math. 3 (1991), no. 4, 309–325. MR 1115949, DOI https://doi.org/10.1515/form.1991.3.309

B. Hughes, L. Taylor, S. Weinberger, and B. Williams, Neighborhoods in stratified spaces with two strata, (preprint).

J. Mather, Notes on topological stability, Harvard Univ., Cambridge, 1970, (photocopied).

Frank Quinn, Ends of maps. II, Invent. Math. 68 (1982), no. 3, 353–424. MR 669423, DOI https://doi.org/10.1007/BF01389410

Frank Quinn, Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988), no. 2, 441–499. MR 928266, DOI https://doi.org/10.1090/S0894-0347-1988-0928266-2

---, 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.

C. P. Rourke and B. J. Sanderson, An embedding without a normal microbundle, Invent. Math. 3 (1967), 293–299. MR 222904, DOI https://doi.org/10.1007/BF01402954

L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123–136; ibid. 47 (1972), 137–163. MR 319207, DOI https://doi.org/10.1007/BF02566793

Mark Steinberger, The equivariant topological $s$-cobordism theorem, Invent. Math. 91 (1988), no. 1, 61–104. MR 918237, DOI https://doi.org/10.1007/BF01404913

Mark Steinberger and James West, Equivariant $h$-cobordisms and finiteness obstructions, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 217–220. MR 776472, DOI https://doi.org/10.1090/S0273-0979-1985-15351-0

R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240–284 (French). MR 239613, DOI https://doi.org/10.1090/S0002-9904-1969-12138-5

Andrei Verona, Stratified mappings—structure and triangulability, Lecture Notes in Mathematics, vol. 1102, Springer-Verlag, Berlin, 1984. MR 771120

Shmuel Weinberger, Constructions of group actions: a survey of some recent developments, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 269–298. MR 780967, DOI https://doi.org/10.1090/conm/036/780967

Shmuel Weinberger, The topological classification of stratified spaces, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR 1308714

Hassler Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, pp. 205–244. MR 0188486

Min Yan, The periodicity in stable equivariant surgery, Comm. Pure Appl. Math. 46 (1993), no. 7, 1013–1040. MR 1223661, DOI https://doi.org/10.1002/cpa.3160460704