Geometry & Topology

Examples of exotic stratifications

Bruce Hughes, Laurence R Taylor, Shmuel Weinberger, and E Bruce Williams

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Abstract

We produce examples of manifold stratified pairs in which the lower strata do not have neighborhoods that are mapping cylinders of fiber bundles, or even block bundles. Moreover, the examples do not improve in this regard under stabilization by products with tori. The examples are locally conelike and the lower strata do have neighborhoods which are mapping cylinders of manifold approximate fibrations. They are constructed by combining the classification of manifold approximate fibrations with the authors’ classification of neighborhood germs.

Article information

Source
Geom. Topol., Volume 11, Number 3 (2007), 1477-1505.

Dates
Revised: 31 May 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799900

Digital Object Identifier
doi:10.2140/gt.2007.11.1477

Mathematical Reviews number (MathSciNet)
MR2326949

Zentralblatt MATH identifier
1136.57012

Subjects
Primary: 57N80: Stratifications
Secondary: 19J99: None of the above, but in this section 55R65: Generalizations of fiber spaces and bundles 57N55: Microbundles and block bundles [See also 55R60, 57Q50]

Keywords
locally conelike stratified space manifold approximate fibration

Citation

Hughes, Bruce; Taylor, Laurence R; Weinberger, Shmuel; Williams, E Bruce. Examples of exotic stratifications. Geom. Topol. 11 (2007), no. 3, 1477--1505. doi:10.2140/gt.2007.11.1477. https://projecteuclid.org/euclid.gt/1513799900


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References

  • D R Anderson, A triangulable space which possesses no natural piecewise linear structure, Topology 17 (1978) 297–302
  • D R Anderson, W C Hsiang, Extending combinatorial ${\rm PL}$ structures on stratified spaces, Invent. Math. 32 (1976) 179–204
  • D R Anderson, W C Hsiang, The functors $K\sb{-1}$ and pseudo-isotopies of polyhedra, Ann. of Math. $(2)$ 105 (1977) 201–223
  • D R Anderson, W C Hsiang, Extending combinatorial piecewise linear structures on stratified spaces. II, Trans. Amer. Math. Soc. 260 (1980) 223–253
  • H Bass, $K$–theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. (1964) 5–60
  • H Bass, J Milnor, J-P Serre, Solution of the congruence subgroup problem for ${\rm SL}\sb{n}\,(n\geq 3)$ and ${\rm Sp}\sb{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. 33 (1967) 59–137
  • W Browder, F Quinn, A surgery theory for $G$–manifolds and stratified sets, from: “Manifolds–-Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973)”, Univ. Tokyo Press, Tokyo (1975) 27–36
  • S Cappell, J L Shaneson, Nonlinear similarity, Ann. of Math. $(2)$ 113 (1981) 315–355
  • S Cappell, S Weinberger, A simple construction of Atiyah–Singer classes and piecewise linear transformation groups, J. Differential Geom. 33 (1991) 731–742
  • R J Daverman, Problems about finite-dimensional manifolds, from: “Open problems in topology”, North-Holland, Amsterdam (1990) 431–455
  • R D Edwards, R C Kirby, Deformations of spaces of imbeddings, Ann. Math. $(2)$ 93 (1971) 63–88
  • F T Farrell, The obstruction to fibering a manifold over a circle, Bull. Amer. Math. Soc. 73 (1967) 737–740
  • F T Farrell, The obstruction to fibering a manifold over a circle, from: “Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2”, Gauthier-Villars, Paris (1971) 69–72
  • F T Farrell, The obstruction to fibering a manifold over a circle, Indiana Univ. Math. J. 21 (1971/1972) 315–346
  • S C Ferry, E K Pedersen, Some mildly wild circles in $S\sp n$ arising from algebraic $K$–theory, K–Theory 4 (1991) 479–499
  • J-C Hausmann, $h$–cobordismes entre variétés homéomorphes, Comment. Math. Helv. 50 (1975) 9–13
  • B Hughes, Bounded homotopy equivalences of Hilbert cube manifolds, Trans. Amer. Math. Soc. 287 (1985) 621–643
  • B Hughes, L R Taylor, S Weinberger, B Williams, Neighborhoods in stratified spaces with two strata, Topology 39 (2000) 873–919
  • B Hughes, L R Taylor, B Williams, Bundle theories for topological manifolds, Trans. Amer. Math. Soc. 319 (1990) 1–65
  • B Hughes, L R Taylor, B Williams, Manifold approximate fibrations are approximately bundles, Forum Math. 3 (1991) 309–325
  • B Hughes, L R Taylor, B Williams, Bounded homeomorphisms over Hadamard manifolds, Math. Scand. 73 (1993) 161–176
  • B Hughes, L R Taylor, B Williams, Rigidity of fibrations over nonpositively curved manifolds, Topology 34 (1995) 565–574
  • L S Husch, Approximating approximate fibrations by fibrations, Canad. J. Math. 29 (1977) 897–913
  • K W Kwun, R H Szczarba, Product and sum theorems for Whitehead torsion, Ann. of Math. $(2)$ 82 (1965) 183–190
  • R Lashof, M Rothenberg, $G$–smoothing theory, from: “Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Part 1”, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I. (1978) 211–266
  • T C Lawson, Inertial $h$–cobordisms with finite cyclic fundamental group, Proc. Amer. Math. Soc. 44 (1974) 492–496
  • W Ling, Translations on $M{\times}\mathbb{R}$, from: “Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Part 2”, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I. (1978) 167–180
  • J N Mather, Notes on topological stability, Harvard Univ., Cambridge, photocopied (1970)
  • J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358–426
  • R Oliver, Whitehead groups of finite groups, London Mathematical Society Lecture Note Series 132, Cambridge University Press, Cambridge (1988)
  • F Quinn, Ends of maps. I, Ann. of Math. $(2)$ 110 (1979) 275–331
  • F Quinn, Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988) 441–499
  • A Ranicki, Lower $K$– and $L$–theory, London Mathematical Society Lecture Note Series 178, Cambridge University Press, Cambridge (1992)
  • J Rosenberg, Algebraic $K$–theory and its applications, Graduate Texts in Mathematics 147, Springer, New York (1994)
  • L C Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972) 271–294
  • L C Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972) 123–136; ibid. 47 (1972), 137–163
  • L Siebenmann, J Sondow, Some homeomorphic sphere pairs that are combinatorially distinct, Comment. Math. Helv. 41 (1966/1967) 261–272
  • R Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969) 240–284
  • F Waldhausen, Algebraic $K$–theory of spaces, a manifold approach, from: “Current trends in algebraic topology, Part 1 (London, Ont., 1981)”, CMS Conf. Proc. 2, Amer. Math. Soc., Providence, R.I. (1982) 141–184
  • C T C Wall, Norms of units in group rings, Proc. London Math. Soc. $(3)$ 29 (1974) 593–632
  • L C Washington, Introduction to cyclotomic fields, second edition, Graduate Texts in Mathematics 83, Springer, New York (1997)
  • S Weinberger, The topological classification of stratified spaces, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1994)
  • M Weiss, B Williams, Automorphisms of manifolds and algebraic $K$–theory. I, K–Theory 1 (1988) 575–626
  • H Whitney, Local properties of analytic varieties, from: “Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse)”, Princeton Univ. Press, Princeton, NJ (1965) 205–244
  • M Yan, The periodicity in stable equivariant surgery, Comm. Pure Appl. Math. 46 (1993) 1013–1040