Extending combinatorial piecewise linear structures on stratified spaces. II
Authors:
Douglas R. Anderson and Wu Chung Hsiang
Journal:
Trans. Amer. Math. Soc. 260 (1980), 223-253
MSC:
Primary 57Q25
DOI:
https://doi.org/10.1090/S0002-9947-1980-0570787-8
MathSciNet review:
570787
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let X be a stratified space and suppose that both the complement of the n-skeleton and the n-stratum have been endowed with combinatorial piecewise linear (PL) structures. In this paper we investigate the problem of “fitting together” these separately given PL structures to obtain a single combinatorial PL structure on the complement of the $(n - 1)$-skeleton. The first main result of this paper reduces the geometrically given “fitting together” problem to a standard kind of obstruction theory problem. This is accomplished by introducing a tangent bundle for the n-stratum and using immersion theory to show that the “fitting together” problem is equivalent to reducing the structure group of the tangent bundle of the n-stratum to an appropriate group of PL homeomorphisms. The second main theorem describes a method for computing the homotopy groups arising in the obstruction theory problem via spectral sequence methods. In some cases, the spectral sequences involved are fairly small and the first few differentials are described. This paper is an outgrowth of earlier work by the authors on this problem.
- Ethan Akin, Manifold phenomena in the theory of polyhedra, Trans. Amer. Math. Soc. 143 (1969), 413–473. MR 253329, DOI https://doi.org/10.1090/S0002-9947-1969-0253329-1
- Douglas R. Anderson and Wu Chung Hsiang, Extending combinatorial ${\rm PL}$ structures on stratified spaces, Invent. Math. 32 (1976), no. 2, 179–204. MR 413114, DOI https://doi.org/10.1007/BF01389961
- 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
- Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491 D. Carter, Negative K-theory for group rings of finite groups, Thesis, Columbia University, 1978.
- Marshall M. Cohen, A general theory of relative regular neighborhoods, Trans. Amer. Math. Soc. 136 (1969), 189–229. MR 248802, DOI https://doi.org/10.1090/S0002-9947-1969-0248802-6
- 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
- F. T. Farrell and W. C. Hsiang, Manifolds with $\pi _{i}=G\times \alpha T$, Amer. J. Math. 95 (1973), 813–848. MR 385867, DOI https://doi.org/10.2307/2373698
- André Haefliger and Valentin Poenaru, La classification des immersions combinatoires, Inst. Hautes Études Sci. Publ. Math. 23 (1964), 75–91 (French). MR 172296
- A. E. Hatcher, Higher simple homotopy theory, Ann. of Math. (2) 102 (1975), no. 1, 101–137. MR 383424, DOI https://doi.org/10.2307/1970977
- A. E. Hatcher, Concordance spaces, higher simple-homotopy theory, and applications, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 3–21. MR 520490
- J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR 0248844
- R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742–749. MR 242166, DOI https://doi.org/10.1090/S0002-9904-1969-12271-8
- Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah; Annals of Mathematics Studies, No. 88. MR 0645390
- J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR 180986, DOI https://doi.org/10.2307/1970498
- C. Lacher, Locally flat strings and half-strings, Proc. Amer. Math. Soc. 18 (1967), 299–304. MR 212805, DOI https://doi.org/10.1090/S0002-9939-1967-0212805-1
- R. Lashof, The immersion approach to triangulation and smoothing, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 131–164. MR 0317332
- R. Lashof and M. Rothenberg, Triangulation of manifolds. I, II, Bull. Amer. Math. Soc. 75 (1969), 750–754; ibid. 75 (1969), 755–757. MR 0247631, DOI https://doi.org/10.1090/S0002-9904-1969-12272-X
- R. Lashof and M. Rothenberg, $G$-smoothing theory, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 211–266. MR 520506
- John Milnor, Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. (2) 74 (1961), 575–590. MR 133127, DOI https://doi.org/10.2307/1970299
- J. Milnor, Topological manifolds and smooth manifolds, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 132–138. MR 0161345
- Julius L. Shaneson, Wall’s surgery obstruction groups for $G\times Z$, Ann. of Math. (2) 90 (1969), 296–334. MR 246310, DOI https://doi.org/10.2307/1970726 L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension $\geqslant \,5$, Thesis, Princeton Univ., Princeton, N. J., 1965.
- L. C. Siebenmann, Pseudo-annuli and invertible cobordisms, Arch. Math. (Basel) 19 (1968), 528–535. MR 239611, DOI https://doi.org/10.1007/BF01898777
- L. C. Siebenmann, Topological manifolds, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 133–163. MR 0423356
- 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
- John R. Stallings, On infinite processes leading to differentiability in the complement of a point, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 245–254. MR 0180983
- C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two, Proc. Cambridge Philos. Soc. 61 (1965), 659–664. MR 184249, DOI https://doi.org/10.1017/s0305004100039001 E. C. Zeeman, Seminar on combinatorial topology, Mimeograph Notes, I.H.E.S., Paris, 1963.
Retrieve articles in Transactions of the American Mathematical Society with MSC: 57Q25
Retrieve articles in all journals with MSC: 57Q25
Additional Information
Keywords:
Stratified space,
locally triangulable space,
immersion theory,
isotopy extension theorem,
algebraic <I>K</I>-theory
Article copyright:
© Copyright 1980
American Mathematical Society