Embeddings of open manifolds
HTML articles powered by AMS MathViewer
- by Nancy Cardim PDF
- Trans. Amer. Math. Soc. 351 (1999), 2353-2373 Request permission
Abstract:
Let $TOP(M)$ be the simplicial group of homeomorphisms of $M$. The following theorems are proved. Theorem A. Let $M$ be a topological manifold of dim $\geq$ 5 with a finite number of tame ends $\varepsilon _{i}$, $1\leq i\leq k$. Let $TOP^{ep}(M)$ be the simplicial group of end preserving homeomorphisms of $M$. Let $W_{i}$ be a periodic neighborhood of each end in $M$, and let $p_{i}: W_{i} \to \mathbb {R}$ be manifold approximate fibrations. Then there exists a map $f: TOP^{ep}(M) \to \prod _{i} TOP^{ep}(W_{i})$ such that the homotopy fiber of $f$ is equivalent to $TOP_{cs}(M)$, the simplicial group of homeomorphisms of $M$ which have compact support. Theorem B. Let $M$ be a compact topological manifold of dim $\geq$ 5, with connected boundary $\partial M$, and denote the interior of $M$ by $Int M$. Let $f: TOP(M)\to TOP(Int M)$ be the restriction map and let $\mathcal {G}$ be the homotopy fiber of $f$ over $id_{Int M}$. Then $\pi _{i} \mathcal {G}$ is isomorphic to $\pi _{i} \mathcal {C} (\partial M)$ for $i > 0$, where $\mathcal {C} (\partial M)$ is the concordance space of $\partial M$. Theorem C. Let $q_{0}: W \to \mathbb {R}$ be a manifold approximate fibration with dim $W \geq$ 5. Then there exist maps $\alpha : \pi _{i} TOP^{ep}(W) \to \pi _{i} TOP(\hat W)$ and $\beta : \pi _{i} TOP(\hat W) \to \pi _{i} TOP^{ep}(W)$ for $i >1$, such that $\beta \circ \alpha \simeq id$, where $\hat W$ is a compact and connected manifold and $W$ is the infinite cyclic cover of $\hat W$.References
- J. Adams, On the triad connectivity theorem, unpublished lecture notes.
- Douglas R. Anderson and Wu Chung Hsiang, Extending combinatorial $\textrm {PL}$ structures on stratified spaces, Invent. Math. 32 (1976), no. 2, 179–204. MR 413114, DOI 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 10.2307/1970997
- Dan Burghelea, Automorphisms of manifolds, 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. 347–371. MR 520511
- Dan Burghelea, Richard Lashof, and Melvin Rothenberg, Groups of automorphisms of manifolds, Lecture Notes in Mathematics, Vol. 473, Springer-Verlag, Berlin-New York, 1975. With an appendix (“The topological category”) by E. Pedersen. MR 0380841, DOI 10.1007/BFb0079981
- T. A. Chapman, Approximation results in topological manifolds, Mem. Amer. Math. Soc. 34 (1981), no. 251, iii+64. MR 634341, DOI 10.1090/memo/0251
- Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88. MR 283802, DOI 10.2307/1970753
- Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
- 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
- E. Rees and J. D. S. Jones (eds.), Homotopy theory, London Mathematical Society Lecture Note Series, vol. 117, Cambridge University Press, Cambridge, 1987. MR 932256
- Wu Chung Hsiang, Geometric applications of algebraic $K$-theory, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 99–118. MR 804679
- C. Bruce Hughes, Approximate fibrations on topological manifolds, Michigan Math. J. 32 (1985), no. 2, 167–183. MR 783571, DOI 10.1307/mmj/1029003184
- Bruce Hughes and Andrew Ranicki, Ends of complexes, Cambridge Tracts in Mathematics, vol. 123, Cambridge University Press, Cambridge, 1996. MR 1410261, DOI 10.1017/CBO9780511526299
- 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 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 10.1515/form.1991.3.309
- C. B. Hughes, L. R. Taylor, and E. B. Williams, Bounded homeomorphisms over Hadamard manifolds, Math. Scand. 73 (1993), no. 2, 161–176. MR 1269255, DOI 10.7146/math.scand.a-12462
- —, Splitting Forget Control Maps, in preparation.
- B. Hughes, L. Taylor, S. Weinberger and B. Williams, Neighborhoods in Stratified Spaces. I. Two Strata (to appear).
- Kiyoshi Igusa, Parametrized Morse theory and its applications, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 643–651. MR 1159251
- Robion C. Kirby, Stable homeomorphisms and the annulus conjecture, Ann. of Math. (2) 89 (1969), 575–582. MR 242165, DOI 10.2307/1970652
- Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390, DOI 10.1515/9781400881505
- J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR 180986, DOI 10.2307/1970498
- N. H. Kuiper and R. K. Lashof, Microbundles and bundles. I. Elementary theory, Invent. Math. 1 (1966), 1–17. MR 216506, DOI 10.1007/BF01389695
- 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
- Andrew J. Nicas, Induction theorems for groups of homotopy manifold structures, Mem. Amer. Math. Soc. 39 (1982), no. 267, vi+108. MR 668807, DOI 10.1090/memo/0267
- Frank Quinn, Ends of maps. I, Ann. of Math. (2) 110 (1979), no. 2, 275–331. MR 549490, DOI 10.2307/1971262
- Frank Quinn, Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988), no. 2, 441–499. MR 928266, DOI 10.1090/S0894-0347-1988-0928266-2
- C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744, DOI 10.1007/978-3-642-81735-9
- T. Benny Rushing, Topological embeddings, Pure and Applied Mathematics, Vol. 52, Academic Press, New York-London, 1973. MR 0348752
- L.Siebenmann, The obstruction to finding a boundary for an open neighborhood, Ph.D. Thesis, Princeton University, Princeton, 1966.
- —, The structure of tame ends, Notices of AMS 13 (1966), 862.
- —, A torsion invariant for bands, Notices of AMS 15 (1968), 811.
- L. C. Siebenmann, A total Whitehead torsion obstruction to fibering over the circle, Comment. Math. Helv. 45 (1970), 1–48. MR 287564, DOI 10.1007/BF02567315
- 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 10.1007/BF02566793
- L. C. Siebenmann, Regular (or canonical) open neighborhoods, General Topology and Appl. 3 (1973), 51–61. MR 370604, DOI 10.1016/0016-660X(73)90030-5
- L. Siebenmann, L. Guillou, and H. Hähl, Les voisinages ouverts réguliers, Ann. Sci. École Norm. Sup. (4) 6 (1973), 253–293 (French). MR 331399, DOI 10.24033/asens.1248
- L. Siebenmann, L. Guillou, and H. Hähl, Les voisinages ouverts réguliers: critères homotopiques d’existence, Ann. Sci. École Norm. Sup. (4) 7 (1974), 431–461 (1975) (French). MR 362324, DOI 10.24033/asens.1275
- Michael Weiss and Bruce Williams, Automorphisms of manifolds and algebraic $K$-theory. I, $K$-Theory 1 (1988), no. 6, 575–626. MR 953917, DOI 10.1007/BF00533787
Additional Information
- Nancy Cardim
- Affiliation: Univeridade Federal Fluminense - UFF, Instituto de Matemática, Departamento de Análise, Niterói, RJ, 24020-005 - Brazil
- Email: ganancy@vm.uff.br
- Received by editor(s): November 20, 1996
- Published electronically: January 27, 1999
- Additional Notes: Partially suported by the CNPq of Brazil
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 2353-2373
- MSC (1991): Primary 57N37; Secondary 57N35, 57N45
- DOI: https://doi.org/10.1090/S0002-9947-99-02430-7
- MathSciNet review: 1641091