Journal of Mathematical Sciences

, Volume 147, Issue 6, pp 7155–7217 | Cite as

Combinatorial fiber bundles and fragmentation of a fiberwise PL homeomorphism



With a compact PL manifold X we associate a category \(\mathfrak{T}(X)\). The objects of \(\mathfrak{T}(X)\) are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence
$$B\mathfrak{T}(X) \approx BPL(X)$$
holds, where PL(X) is the simplicial group of PL homeomorphisms. Thus the space \(B\mathfrak{T}(X)\) is a canonical countable (as a CW-complex) model of BPL (X). As a result, we obtain functorial pure combinatorial models for PL fiber bundles with fiber X and a PL polyhedron B as the base. Such a model looks like a \(\mathfrak{T}(X)\)-coloring of some triangulation K of B. The vertices of K are colored by objects of \(\mathfrak{T}(X)\), and the arcs are colored by morphisms in such a way that the diagram arising from the 2-skeleton of K is commutative. Comparing with the classical results of geometric topology, we obtain combinatorial models of the real Grassmannian in small dimensions: \(B\mathfrak{T}(S^{n - 1} ) \approx BO(n)\) for n = 1, 2, 3, 4. The result is proved in a sequence of results on similar models of BPL (X). Special attention is paid to the main noncompact case X = ℝn and to the tangent bundle and Gauss functor of a combinatorial manifold. The trick that makes the proof possible is a collection of lemmas on “fragmentation of a fiberwise homeomorphism,” a generalization of the folklore lemma on fragmentation of an isotopy. Bibliography: 34 titles.


Simplicial Complex Graph System Trivial Bundle Minimal Base Ball Complex 
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  1. 1.
    D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk, “Torification and factorization of birational maps,” J. Amer. Math. Soc., 15,No. 3, 531–572 (2002).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. W. Alexander, “The combinatorial theory of complexes,” Ann. Math. (2), 31,No. 2, 292–320 (1930).CrossRefGoogle Scholar
  3. 3.
    P. S. Alexandroff, “Discrete Räume,” Mat. Sb. (N.S.), 2, 501–518 (1937).MATHGoogle Scholar
  4. 4.
    L. Anderson and N. Mnev, “Triangulations of manifolds and combinatorial bundle theory: an announcement,” Zap. Nauchn. Semin. POMI, 267, 46–52 (2000).Google Scholar
  5. 5.
    F. G. Arenas, “Alexandroff spaces,” Acta Math. Univ. Comenian. (N.S.), 68,No. 1, 17–25 (1999).MATHMathSciNetGoogle Scholar
  6. 6.
    D. K. Biss, “The homotopy type of the matroid Grassmannian,” Ann. Math. (2), 158,No. 3, 929–952 (2003).MATHMathSciNetGoogle Scholar
  7. 7.
    A. Björner, “Posets, regular CW complexes, and Bruhat order,” European J. Combin., 5,No. 1, 7–16 (1984).MATHMathSciNetGoogle Scholar
  8. 8.
    E. H. Brown, Jr, “Cohomology theories,” Ann. of Math. (2), 75, 467–484 (1962).CrossRefMathSciNetGoogle Scholar
  9. 9.
    A. J. Casson, “Generalisations and applications of block bundles,” in: The Hauptvermutung Book, Kluwer Acad. Publ., Dordrecht (1996), pp. 33–67.Google Scholar
  10. 10.
    M. M. Cohen, “Simplicial structures and transverse cellularity,” Ann. Math. (2), 85, 218–245 (1967).CrossRefGoogle Scholar
  11. 11.
    P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Birkhäuser Verlag, Basel (1999).MATHGoogle Scholar
  12. 12.
    A. E. Hatcher, “Higher simple homotopy theory,” Ann. Math. (2), 102,No. 1, 101–137 (1975).CrossRefMathSciNetGoogle Scholar
  13. 13.
    A. E. Hatcher, “A proof of a Smale conjecture, Diff(S 3) ≃ O(4),” Ann. Math. (2), 117,No. 3, 553–607 (1983).CrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Heller, “Homotopy resolutions of semi-simplicial complexes,” Trans. Amer. Math. Soc., 80, 299–344 (1955).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. F. P. Hudson, Piecewise Linear Topology, W. A. Benjamin, New York-Amsterdam (1969).MATHGoogle Scholar
  16. 16.
    J. L. Kelley, General Topology, Van Nostrand Company, Toronto-New York-London (1955).MATHGoogle Scholar
  17. 17.
    N. H. Kuiper and R. K. Lashof, “Microbundles and bundles. I. Elementary theory,” Invent. Math., 1, 1–17 (1966).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    N. H. Kuiper and R. K. Lashof, “Microbundles and bundles. II. Semisimplicial theory,” Invent. Math., 1, 243–259 (1966).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    N. Levitt, Grassmannians and Gauss Maps in Piecewise-Linear Topology. Lect. Notes Math., Vol. 1366, Springer-Verlag, Berlin (1989).MATHGoogle Scholar
  20. 20.
    J.-L. Loday, Cyclic Homology, 2nd edition, Springer-Verlag, Berlin (1998).MATHGoogle Scholar
  21. 21.
    A. Lundell and S. Weingram, The Topology of CW-Complexes, Van Nostrand Reinhold, New York (1969).MATHGoogle Scholar
  22. 22.
    J. P. May, Classifying Spaces and Fibrations, Mem. Amer. Math. Soc., 1,No. 155 (1975).Google Scholar
  23. 23.
    M. C. McCord, “Singular homology groups and homotopy groups of finite topological spaces,” Duke Math. J., 33, 465–474 (1966).MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    D. McDuff, “On the classifying spaces of discrete monoids,” Topology, 18,No. 4, 313–320 (1979).MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    J. Milnor, Microbundles and Differential Structures (Mimeographed notes), Princeton University, September 1961.Google Scholar
  26. 26.
    N. E. Mnëv and G. M. Ziegler, “Combinatorial models for the finite-dimensional Grassmannians,” Discrete Comput. Geom., 10,No. 3, 241–250 (1993).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    R. Morelli, “The birational geometry of toric varieties,” J. Algebraic Geom., 5,No. 4, 751–782 (1996).MATHMathSciNetGoogle Scholar
  28. 28.
    D. Quillen, “Homotopy properties of the poset of nontrivial p-subgroups of a group,” Adv. Math., 28,No. 2, 101–128 (1978).MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    C. M. Rourke and B. J. Sanderson, “Δ-sets I: Homotopy theory,” Quart. J. Math., 22, 321–338 (1971).MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Springer-Verlag, New York (1972).MATHGoogle Scholar
  31. 31.
    M. Steinberger, “The classification of PL fibrations,” Michigan Math. J., 33,No. 1, 11–26 (1986).MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    J. H. C. Whitehead, “Simplicial spaces, nuclei and m-groups,” Proc. London Math. Soc., 45, 243–327 (1939).MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    E. C. Zeeman, Seminar on Combinatorial Topology (Mimeographed notes), I.H.E.S., Paris (1963).Google Scholar
  34. 34.
    E. C. Zeeman, “Relative simplicial approximation,” Proc. Cambridge Philos. Soc., 60, 39–43 (1964).MATHMathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.St.Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

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