Journal of Mathematical Sciences

, Volume 200, Issue 6, pp 710–717 | Cite as

A Note on the Tangent Bundle and Gauss Functor of Posets and Manifolds

Article
  • 30 Downloads

We introduce a notion of the tangent bundle of a poset. In the case where the poset is the poset of simplices of a combinatorial manifold, the construction produces the best possible combinatorial model for the geometric compactified tangent bundle.

Keywords

■■■  

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Björner, “Posets, regular CW complexes and Bruhat order,” European J. Combin., 5, No. 1, 7–16 (1984).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    A. Björner, “Topological methods,” in: Handbook of Combinatorics, Vol. 1, 2, Elsevier, Amsterdam (1995), pp. 1819–1872.Google Scholar
  3. 3.
    P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Birkhäuser Verlag, Basel (1999).CrossRefMATHGoogle Scholar
  4. 4.
    A. E. Hatcher, “Higher simple homotopy theory,” Ann. Math. (2), 102, 101–137 (1975).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    J. F. P. Hudson, Piecewise Linear Topology, W. A. Benjamin, New York–Amsterdam (1969).Google Scholar
  6. 6.
    N. H. Kuiper and R. K. Lashof, “Microbundles and bundles. I. Elementary theory,” Invent. Math., 1, 1–17 (1966).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    N. H. Kuiper and R. K. Lashof, “Microbundles and bundles. II. Semisimplical theory,” Invent. Math., 1, 243–259 (1966).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    N. Levitt, Grassmannians and Gauss Maps in Piecewise-Linear Topology, Lect. Notes Math., 1366, Springer-Verlag, Berlin (1989).Google Scholar
  9. 9.
    R. MacPherson, “The combinatorial formula of Gabrielov, Gelfand and Losik for the first Pontrjagin class,” in: Séminaire Bourbaki, 29e année (1976/77), Lect. Notes Math., 677, Springer, Berlin (1978), Exp. No. 497, pp. 105–124.Google Scholar
  10. 10.
    R. MacPherson, “Combinatorial differential manifolds,” in: Topological Methods in Modern Mathematics (Stony Brook, 1991), Publish or Perish, Houston (1993), pp. 203–221.Google Scholar
  11. 11.
    J. P. May, Classifying Spaces and Fibrations, Mem. Amer. Math. Soc., 155 (1975).Google Scholar
  12. 12.
    J Milnor, Microbundles and Differential Structures (mimeographed notes), Princeton University (1961).Google Scholar
  13. 13.
    N. Mnëv, “Combinatorial fiber bundles and fragmentation of PL fiberwise homeomorphism,” Zap. Nauchn. Semin. POMI, 344, 56–173 (2007); arXiv:0708.4039.Google Scholar
  14. 14.
    N. E. Mnëv and G. M. Ziegler, “Combinatorial models for the finite-dimensional Grassmannians,” Discrete Comput. Geom., 10, No. 3, 241–250 (1993).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    M. Steinberger, “The classification of PL fibrations,” Michigan Math. J., 33, No. 1, 11–26 (1986).CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    J. H. C. Whitehead, “Simplicial spaces, nuclei and m-groups,” Proc. London Math. Soc., 45, 243–327 (1939).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.St.Petersburg Department of Steklov Mathematical Institute; Chebyshev LaboratorySt.Petersburg State UniversitySt.PetersburgRussia

Personalised recommendations