Transverse fundamental group and projected embeddings



For a generic degree d smooth map f: N n M n we introduce its “transverse fundamental group” π(f), which reduces to π1(M) in the case where f is a covering, and in general admits a monodromy homomorphism π(f) → S |d|; nevertheless, we show that π(f) can be nontrivial even for rather simple degree 1 maps S n S n . We apply π(f) to the problem of lifting f to an embedding NM × ℝ2: for such a lift to exist, the monodromy π(f) → S |d| must factor through the group of concordance classes of |d|-component string links. At least if |d| < 7, this requires π(f) to be torsion-free.


STEKLOV Institute Double Point Braid Group Inverse Limit Free Abelian Group 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. M. Akhmet’ev, “On isotopic and discrete realizations of maps of an n-dimensional sphere in Euclidean space,” Mat. Sb. 187 (7), 3–34 (1996)MathSciNetCrossRefGoogle Scholar
  2. 1.
    P. M. Akhmet’ev, Sb. Math. 187, 951–980 (1996).MATHMathSciNetCrossRefGoogle Scholar
  3. 2.
    P. M. Akhmetiev, D. Repovš, and A. B. Skopenkov, “Obstructions to approximating maps of n-manifolds into R2n by embeddings,” Topology Appl. 123, 3–14 (2002).MATHMathSciNetCrossRefGoogle Scholar
  4. 3.
    S. Buoncristiano, C. P. Rourke, and B. J. Sanderson, A Geometric Approach to Homology Theory (Cambridge Univ. Press, Cambridge, 1976), LMS Lect. Note Ser. 18.MATHGoogle Scholar
  5. 4.
    A. Clark and R. Fokkink, “Embedding solenoids,” Fundam. Math. 181, 111–124 (2004).CrossRefGoogle Scholar
  6. 5.
    P. F. Duvall and L. S. Husch, “Embedding finite covering spaces into bundles,” Topol. Proc. 4, 361–370 (1979).Google Scholar
  7. 6.
    D. L. Goldsmith, “Homotopy of braids—in answer to a question of E. Artin,” in Topology Conference: Virginia Polytech. Inst. and State Univ., 1973 (Springer, Berlin, 1974)Google Scholar
  8. 6.
    D. L. Goldsmith, Lect. Notes Math. 375, pp. 91–96.Google Scholar
  9. 7.
    N. Habegger and X.-S. Lin, “The classification of links up to link-homotopy,” J. Am. Math. Soc. 3, 389–419 (1990).CrossRefGoogle Scholar
  10. 8.
    N. Habegger and X.-S. Lin, “On link concordance and Milnor’s ¯µ invariants,” Bull. London Math. Soc. 30, 419–428 (1998).MathSciNetCrossRefGoogle Scholar
  11. 9.
    V. L. Hansen, “Polynomial covering spaces and homomorphisms into the braid groups,” Pac. J. Math. 81, 399–410 (1979).CrossRefGoogle Scholar
  12. 10.
    V. L. Hansen, Braids and Coverings: Selected Topics (Cambridge Univ. Press, Cambridge, 1989), LMS Stud. Texts 18.Google Scholar
  13. 11.
    J. Hillman, Algebraic Invariants of Links, 2nd ed. (World Sci., Singapore, 2012), Ser. Knots Everything 52.Google Scholar
  14. 12.
    S. P. Humphries, “Torsion-free quotients of braid groups,” Int. J. Algebra Comput. 11, 363–373 (2001).CrossRefGoogle Scholar
  15. 13.
    J. Keesling and D. C. Wilson, “Embedding T n-like continua in Euclidean space,” Topology Appl. 21, 241–249 (1985).MathSciNetCrossRefGoogle Scholar
  16. 14.
    S. A. Melikhov, “Sphere eversions and realization of mappings,” Tr. Mat. Inst. im. V. A. Steklova, Ross. Akad. Nauk 247, 159–181 (2004)Google Scholar
  17. 14.
    S. A. Melikhov, Proc. Steklov Inst. Math. 247, 143–163 (2004)].Google Scholar
  18. 15.
    S. A. Melikhov, “The van Kampen obstruction and its relatives,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 266, 149–183 (2009)Google Scholar
  19. 15.
    S. A. Melikhov, Proc. Steklov Inst. Math. 266, 142–176 (2009)].MathSciNetCrossRefGoogle Scholar
  20. 16.
    S. A. Melikhov and E. V. Shchepin, “The telescope approach to embeddability of compacta,” arXiv:math/0612085 [math.GT].Google Scholar
  21. 17.
    P. V. Petersen, “Fatness of covers,” J. Reine Angew. Math. 403, 154–165 (1990).MathSciNetGoogle Scholar
  22. 18.
    M. Yamamoto, “Lifting a generic map of a surface into the plane to an embedding into 4-space,” Ill. J. Math. 51, 705–721 (2007).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations