Topological Methods in Nonlinear Analysis

The trivial homotopy class of maps from two-complexes into the real projective plane

Marcio Colombo Fenille

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study reasons related to two-dimensional CW-complexes which prevent an extension of the Hopf–Whitney Classification Theorem for maps from those complexes into the real projective plane, even in the simpler situation in which the complex has trivial second integer cohomology group. We conclude that for such a two-complex $K$, the following assertions are equivalent: (1) Every based map from $K$ into the real projective plane is based homotopic to a constant map; (2) The skeleton pair $(K,K^1)$ is homotopy equivalent to that of a model two-complex induced by a balanced group presentation; (3) The number of two-dimensional cells of $K$ is equal to the first Betti number of its one-skeleton; (4) $K$ is acyclic; (5) Every based map from $K$ into the circle $S^1$ is based homotopic to a constant map.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 603-615.

Dates
First available in Project Euclid: 21 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1458588653

Digital Object Identifier
doi:10.12775/TMNA.2015.060

Mathematical Reviews number (MathSciNet)
MR3494960

Zentralblatt MATH identifier
1370.55003

Citation

Fenille, Marcio Colombo. The trivial homotopy class of maps from two-complexes into the real projective plane. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 603--615. doi:10.12775/TMNA.2015.060. https://projecteuclid.org/euclid.tmna/1458588653


Export citation

References

  • C. Aniz, Strong surjectivity of mappings of some 3-complexes into $3$-manifolds, Fund. Math. 192 (2006), 195–214.
  • ––––, Strong surjectivity of mappings of some 3-complexes into $M_{Q_8}$, Cent. Eur. J. Math. 6 (4), (2008), 497–503.
  • R.H. Crowell and R.H. Fox, Introduction to Knot Theory, Dover Publications, Inc., Mineola, New York, 2008.
  • J.F. Davis and P. Kirk, Lectures Notes in Algebraic Topology, Graduate Studies in Mathematics, Volume 35, American Mathematical Society, 2001.
  • M.C. Fenille and O.M. Neto, Strong surjectivity of maps from $2$-complexes into the $2$-sphere, Cent. Eur. J. Math. 8 (3), (2010), 421–429.
  • Sze-Tsu Hu, Homotopy Theory, Academic Press, Inc., New York, 1959.
  • A.J. Sieradski, Algebraic Topology for Two-Dimensional Complexes, Two-dimensional Homotopy and Combinatorial Group Theory (C. Hog-Angeloni, W. Metzler and A.J. Sieradski, eds.), 51–96, Cambridge University Press, 1993.
  • G.W. Whitehead, Elements of Homotopy Theory, Springer–Verlag New York Inc., 1978.