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Topological Methods in Nonlinear Analysis

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

Authors

  • Marcio Colombo Fenille

DOI:

https://doi.org/10.12775/TMNA.2015.060

Keywords

Hopf--Whitney Classification Theorem, two-dimensional CW complexes, group presentation, homotopy class, cohomology group, real projective plane

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.

References

C. Aniz, Strong surjectivity of mappings of some 3-complexes into $3$-manifolds, Fund. Math. 192 (2006), 195-214.

C. Aniz, 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.

Vol 46, No 2 (December 2015)

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Published

2015-12-01

How to Cite

1.
FENILLE, Marcio Colombo. The trivial homotopy class of maps from two-complexes into the real projective plane. Topological Methods in Nonlinear Analysis [online]. 1 December 2015, T. 46, nr 2, s. 603–616. [accessed 1.4.2023]. DOI 10.12775/TMNA.2015.060.
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Vol 46, No 2 (December 2015)

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