## Topological Methods in Nonlinear Analysis

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

### The trivial homotopy class of maps from two-complexes into the 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.

#### 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