## Topological Methods in Nonlinear Analysis

### Maps on graphs can be deformed to be coincidence free

P. Christopher Staecker

#### Abstract

We give a construction to remove coincidence points of continuous maps on graphs ($1$-complexes) by changing the maps by homotopies. When the codomain is not homeomorphic to the circle, we show that any pair of maps can be changed by homotopies to be coincidence free. This means that there can be no nontrivial coincidence index, Nielsen coincidence number, or coincidence Reidemeister trace in this setting, and the results of our previous paper A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles'' are invalid.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 37, Number 2 (2011), 377-381.

Dates
First available in Project Euclid: 20 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461184791

Mathematical Reviews number (MathSciNet)
MR2849828

Zentralblatt MATH identifier
1233.55003

#### Citation

Staecker, P. Christopher. Maps on graphs can be deformed to be coincidence free. Topol. Methods Nonlinear Anal. 37 (2011), no. 2, 377--381. https://projecteuclid.org/euclid.tmna/1461184791

#### References

• R. Brooks, On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy , Pacific J. Math., 139 , 45–52 (1971) \ref\key 2
• D. L. Gonçalves, Coincidence theory for maps from a complex into a manifold , Topology Appl., 92 , 63–77 (1999) \ref\key 3
• P. C. Staecker, A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles , Topol. Methods Nonlinear Anal., 33 , 41–50 (2009)