Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 75848, 15 pages
doi:10.1155/FPTA/2006/75848

Wecken type problems for self-maps of the Klein bottle

D. L. Gonçalves1 and M. R. Kelly2

1Departamento de Matemática, IME-USP, Caixa Postal 66281, Ag. Cidade de São Paulo, São Paulo 05315-970, SP, Brazil
2Department of Mathematics and Computer Science, Loyola University, 6363 St. Charles Avenue, New Orleans 70118, LA, USA

Received 6 October 2004; Revised 1 March 2005; Accepted 21 July 2005

Copyright © 2006 D. L. Gonçalves and M. R. Kelly. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider various problems regarding roots and coincidence points for maps into the Klein bottle K. The root problem where the target is K and the domain is a compact surface with non-positive Euler characteristic is studied. Results similar to those when the target is the torus are obtained. The Wecken property for coincidences from K to K is established, and we also obtain the following 1-parameter result. Families fn,g:K→K which are coincidence free but any homotopy between fn and fm, n≠m, creates a coincidence with g. This is done for any pair of maps such that the Nielsen coincidence number is zero. Finally, we exhibit one such family where g is the constant map and if we allow for homotopies of g, then we can find a coincidence free pair of homotopies.