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

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.