Wecken type problems for self-maps of the Klein bottle

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 f n ,g : K → K which are coincidence free but any homotopy between f n and f m , 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.

Coincidence Properties for Maps from the Torus to the Klein Bottle

“…That is, when we have g 1 = g 2 and all homotopies under consideration keep the second leg fixed at the map g 1 . In [9] we see that this problem is much more difficult for self-coincidences on the Klein bottle than it was for the torus. The fact that the torus has a multiplicative structure was crucial in resolving the minimal coincidence problem, and also implies that the restricted problem is equivalent.…”

“…Much of this work is an extension of the ideas developed in [8,9] for self-coincidences of maps on the torus and on the Klein bottle. As was done in [9], we try to understand how Nielsen theoretic properties for these maps compare with those of maps from the torus into itself.…”

“…Much of this work is an extension of the ideas developed in [8,9] for self-coincidences of maps on the torus and on the Klein bottle. As was done in [9], we try to understand how Nielsen theoretic properties for these maps compare with those of maps from the torus into itself. One objective of this work is to present a computation of the Nielsen theoretic data for pairs of maps from the torus, denoted by T in this paper, to the Klein bottle, denoted by K. We compute the Reidemeister classes, characterize defective classes and give a formula for the Nielsen coincidence number.…”

“…This problem, following the work in [9], can be described as follows: Can one find a pair H, G of homotopies from f 1 to f 2 and g 1 to g 2 respectively, such that #Coin(H( · , t), G( · , t)) = MC[f 1 , g 1 ] for all t ∈ [0, 1]? We refer to this as the 1-parameter minimal coincidence problem or simply the minimal coincidence problem when the context is clear.…”

“…In [7][8][9] the authors study the 1-parameter minimal coincidence problem, and also the related restricted minimal coincidence problem in the setting of surface mappings. That is, when we have g 1 = g 2 and all homotopies under consideration keep the second leg fixed at the map g 1 .…”