Fixed Point Theory and Applications
Volume 2004 (2004), Issue 4, Pages 273-307

Roots of mappings from manifolds

Robin Brooks

Department of Mathematics, Bates College, 2 Andrews Road, Lewiston 04240 -6028, ME, USA

Received 15 June 2004

Copyright © 2004 Robin Brooks. 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.


Assume that f:XY is a proper map of a connected n-manifold X into a Hausdorff, connected, locally path-connected, and semilocally simply connected space Y, and y0Y has a neighborhood homeomorphic to Euclidean n-space. The proper Nielsen number of f at y0 and the absolute degree of f at y0 are defined in this setting. The proper Nielsen number is shown to a lower bound on the number of roots at y0 among all maps properly homotopic to f, and the absolute degree is shown to be a lower bound among maps properly homotopic to f and transverse to y0. When n>2, these bounds are shown to be sharp. An example of a map meeting these conditions is given in which, in contrast to what is true when Y is a manifold, Nielsen root classes of the map have different multiplicities and essentialities, and the root Reidemeister number is strictly greater than the Nielsen root number, even when the latter is nonzero.