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Open Access Research Article

Roots of mappings from manifolds

Robin Brooks

Author Affiliations

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

Fixed Point Theory and Applications 2004, 2004:643139  doi:10.1155/S1687182004406093

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2004/4/643139


Received:15 June 2004
Published:26 December 2004

© 2004 Brooks

Assume that is a proper map of a connected -manifold into a Hausdorff, connected, locally path-connected, and semilocally simply connected space , and has a neighborhood homeomorphic to Euclidean -space. The proper Nielsen number of at and the absolute degree of at are defined in this setting. The proper Nielsen number is shown to a lower bound on the number of roots at among all maps properly homotopic to , and the absolute degree is shown to be a lower bound among maps properly homotopic to and transverse to . When , 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 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.

Research Article