Open Access

Roots of mappings from manifolds

Fixed Point Theory and Applications20042004:643139

DOI: 10.1155/S1687182004406093

Received: 15 June 2004

Published: 26 December 2004

Abstract

Assume that https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq1_HTML.gif is a proper map of a connected https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq2_HTML.gif -manifold https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq3_HTML.gif into a Hausdorff, connected, locally path-connected, and semilocally simply connected space https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq4_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq5_HTML.gif has a neighborhood homeomorphic to Euclidean https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq6_HTML.gif -space. The proper Nielsen number of https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq7_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq8_HTML.gif and the absolute degree of https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq9_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq10_HTML.gif are defined in this setting. The proper Nielsen number is shown to a lower bound on the number of roots at https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq11_HTML.gif among all maps properly homotopic to https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq12_HTML.gif , and the absolute degree is shown to be a lower bound among maps properly homotopic to https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq13_HTML.gif and transverse to https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq14_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq15_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2FS1687182004406093/MediaObjects/13663_2004_Article_928_IEq16_HTML.gif 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.

Authors’ Affiliations

(1)
Department of Mathematics, Bates College

Copyright

© Brooks 2004