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Roots of mappings from manifolds

Abstract

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.

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Correspondence to Robin Brooks.

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Brooks, R. Roots of mappings from manifolds. Fixed Point Theory Appl 2004, 643139 (2004). https://doi.org/10.1155/S1687182004406093

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