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.
Highlights
Let f : X → Y be a map of topological spaces and y0 ∈ Y
In Nielsen root theory, by analogy with Nielsen fixed-point theory, the roots of f are grouped into Nielsen classes, a notion of essentiality is defined, and the Nielsen root number is defined to be the number of essential root classes
We show that if f : X → Y is a proper map of a connected n-manifold X into a Hausdorff space Y
Summary
Let f : X → Y be a map of topological spaces and y0 ∈ Y. Assume y0 ∈ Y has a neighborhood homeomorphic to Euclidean n-space Rn. every map properly homotopic to f and transverse to y0 has at least Ꮽ( f , y0) roots, where Ꮽ( f , y0) denotes the absolute degree of f at y0. Assume y0 ∈ Y has a neighborhood homeomorphic to Euclidean n-space Rn. every map properly homotopic to f has at least PNR( f , y0) roots at y0, where PNR( f , y0) denotes the proper Nielsen root number of f at y0, and every Nielsen root class of f at y0 with nonzero multiplicity is properly essential. K} × Sn ⊂ Z × Sn induces an injection i : kSn ZSn. Let Sn/{S, N} denote the space formed from Sn by identifying the north and south poles. The last section completes the proofs of Theorems 1.1 and 1.2
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
