Roots of mappings from manifolds

• Main Author: Brooks Robin
• Format: Article
• Language: English
• Published: SpringerOpen 2004-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2004/643139

<p/> <p>Assume that <inline-formula><graphic file="1687-1812-2004-643139-i1.gif"/></inline-formula> is a proper map of a connected <inline-formula><graphic file="1687-1812-2004-643139-i2.gif"/></inline-formula>-manifold <inline-formula><graphic file="1687-1812-2004-643139-i3.gif"/></inline-formula> into a Hausdorff, connected, locally path-connected, and semilocally simply connected space <inline-formula><graphic file="1687-1812-2004-643139-i4.gif"/></inline-formula>, and <inline-formula><graphic file="1687-1812-2004-643139-i5.gif"/></inline-formula> has a neighborhood homeomorphic to Euclidean <inline-formula><graphic file="1687-1812-2004-643139-i6.gif"/></inline-formula>-space. The proper Nielsen number of <inline-formula><graphic file="1687-1812-2004-643139-i7.gif"/></inline-formula> at <inline-formula><graphic file="1687-1812-2004-643139-i8.gif"/></inline-formula> and the absolute degree of <inline-formula><graphic file="1687-1812-2004-643139-i9.gif"/></inline-formula> at <inline-formula><graphic file="1687-1812-2004-643139-i10.gif"/></inline-formula> are defined in this setting. The proper Nielsen number is shown to a lower bound on the number of roots at <inline-formula><graphic file="1687-1812-2004-643139-i11.gif"/></inline-formula> among all maps properly homotopic to <inline-formula><graphic file="1687-1812-2004-643139-i12.gif"/></inline-formula>, and the absolute degree is shown to be a lower bound among maps properly homotopic to <inline-formula><graphic file="1687-1812-2004-643139-i13.gif"/></inline-formula> and transverse to <inline-formula><graphic file="1687-1812-2004-643139-i14.gif"/></inline-formula>. When <inline-formula><graphic file="1687-1812-2004-643139-i15.gif"/></inline-formula>, 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 <inline-formula><graphic file="1687-1812-2004-643139-i16.gif"/></inline-formula> 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.</p>