On the space of equivariant local maps
The author's abstract reads: ``We introduce the space of equivariant local maps and present the full proof of the splitting theorem for the set of otopy classes of local maps in the case where we have a representation of a compact Lie group.'' The author's bibliography contains several papers which mention the word otopy. The paper first defining otopy is \textit{D. H. Gottlieb} and \textit{G. Samaranayake} [New York J. Math. 1, 130--148 (1995; Zbl 0883.57025)]. Otopy is a modification of homotopy given by eliminating the ``hom'' from homotopy, and the leading O representing open domains. (Some topologists thought the word was too ugly to use.) The author hopes ``it is worth pointing out that the main motivation for studying the space of local maps is that it forms the natural environment for topological degree theories.'' This is certainly true of local vector fields, where the local index defines the otopy vector field classes. Suppose \(V\) is a vector field on a compact manifold \(M\) with or without boundary \(\partial M\). There is a formula, originally due to \textit{M. Morse} [Am. J. Math. 51, 165--178 (1929; JFM 55.0972.02)], which was not as widely known as it should be: So let \(V\) be a continuous vector field on \(M\) with no zeros on \(\partial M\). Then \(\mathrm{Ind}(V)+\mathrm{Ind}(\partial-V)=\xi(M)\) (my notation) where \(\partial-V\) is the vector field \(V\) restricted to \(\partial M\) and projected onto \(\partial-M\) which is the portion of \(\partial M\) where \(V\) points inside. The local vector field \(\partial-V\) changes by an otopy as \(V\) changes by a homotopy.
scientific article; zbMATH DE number 6700685
scientific article