Let G be a compact connected Lie group, K a closed subgroup and M = G/K the homogeneous space of right cosets. Suppose that M is orientable. We show that for any selfmap f: M → M, L(f) = 0 ⇒ N(f) = 0 and L(f) ≠ 0 ⇒ N(f) = R(f) where L(f), N(f), and R(f) denote the Lefschetz, Nielsen, and Reidemeister numbers of f, respectively. In particular, this implies that the Lefschetz number is a complete invariant, i.e., L(f) = 0 iff f is deformable to be fixed point free. This was previously known under the hypothesis that p⁎: Hₙ(G) → Hₙ(M) is nontrivial where n = dim M. A simple formula using equivariant degree is given for the Reidemeister trace of a selfmap f: M → M.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-2-4,
author = {Peter Wong},
title = {Fixed point theory for homogeneous spaces, II},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {161-175},
zbl = {1081.55005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-2-4}
}
Peter Wong. Fixed point theory for homogeneous spaces, II. Fundamenta Mathematicae, Tome 185 (2005) pp. 161-175. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-2-4/