Let G be a compact group and X a G-ANR. Then X is a G-AR iff the H-fixed point set is homotopy trivial for each closed subgroup H ⊂ G.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-1,
author = {Sergey A. Antonyan},
title = {G-ANR's with homotopy trivial fixed point sets},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {1-16},
zbl = {1149.54011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-1}
}
Sergey A. Antonyan. G-ANR's with homotopy trivial fixed point sets. Fundamenta Mathematicae, Tome 193 (2007) pp. 1-16. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-1/