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/