Orbit spaces of Hilbert manifolds

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• 作者：Sergey A. Antonyan ; ^{antonyan@unam.mx" class="auth_mail" title="E-mail the corresponding author} ; Natalia Jonard-Pé ; rez ^{nat@ciencias.unam.mx" class="auth_mail" title="E-mail the corresponding author} ; Saú ; l Juá ; rez-Ordó ; ñ ; ez ^{sjo@ciencias.unam.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Polish group ; Hyperspace of compact convex sets ; Affine action ; Orbit space ; ℓ_{2ℓ2-manifold}
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：439
• 期：2
• 页码：725-736
• 全文大小：350 K

文摘

Let G be a compact group acting on a Polish group X   by means of automorphisms. It is proved that the orbit space X/G$X/G$ is an ℓ₂${\ell }_2$-manifold (resp., homeomorphic to ℓ₂${\ell }_2$) provided X is a G-ANR (resp., G  -AR) and the fixed point set X^G$X^G$ is not locally compact. It is also proved that if a compact group G acts affinely on a separable closed convex subset K   of a Fréchet space with a non-locally compact fixed point set 5b20bdccf37c5d38ee855" title="Click to view the MathML source">K^G$K^G$, then the orbit space K/G$K/G$ is homeomorphic to ℓ₂${\ell }_2$. In particular, (1) if C(Y,X)$C(Y,X)$ denotes the space of all maps from a compact metric G-space Y to a non-locally compact Polish ANR (resp., AR) group X, endowed with the compact-open topology and the induced action of G  , then the orbit space C(Y,X)/G$C(Y,X)/G$ is an ℓ₂${\ell }_2$-manifold (resp., homeomorphic to ℓ₂${\ell }_2$), and (2) if X is an infinite-dimensional separable Fréchet G  -space and cc(X)$cc(X)$ denotes the hyperspace of all non-empty compact convex subsets of X, endowed with the Hausdorff metric topology and the induced action of G  , then the orbit space 05b646a0eb962d6" title="Click to view the MathML source">cc(X)/G$cc(X)/G$ is homeomorphic to ℓ₂${\ell }_2$, whenever the fixed point set cc(X)^G$cc{(X)}^G$ is not locally compact.