Twisted products: Enveloping actions and equivariant absolute neighborhood extensors
Luis Martínez, Héctor Pinedo
TL;DR
The paper investigates how partial actions of a topological group $G$ on a space $X$ relate to global actions via twisting constructions and enveloping spaces. It develops the twisted-product framework $G\\times_K X$ and proves adjunctions that connect $\\mathcal{PA}_K$ and $\\mathcal{A}_G$, as well as metrizability criteria for $G\\times_K X$. It then studies how homotopy and contractibility behave under globalization, proving that $X_G$ inherits $G$-contractibility from $X$ and that homotopies are preserved by the globalization functor $E$. Finally, it provides sufficient conditions for $X_G$ to be a $G$-ANE (and hence for $X$ to be an ANE under niceness) in the setting of compact Lie groups and finite structures, linking local and global equivariant retract properties. These results advance equivariant topology by clarifying when and how partial-action data extend to well-behaved equivariant topological objects.
Abstract
The classical notion of twisted product is studied in the context of partial actions, in particular, we show that the globalization of a partial action is a twisted product. In addition, we establish conditions for the metrizability of twisted products, and some homotopy and categorical properties are proved. Furthermore, sufficient conditions for the enveloping space to be an equivariant absolute neighborhood extensor are also studied.