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Projecting homeomorphisms from covering spaces. (English) Zbl 0704.57013

Groups of self-equivalences and related topics, Proc. Conf., Montréal/Can. 1988, Lect. Notes Math. 1425, 114-132 (1990).
[For the entire collection see Zbl 0695.00020.]
Let X and Y be locally compact Hausdorff spaces, and let Homeo(X) and Homeo(Y) be their homeomorphism groups. When equipped with the compact- open topology, these groups are Hausdorff topological groups. Let \(p: X\to Y\) be a continuous surjection. Denote by \(Homeo_ p(X)\), respectively \(Homeo^ p(Y)\), the subspace in Homeo(X), respectively Homeo(Y), consisting of the p-projectable homeomorphisms of X, respectively the p-liftable homeomorphisms of Y. A pair of homeomorphisms \(\tilde f\in Homeo(X)\) and \(f\in Homeo(Y)\) belong to \(Homeo_ p(X)\), respectively \(Homeo^ p(Y)\), if \(fp=p\tilde f\). Obviously \(Homeo_ p(X)\) and \(Homeo^ p(Y)\) are themselves topological groups when given the subspace topology, and there is a natural epimorphism \(\rho[p]:\) Homeo\(_ p(X)\to Homeo^ p(Y)\). The main question studied in this interesting and well written paper is whether the projecting homomorphism \(\rho\) [p] is a topological group quotient map. First it is shown that \(\rho\) [p] is continuous if p is quasiproper, i.e. for each compact set \(C\subseteq Y\) there is a compact set \(\tilde C\subseteq X\) so that \(p(\tilde C)=C\) (a fairly mild condition). It is a much more delicate question whether \(\rho[p]\) is an open map, which is equivalent to being a quotient map. Therefore Miller restricts to the special case where p is a covering map. But even in this setting \(\rho\) [p] can fail to be open, and he provides an interesting example of a covering map p between noncompact 2-manifolds where this happens. On the other hand he proves in one of the main results that \(\rho[p]\) is an open map for a wide range of covering maps p including regular coverings and coverings over a compact base. He also proves that the projecting homomorphism \(\rho[p]\) is always a fibration with unique path lifting.If in addition it is an open map, it is in fact a covering map, and he presents some applications of this. Finally, he shows that similar results hold for spaces of self-homotopy- equivalences.
Reviewer: V.L.Hansen

MSC:

57N20 Topology of infinite-dimensional manifolds
57M10 Covering spaces and low-dimensional topology
58B05 Homotopy and topological questions for infinite-dimensional manifolds
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds

Citations:

Zbl 0695.00020