The space of cross sections of a bundle
The space of cross sections of a bundle
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束的横截面空间
DOI：
10.1090/s0002-9939-1988-0947690-7
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发表时间：
1988
期刊：
影响因子：
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通讯作者：
K. Sakai
中科院分区：
文献类型：
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作者：
K. Sakai
设B是非离散紧的，Y是无孤立点的可分完全度量化的ANR，p：X-B是局部平凡丛，纤维Y允许截面。证明了p：X-B的所有横截面的空间r(X)是12维流形。0。导言。通过这篇文章，空间是可分的，可度量的，映射是连续的。设p：X-*B是一个具有纤维Y的局部平凡丛，即每个点bEB都有一个邻域U和一个同胚群p：UxY-，p-1(U)使得p‘p=ru，映射到U的投影S：B-*X称为p：X-*B的横截面.具有紧-开拓扑的p：X-*B的所有横截面空间记为17(X)。则17(X)是从B到X的所有映射的空间C(B，X)的闭子空间.如果B是紧的，d是X的相容度量，则17(X)(和C(B，X))的拓扑由超度量d(f，g)=sup{d(f(B)，g(B))i b E B}诱导.以希尔伯特空间12为模型的流形称为12-流形。在本文中，我们证明了以下主要定理。设B是非离散紧的，Y是没有孤立点的完全可度量化的ANR，p：xB是局部平凡丛，纤维Y允许截面.则17(X)是12-流形。对于平凡丛7RB：B×Y-*B，空间F(B×Y)可视为空间C(B，Y)。因此，如果B是非离散紧空间，Y是没有孤立点的完全度量化的ANR，则C(B，Y)是12维流形。这是Eells-Geoghegan-Torunczyk[E，Ge，to 1]结果的推广。作者要感谢Doug Curtis的有益评论。1.预赛。我们的证明基于以下内容：Toru？NCZYK关于12-流形的刻画定理[TO2](参看.[至3]))。一个完全度量化的ANR X是一个12-流形当且仅当X具有离散逼近性质，即对n个单元(n，E，N)到X的自由并的每个映射f：eDnEN I‘X和每个映射E：X-+(0，1)有一个由编辑于1987年2月27日收到的，并经修订后的1987年4月22日的形式。1988年4月2日提交给日本数学学会。1980年《数学学科分类》。主58D15、57N20、55F10。
Let B be a nondiscrete compactum, Y a separable complete metrizable ANR with no isolated point and p: X -B a locally trivial bundle with fiber Y admitting a section. It is proved that the space r(X) of all cross sections of p: X -B is an 12-manifold. 0. Introduction. Through the paper, spaces are separable metrizable and maps are continuous. Let p: X -* B be a locally trivial bundle with fiber Y, that is, each point b E B has a neighborhood U and a homeomorphism p: U x Y -, p-1 (U) such that p'p = ru, the projection to U. A map s: B -* X is called a cross section of p: X -* B provided ps = id. The space of all cross sections of p: X -* B with compact-open topology is denoted by 17(X). Then 17(X) is a closed subspace of the space C(B, X) of all maps from B into X. If B is compact and d is a compatible metric for X, the topology of 17(X) (and C(B, X)) is induced by the sup-metric d(f, g) = sup{d(f (b), g(b)) I b E B}. A manifold modeled on Hilbert space 12 is called an 12-manifold. In this note, we prove the following MAIN THEOREM. Let B be a nondiscrete compactum, Y a complete metrizable ANR with no isolated point and p: X B a locally trivial bundle with fiber Y admitting a section. Then 17(X) is an 12-manifold. For the trivial bundle 7rB: B x Y -* B, the space F(B x Y) can be regarded as the space C(B, Y). Thus the space C(B, Y) is an 12-manifold if B is a nondiscrete compactum and Y is a complete-metrizable ANR with no isolated point. This is a generalization of Eells-Geoghegan-Torunczyk's result [E, Ge, To1]. The author would like to thank Doug Curtis for helpful comments. 1. Preliminaries. Our proof is based on the following: TORU?NCZYK'S CHARACTERIZATION THEOREM FOR 12-MANIFOLDS [TO2] (CF. [To3]). A complete-metrizable ANR X is an 12-manifold if and only if X has the discrete approximation property, that is, for each map f: eDnEN I' X of the free union of n-cells (n E N) into X and each map E: X -+ (0, 1) there is a Received by the editors February 27, 1987 and, in revised form, April 22, 1987. Presented to the Mathematical Society of Japan, April 2, 1988. 1980 Mathematics Subject Classification. Primary 58D15, 57N20, 55F10.