The signature of fiber bundles
The signature of fiber bundles
复制标题
纤维束的签名
DOI：
10.1090/s0002-9939-1972-0324696-0
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发表时间：
1972
期刊：
影响因子：
--
通讯作者：
J. Schafer
中科院分区：
文献类型：
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作者：
J. Schafer
设Y-*Z-* X是定向拓扑流形范畴中的局部平凡纤维丛.证明了若结构群G的单位分支有有限指数，则（Z的签名）=（X的签名）*（Y的签名）.设F-*EAB是局部平凡纤维丛，使得（1）E，F，B是闭的定向拓扑流形. (2)E、F、B是相干取向的，即F和B的取向决定E的取向。在这种情况下，是否可以得出a（E）=a（B）.a（F），其中a（）表示签名同态？当科代拉[4]和Atiyah [1]产生复曲面对复曲面的局部平凡映射使得全空间具有非零签名时，附加条件是必要的。事实上，在光滑的情况下，阿蒂亚给出了一个计算r（E）的公式，并显示了对B的基本群的依赖性。本文的方法是从丛的结构群G出发，确定G上的条件，从而得到上述问题的肯定答案。若G是任意拓扑群，令r=G/Go，其中Go是单位元的连通分支。主要结果是定理。设G是一个局部紧的有限维拓扑群，使得IFl是有限的。如果FEA。B是具有结构群G的任意定向局部平凡拓扑纤维丛，则aE= oi B aF。如果丛的结构群更不用说不是G，但可以约化为G，则该定理显然仍然有效。注释2.假设G是局部紧的，有限维的存在只是为了确保G-r具有局部截面。保证这一点的任何其他关于G的假设都同样有效。见[3]。提案1.如果F =（e），则qE= oB * oF。编辑于1970年3月4日收到，修订版于1971年7月20日收到。AMS 1969主题类flactions。小学5730，5560。
Let Y-*Z-* X be a locally trivial fiber bundle in the category of oriented topological manifolds. It is shown that if the identity component of the structure group G has finite index, then (signature of Z) = (signature of X) * (signature of Y). Let F-*EAB be a locally trivial fiber bundle such that (1) E, F, B are closed, oriented topological manifolds. (2) E, F, B are coherently oriented, that is, the orientation of F and B determine that of E. In this situation, does it follow that a(E)=a(B).a(F), where a( ) denotes the signature homomorphism? That additional conditions are necessary is shown both by Kodaira [4] and Atiyah [1] when they produce a locally trivial fibering of a complex surface by a complex surface such that the total space has a nonzero signature. In fact, in the smooth case, Atiyah produces a formula computing r(E) and showing the dependency on the fundamental group of B. The approach of this paper is to look at the structure group G of the bundle and determine conditions on G in order to obtain an affirmative answer to the above question. If G is any topological group, let r=G/Go, where Go is the connected component of the identity. The main result is the THEOREM. Let G be a locally compact, finite dimensional topological group such that I Fl is finite. If FEA. B is any oriented locally trivial topologicalfiber bundle with structure group G, then aE= oiB aF. REMARK 1. The theorem obviously remains valid if the structure group of the bundle is not, a fortiori, G, but can be reduced to G. REMARK 2. The hypothesis that G be locally compact, finite dimensional only exists to insure that G-r possesses a local cross section. Any other hypothesis on G insuring this is equally valid. See [3], for instance. PROPOSITION 1. If F = (e), then qE= oB * oF. Received by the editors March 4, 1970 and, in revised form, July 20, 1971. AMS 1969 subject class flcations. Primary 5730, 5560.