Bounded cohomology of certain groups of homeomorphisms
Bounded cohomology of certain groups of homeomorphisms
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某些同胚群的有界上同调
DOI：
10.1090/s0002-9939-1985-0787909-6
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发表时间：
1985
影响因子：
3.1
通讯作者：
S. Morita
中科院分区：
文献类型：
--
作者：
S. Matsumoto;S. Morita
我们考虑了有界上同调注入普通上同调的条件，证明了RW的所有紧支撑上同胚群的有界上同调为零。导言。本文考虑空间或群的有界上同调、普通实上同调和1同调之间的关系。特别地，我们给出了有界上同调注入普通上同调的一个充要条件，并利用它证明了R‘7上紧支撑的所有同胚群HomeoKR’的有界上同调和1同调为零。我们还确定了SL2R的第二有界上同调。1.有界上同调。让我们快速回顾一下由Gromov发展的有界上同调理论[2](另见Brooks[1]和Mitsum atsu[5])。设X是一个拓扑空间，W‘*(X)={CQ(X)，AQ}是X的实系数奇链复形。定义CQ(X)上的范数Liin=1aaiill=，-=JOJAR。这样，微分a就是有界线性算子。设W‘((X)={(C’(X)，AQ})是W‘，(X)的范数完备。因此，Cql(X)(Y？=1laiailD%1jar<；00)是Banach空间。通过到对偶Banach空间，我们得到了一个余链复形，如果b(X)=(Cq(X)，bq是由有界余链组成的普通奇异余链复形的一个子复形。用hl(X)表示的W*‘(X)的同调称为X的11同调，用Hb*(X)表示的Wb(X)的上同调称为X的有界上同调.包含诱导同态H*(X)-*H*(X)和HB*(X)-*H*(X).由于有界算子的像不一定是闭子空间，所以在H*(X)或HB*(X)上诱导的伪范数可能不是范数。在MitSumatsu[5]的基础上，我们定义了hl*(X)(分别为H，b*(X))为hl(X)的商(分别Hb*(X))的伪范零子空间。换言之，H“(X)Z(‘(X)/Bl$(X)and Hbq(X)=ZB(X)/Bbf(X)，其中Z或B表示相应复形的(Co)圈或(Co)边界的空间，B表示B的闭包。请注意，H’(X)和Hbq(X)是Banach空间。
We consider the condition when bounded cohomology injects into ordinary cohomology and prove the vanishing of bounded cohomology of the group of all compactly supported homeomorphisms of RW. Introduction. In this note we consider relations among bounded cohomology, ordinary real cohomology and 1 homology of spaces or groups. In particular we present a necessary and sufficient condition under which bounded cohomology injects into ordinary cohomology and by using it prove the vanishing of bounded cohomology and 1 homology of HomeoKR', the group of all homeomorphisms of R'7 with compact support. We also determine the second bounded cohomology of SL2R. 1. Bounded cohomology. Let us quickly review the theory of bounded cohomology developed by Gromov [2] (see also Brooks [1] and Mitsumatsu [5]). Let X be a topological space and let W'*(X) = {CQ(X), aq} be the singular chain complex of X with real coefficients. Define a norm on Cq(X) by lIIn=1aaiill = ,-=jail. The differentials a are then bounded linear operators. Let W'((X) = {(C'(X), aq} be the norm completion of W',(X). Thus Cql(X) (Y?=1laiailD%1jail < 00) is a Banach space. Passing to the dual Banach spaces, we obtain a cochain complex if b(X) = (Cq( X), b q It is a subcomplex of the ordinary singular cochain complex consisting of bounded cochains. The homology of W*' (X), denoted by Hl (X), is called 11 homology of X and the cohomology of Wb (X), denoted by Hb*(X), is called bounded cohomology of X. The inclusions induce homomorphisms H*(X) -* H* (X) and Hb*( X) -* H *(X). Since the image of a bounded operator is not necessarily a closed subspace, it may happen that the pseudonorms induced on H* (X) or Hb*( X) are not norms. Following Mitsumatsu [5], we define Hl*(X) (resp. H,b*(X)) to be the quotient of Hl (X) (resp. Hb*(X)) by the subspace of pseudonorm zero. In other words, H"'(X) Z('( X)/Bl$ ( X) and Hbq( X) = Zb( X)/Bbf( X), where Z or B denotes the spaces of (co)cycles or (co)boundaries of the corresponding complex and B denotes the closure of B. Notice that H' (X) and Hbq(X) are Banach spaces. There is a Received by the editors July 19, 1984. 1980 Mathematics Subject Classification. Primary 55N99; Secondary 57T99.