On almost rational co-H-spaces
On almost rational co-H-spaces
复制标题
在几乎有理余 H 空间上
DOI：
10.1090/s0002-9939-1983-0677254-x
复制
发表时间：
1983
影响因子：
4
通讯作者：
H. Henn
中科院分区：
文献类型：
--
作者：
H. Henn
设X是0-连通余-Il-空间，其同伦群7T，，(X)是q-向量空间当且其基本群7T(X)是任意的。证明了X同伦等价于两维以上有理球面的楔形和普通一维球面的楔形。导言。我们研究了0-连通但不一定是1-连通余H-空间X与7r，(X)是n>；2的有理向量空间。我们称这种空间为0-连通几乎有理余H-空间(-r，(X)不一定也不是有理向量空间，只要它不是平凡的)。我们证明了这样的空间X是同伦的，等价于维度大于一的有理球面和普通一维球面的楔形。Berstein[Be]和Tomer[To]证明了满足一定有限性条件的单连通空间的类似结果。我们注意到，具有7R，(X)自由和，(X)Q向量空间n>；2的空间经常出现，即作为具有自由基本群的空间的泛覆盖的有理化的商[Co，p.395F]。本文组织如下：在文献[1]中，我们证明了1-连通有理空间的悬挂结果。利用这一点，当X是0-连通的几乎有理空间(？2)时，我们能够证明SQX的结果。如果X是余H-空间，则它是S2X的收缩，并且这一事实允许证明一般结果(？3)。0。记号和惯例。(A)假设所有空间都是CW-复形的同伦型，并且所有的构造，如乘积等，都是在紧生成范畴中进行的。(B)同调群为有理向量空间的幂零空间称为有理空间。关于本地化的基本性质，特别是合理化，读者可参考[SU或Hi-Mi-Ro，1]。(C)球面S(n>；0)的有理化称为有理球面。显然，单连通空间是有理球面当且仅当它具有有理球面的同调。1.命题。设X是单连通有理空间。则SX是同伦的，是有理球面的楔形。由编辑于1981年5月8日收到。1980年数学学科分类i。主要55P45；辅助55P62。Kei‘s单词和短语。几乎有理co-H-空间。‘部分得到了德国伏尔克斯研究中心的支持。？1983美国数学学会0002-9939/82/OOO-0695/$02.00 164本内容从太阳的40.77.167.2下载，2016年9月18日05：28：03 UTC所有使用均受http://about.jstor.org/terms关于几乎有理CO-H-空间165证明的约束。我们使用X的同调分解(参见[hi])X2CX3CX4C...CuXn=X.Na2我们有(1)Xoo XI(2)所有Xn都是1-连通的，(3)HR(Xn)=Oforr>；n，(4)HR(X，)HR(X00)对r<；n，其中i表示包含，(5)X+L是由映射An附加一个Moore空间M(hn+(X)，n)得到的同伦。从(5)中我们得到了一个Puppe序列M(Hn+L(X)，n)on Xn Xn+1--3M(Hn+i(X)，n+)SANSN(注：M(H+？(X)，n+1)SM(Hn 1(X)，n).因为Hn+？(X)是有理向量空间，所以M(H+if(X)，n+1)是有理球面的楔形同伦。我们将通过证明SXn+L同伦于有理球面的楔形，从而归纳地证明SXn+SXn是同伦的。(感应开始是因为X2 M(H2(X)，2)。)然后就是那个SX。是同伦的有理球面的楔形。它由(4)推出，即Hf*(An)=0。因此，我们的命题将由引理引出。如果X，Y是有理空间之间的映射，使得H*(F)0，则SF 0。证据。证明了X、Y、USY是零同伦的，其中I是正则映射。现在QSY是0-连通有理H-空间。在这样的空间中，所有的波斯尼科夫不变量都是平凡的(见[Mi-Mo，第263页])，因此
Let X be a 0-connected co-Il-space whose homotopy groups 7T,,( X) are Q vector spaces if ni > I and whose fundamcntal group 7T( X) is arbitrary. We prove that X is homotopy equivalent to a wedge of rational spheres of dimension at least two and of ordinary one-dimensional spheres. Introduction. We investigate 0-connected but not necessarily 1-connected co-Hspaces X with 7r,( X) a rational vector space for n > 2. We call such spaces 0-connected almost rational co-H-spaces (-r,(X) need not be and indeed is not a rational vector space provided it is not trivial). We prove that such a space X is homotopy equivalent to a wedge of rational spheres of dimension bigger than one and of ordinary one-dimensional spheres. Analogous results for simply connected spaces which satisfy certain finiteness conditions were proved by Berstein [Be] and Toomer [To]. We remark that spaces with 7r,(X) free and ,,( X) a Q vector space for n > 2 occur quite often, namely as quotients of a rationalization of the universal cover of a space with free fundamental group [Co, p. 395f]. The paper is organized as follows: In ?1 we prove the result for suspensions of 1-connected rational spaces. Using this we are able to prove the result for SQX, if X is a 0-connected almost rational space (?2). If X is a co-H-space, it is a retract of S2X and this fact allows a proof of the general result (?3). 0. Notations and conventions. (a) All spaces are assumed to be of the homotopy type of a CW-complex and all constructions like products etc. are performed in the compactly generated category. (b) A nilpotent space whose homology groups are rational vector spaces is called a rational space. For basic properties of localization, in particular rationalization, the reader is referred to [Su or Hi-Mi-Ro, 1]. (c) The rationalization of a sphere S' (n > 0) is called a rational sphere SQ. It is obvious that a simply connected space is a rational sphere if and only if it has the homology of a rational sphere. 1. PROPOSITION. Let X be a simply connected rational space. Then SX is up to homotopy a wedge of rational spheres. Received by the editors May 8, 1981. 1980 Mathenmatics Subject Classificwationi. Primary 55P45; Secondary 55P62. Kei' words and phrases. Almost rational co-H-spaces. 'Partially supported by the Studienstiftung des deutschen Volkes. ?1983 American Mathematical Society 0002-9939/82/OO0-0695/$02.00 164 This content downloaded from 40.77.167.2 on Sun, 18 Sep 2016 05:28:03 UTC All use subject to http://about.jstor.org/terms ON ALMOST RATIONAL CO-H-SPACES 165 PROOF. We use a homology decomposition of X (cf. [Hi]) X2 C X3 C X4 C ...CU Xn = X. na2 We have (1) Xoo XI (2) all Xn are 1-connected, (3)Hr(Xn) = Oforr>n, (4) Hr( X,) Hr( X00) for r < n where i denotes the inclusion, (5) X+l is up to homotopy obtained from Xn by attaching a Moore space M(Hn+ (X), n) by a map an. From (5) we obtain a Puppe sequence M(Hn+ l(X), n) onXn Xn+ 1--3 M(Hn+ i(X), n + )San nSn (Note that M(H+ ?(X), n + 1) SM(Hn 1(X), n).) Now M(H+ If(X), n + 1) is up to homotopy a wedge of rational spheres because Hn + ?( X) is a rational vector space. We will prove inductively that SXn+l is up to homotopy a wedge of rational spheres by showing that San 0. (Induction starts because X2 M(H2(X), 2).) Then it follows that SX. is up to homotopy a wedge of rational spheres. It follows from (4) that Hf*(an) = 0. Hence our proposition will follow from LEMMA. If X Y is a map between rational spaces such that H*( f) 0 then Sf 0. PROOF. It suffices to show that X Y USY is null-homotopic where i is the canonical map. Now QSY is a 0-connected rational H-space. In such a space all Postnikov invariants are trivial (see [Mi-Mo, p. 263]) and thus