Puppe sequence

Abstract: 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

Abstract: Smooth circle actions are constructed on certain homotopy spheres not previously known to admit such actions. In this paper we shall prove the following two results: PROPOSITION A. Let ?8 be any homotopy 8-sphere. Then there is a smooth semifree circle action on ?8 with S4 as itsfixed point set. PROPOSITION B. Let 210 be any homotopy 10-sphere bounding a spin nanifold. Then there is a smooth semifree circle action on ?210 with S4 as its fixed point set. Combining these with other results, we know that any smooth manifold ?lt which is piecewise-differentiably homeomorphic to Sn, bounds a spin manifold, and satisfies n _ 13 has a smooth circle action. The above propositions imply the cases n= 8, 10, while the cases n=7, 11, 12 follow because Ir? 0=OP1 in these cases and every homotopy sphere in OP,+1 (n>5) has a semifree circle action with a homotopy (n-4)-sphere as its fixed point set (e.g., see [3]). Finally, the cases n=9, 13 follow from the above remark on OP,+1 and results of Bredon [1]. Undoubtedly, the central difficulty in obtaining connected Lie group actions on homotopy spheres is the lack of a manageable construction for an arbitrary such manifold. The value of such a realization is obvious in the construction of large orthogonal group actions on homotopy spheres bounding nr-manifolds. Bredon's construction of smooth S1 and S3 actions on homotopy spheres in the image of the Milnor-MunkresNovikov pairing [1] is another illustration of the usefulness of an explicit construction for a given homotopy sphere. In this paper we shall show that certain homotopy spheres in the image of the Milnor plumbing pairing 0: (SO) x 7Tp(SOqF),+,+l (see [4] or [5]) also have smooth circle Received by the editors July 26, 1971 and, in revised form, March 24, 1972. AMS 1970 subject classf/lcations. Primary 57D60, 57E15, 57E25; Secondary 57D50, 57D55.

Abstract: Basic homotopical algebra is developed in a setting consisting of a cubical monad [G3], i.e. a cylinder endofunctor I, equipped with connections g−, g+: I2→ I, and— possibly—with symmetries extending the reversion r: I→I and the interchange s:I2→I2 of the standard topological case. Our study is mostly concerned with the Puppe sequence of a map f and its comparison with the sequence of iterated homotopy cokernels off. As an application, the homotopy structure of cochain algebras is studied in the present frame, through the cubical co-monad of the path functor P and the left adjoint cylinder functor I.