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Elliptic spaces with the rational homotopy type of spheres. (English) Zbl 0919.55005

A topological space \(Z\) is said to be \(p\)-elliptic if it has the \(p\)-local homotopy type of a finite , 1-connected CW-complex and if the loop space homology \(H_*(\Omega Z;\mathbb{F}_p)\), where \(\mathbb{F}_p\) is the prime field of characteristic \(p\), is finitely-generated as an algebra and nilpotent as a Hopf algebra. Any \(p\)-elliptic space \(Z\) is \(\mathbb{Q}\)-elliptic, and \(\mathbb{Q}\)-elliptic spaces are those spaces which have the rational homotopy type of a finite, 1-connected CW-complex and have finite total rational homotopy rank. Elliptic spaces and some important properties of them were studied by Y. Félix, S. Halperin and J.-C. Thomas [Bull. Am. Math. Soc., New Ser. 25, No. 1, 69-73 (1991; Zbl 0726.55006); Enseign. Math., II. Sér. 39, No. 1-2, 25-32 (1993; Zbl 0786.55006)]. Examples of \(p\)-elliptic spaces are finite, 1-connected \(H\)-spaces and spheres. The author is interested in the study of those \(p\)-elliptic spaces which have the rational homotopy type of a sphere \(S^N\). For that he assumes that \(p\) is a large prime for \(Z\). That means: If \(Z\) is an \(r\)-connected, \(n\)-dimensional CW-complex, then \(p\geq{r\over n}\) or \(p=0\) \((\mathbb{F}_0= \mathbb{Q})\). The main result is: Let \(Z\) be a \(p\)-elliptic, 1-connected space which has the rational homotopy type of \(S^N\), \(N\geq 2\), and let \(p\) be a large prime for \(Z\). If \(N=2n\), then \(Z\) has the \(p\)-local homotopy type of \(S^N\). If \(N=2 n+1\), then \(Z\) is \(p\)-formal (a minimal Adams-Hilton model \({\mathcal A}=T(V)\) for \(Z\) over \(\mathbb{F}_p\) has a quadratic differential), and \(H^*(Z;\mathbb{F}_p) \approx \Lambda a\otimes B\), where \(\deg a=2t-1\) and \(B\) is an algebra with the same Hilbert series as \(\mathbb{F}_p[b] |_{(b^m)}\), with \(\deg b=2t\), \(m\geq 1\), and \(N\doteq 2mt-1\). In addition, in this case \(N=2n+1\), the author gives a description of the Adams-Hilton model for a \(p\)-minimal decomposition of \(Z\) over \(\mathbb{Z}_{(p)}\).

MSC:

55P62 Rational homotopy theory
55P15 Classification of homotopy type
55P60 Localization and completion in homotopy theory
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
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