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Smooth free involutions on homotopy 4k-spheres. (English) Zbl 0543.57023

As stated in the title the authors study smooth free involutions on homotopy 4k-spheres or equivalently, smooth manifolds of the homotopy type of \({\mathbb{R}}P(4k)\). Generalizing their construction from [Ann. Math., II. Ser. 113, 357-365 (1981; Zbl 0474.57014)] they define an invariant \(\rho\) : it consists of a \({\mathbb{Q}}/{\mathbb{Z}}\)-linear combination of the Eells-Kuiper invariant \(\mu\) of a spin structure on a characteristic homotopy projective space and the \(\alpha\)-invariant of its double cover. As an application they compute a lower bound for the number of distinct homotopy \({\mathbb{R}}P(4k)'s\). From the introduction: ”We obtain no more smooth homotopy \({\mathbb{R}}P(4k)'s\) than claimed earlier by, for instance, C. H. Giffen [Bull. Am. Math. Soc. 75, 509-513 (1969; Zbl 0195.533)] or the reviewer [Math. Z. 170, 233-246 (1980; Zbl 0425.57013)]”.
Reviewer’s remark: This is not surprising. In the language of the reviewer’s paper they compute the standard involutions. (This is the \({\mathbb{Z}}/2\)-analogue of Kervaire-Milnor’s b \(P_{n+1}.)\) With loc.cit. one can easily compute all of them: there are \(a_ k\cdot 2^{2k-1}\) standard involutions (in dimension 4k), \(a_ k=4/(3+(-1)^ k)\), \(k>1\). (Proposition 2.14 in my paper is sligthly incorrect.)
Reviewer: P.Löffler

MSC:

57S17 Finite transformation groups
57S25 Groups acting on specific manifolds
57R20 Characteristic classes and numbers in differential topology
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