## 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

### Citations:

Zbl 0474.57014; Zbl 0195.533; Zbl 0425.57013
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