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Free actions of some compact groups on Milnor manifolds. (English) Zbl 1429.57035

Let \(r\) and \(s\) be integers such that \(0\leq s\leq r\). Denote by \(\mathbb{R}H_{r,s}\), \(\mathbb{C}H_{r,s}\) the real and complex Milnor manifolds, respectively. For certain values of \(r\) and \(s\) these manifolds admit free \(\mathbb{Z}_2\)- and \(\mathbb{S}^1\)-actions.
The paper under review considers the following two questions: a) For which values of \(r\) and \(s\) these spaces admit free actions; b) Once we have a free action, what is the cohomology with \(\mathbb{Z}_2\) coefficients of the orbit space.
Before showing the main results, the authors describe the state of art of this problem and contextualize the Milnor manifolds.
Relative to free actions the main results are:
Corollary 5.6. Let \(1<s<r\) and \( r\not\equiv 2\) (mod 4). Then, \(\mathbb{R}H_{r,s}\) admits a free involution if and only if both \(r\) and \(s\) are odd.
Corollary 5.8. Let \(1<s<r\) and \( r\not\equiv 2\) (mod 4). Then, \(\mathbb{C}H_{r,s}\) admits a free involution if and only if both \(r\) and \(s\) are odd.
A slight modification of the results above holds for spaces \(X\) having the same mod-2 homotopy type of the Milnor manifolds above, considered in the paper. Concerning \(\mathbb{S}^1\) free actions we have:
Proposition 5.9. Let \(1\leq s \leq r\). Then \(\mathbb{S}^1\) acts freely on \(\mathbb{R}H_{r,s}\) if and only if both \(r\) and \(s\) are odd.
Then come the main results, which are the description of \(H^*(X/G; \mathbb{Z}_2)\) when: a) \(X\simeq_2 \mathbb{R}H_{r,s}\), \(G=\mathbb{Z}_2\); b) \(X\simeq_2 \mathbb{C}H_{r,s}\), \(G=\mathbb{Z}_2\); c) \(X\simeq_2 \mathbb{R}H_{r,s}\), \(G=\mathbb{S}^1\). To exemplify we state the main result in the case a).
Theorem 6.1. Let \(G=\mathbb{Z}_2\) act freeely on a compact Hausdorff space \(X\simeq_2\mathbb{R}H_{r,s}\) such that induced action on mod 2 cohomology is trivial. Then, \[H^*(X/G;\mathbb{Z}_2)\cong \mathbb{Z}_2[x,y,z,w]/I,\] where \begin{multline*} I=\langle z^2, w^2-\gamma_1zw-\gamma_2x-\gamma_3y, x^{(s+1)/2}+\alpha_0zwx^{(s-1)/2}+\alpha_1zwx^{(s-3)/2}y+\cdots+\alpha_{(s-1)/2}zwy^{(s-1)/2},\\ (w+\beta_0z)y^{(r-1)/2} + (w+\beta_1z)xy^{(r-3)/2}+\cdots +(w+\beta_{(s-1)/2}z)x^{(s-1)/2}y^{(r-s)/2}\rangle, \end{multline*} with \(\deg(x)=2\), \(\deg(y)=2\), \(\deg(z)=1\) and \(\alpha_i, \beta_i, \gamma_i \in \mathbb{Z}_2\).
A main tool used is the Leray-Serre spectral sequence for certain Borel fibrations.
The paper is well organised and at the end some applications to equivariant maps are presented. They are concerned with maps between spheres and spaces which have the mod (2) homotopy type of a Milnor manifold. Some of the applications are stated in terms of Schwartz genus of a free action.

MSC:

57S25 Groups acting on specific manifolds
57S10 Compact groups of homeomorphisms
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
55M20 Fixed points and coincidences in algebraic topology
57R91 Equivariant algebraic topology of manifolds
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