Abstract
A representation G ⊂ U(n) of degree n has fixity equal to the smallest integer f such that the induced action of G on U(n) /U(n-f-1) is free. Using bundle theory we show that if G admits a representation of fixity one, then it acts freely and smoothly on \(\mathbb S^{2n-1}\times\mathbb S^{4n-5}.\) We use this to prove that a finite p-group (for p > 3) acts freely and smoothly on a product of two spheres if and only if it does not contain (ℤ /p)3 as a subgroup.
We use propagation methods from surgery theory to show that a representation of fixity f < n - 1 gives rise to a free action of G on a product of f + 1 spheres provided the order of G is relatively prime to (n - 1)!. We give an infinite collection of new examples of finite p-groups of rank r which act freely on a product of r spheres, hence verifying a strong form of a well-known conjecture for these groups. In addition we show that groups of fixity two act freely on a finite complex with the homotopy type of a product of three spheres. A number of examples are explicitly described.
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Adem, A., Davis, J. & Ünlü, Ö. Fixity and free group actions on products of spheres . Comment. Math. Helv. 79, 758–778 (2004). https://doi.org/10.1007/s00014-004-0810-4
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DOI: https://doi.org/10.1007/s00014-004-0810-4