Free actions of finite groups on products of even-dimensional spheres - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:07:37Z http://mathoverflow.net/feeds/question/19773 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19773/free-actions-of-finite-groups-on-products-of-even-dimensional-spheres Free actions of finite groups on products of even-dimensional spheres Zbigniew Błaszczyk 2010-03-29T21:30:26Z 2010-03-29T22:40:35Z <p>Suppose a finite 2-group <em>G</em> acts freely on <em>X</em> = $\prod_{i=1}^k$ <em>S</em>$^{2n_i}$, a product of <em>k</em> even-dimensional spheres, <em>k</em> > 2. Is it possible for <em>G</em> to be non-abelian? What if we additionally assume that spheres in the product are equidimensional?</p> <p>Some comments: The equality 2<sup>k</sup> = $\chi(X)$ = |G|$\chi(X/G)$ coming from the covering <em>X</em> $\to$ <em>X/G</em> ensures that a finite group acting freely on <em>X</em> is a 2-group and also answers the question for <em>k</em> = 1, 2. (Actually, for <em>k</em> = 1 this gives a proof of a classical theorem: the only group which can act freely on an even-dimensional sphere is the cyclic group of order 2. Does anyone know who is this result originally due to? Sorry for a question inside the question; perhaps someone can comment on this one.) Hence if one would like to construct a sort of "minimal" example, it should involve an action of either the quaternion group Q<sub>8</sub> or the dihedral group Dih<sub>4</sub> for <em>k</em> = 3. I thought about this a little bit, but I feel like I'm not comfortable enough with non-abelian groups.</p> <p>I've stumbled across some papers where authors characterize arbitrary finite groups acting freely on <em>X</em> in terms of existence of particular representations, but explicit examples are given only in the abelian case.</p> http://mathoverflow.net/questions/19773/free-actions-of-finite-groups-on-products-of-even-dimensional-spheres/19777#19777 Answer by Mariano Suárez-Alvarez for Free actions of finite groups on products of even-dimensional spheres Mariano Suárez-Alvarez 2010-03-29T21:46:38Z 2010-03-29T22:40:35Z <p>[Adem, Alejandro; Davis, James F. Topics in transformation groups. Handbook of geometric topology, 1--54, North-Holland, Amsterdam, 2002. MR1886667] mentions the fact that a finite $2$-group such that every element of order $2$ is central acts freely on $(S^{|G|/2-1})^k$, where $k$ is the rank of $G$, that is, the rank of the biggest subgroup of the form $(\mathbb Z_2)^{(r)}$: «The action is built by inducing up sign representations on $k$ elements of order $2$ which span the unique central elementary abelian subgroup of $G$ and then taking their product».</p> <p>Using GAP I find a smallest non abelian example $G$ of order 32, which it describes as $(C_4\times C_2):C_4$. </p>