\(\mathbb{Z}_{/p} \times \mathbb{Z}_{/p}\) actions on \(S^n \times S^n\) (Q6596243)

The group \(\mathbb{Z}_{{/}p}\times \mathbb{Z}_{{/}p}\) acts on \(S^n\times S^n\) and there are many actions. The work under review studies the homotopy type of such orbit spaces. The first main result is\N\NTheorem 3.3. Let \(X\) and \(Y\) be quotients of free \(\mathbb{Z}_{{/}p}\times \mathbb{Z}_{{/}p}\) actions on \(S^n \times S^n\) with odd \(n \geq 3\), where \(p > 3\) satisfies \(2p+n-3 > 2n\), and let \(k_{X}^{n+1}\) and \(k_{Y}^{n+1}\) denote the first nontrivial \(k\)-invariant. The spaces \(X\) and \(Y\) are homotopy equivalent if and only if there are isomorphisms \(g_1:\pi_1X \to \pi_1Y\) and \(g_n:\pi_nX \to \pi_nY\) with \(g_n\in G_n\subset GL_2(\mathbb{Z})\) (a certain subgroup) and such that \N\[\N\begin{tikzcd} K(\pi_1(X),1) \ar[r]{k_{Xn+1}} \ar[d]{g_{*1}} & K(\pi_n(X),n+1) \ar[d]{g_{*n}}\\\NK(\pi_1(Y),1) \ar[r]{k_{Y}^{n+1}} & K(\pi_n(Y),n+1) \end{tikzcd}\N\]\Ncommutes up to homotopy, i.e. \(k_{X}^{n+1}\in H^{n+1}(\pi_1X; \pi_n(X))\) and \(k_{Y}^{n+1}\in H^{n+1}(\pi_1Y; \pi_n(Y))\) are identified through the maps induced by \(g_1\) and \(g_n\).\N\NAlso they show, see Proposition 4.1, that the Postnikov invariant \(k_{X}^{n+1}\) of an orbit space \(X/(\mathbb{Z}_{{/}p}\times \mathbb{Z}_{{/}p})\) must satisfy a certain condition. Examples of orbit spaces are constructed generalizing the construction of the lens space. Then they specialize to the case of \(S^3\times S^3\). The main result for this case is\N\NTheorem 6.6. Let \(p > 3\) be prime. If \(p \equiv 1\) mod \(4\), then there are four homotopy classes of quotients of \(S^3 \times S^3\) by free \(\mathbb{Z}_{{/}p}\times \mathbb{Z}_{{/}p}\) actions. If \(p\equiv 3\bmod 4\), then there are two classes.\N\NAs applications it is shown that certain products of two \(3\)-dimensionial lens spaces have the same homotopy type. At the end, making use of the condition that the Postnikov invariant must satisfy, they provide an example of a finite group \(G\supset \mathbb{Z}_{{/}p}\times \mathbb{Z}_{{/}p}\) which can not act freely on \(S^n\times S^n\), which is related to the open question which finite groups act freely on \(S^n\times S^n\).