A necessary condition for the Smith equivalence of \(G\)-modules and its sufficiency (Q2836151)

Let \(G\) be a finite group. The real \(G\)-modules \(V\) and \(W\) are called Smith equivalent if there exists a homotopy sphere \(\Sigma \) with a smooth \(G\)-action such that \(\Sigma ^G=\{a,b\}\), the tangential \(G\)-modules \(T_a(\Sigma)\) and \(T_b(\Sigma)\) are isomorphic. The author gives two theorems concerning necessary conditions for the Smith equivalence of \(G\)-modules: The first one is that if a \(2\)-Sylow subgroup \(G_2\) of \(G\) is normal and the real \(G\)-modules \(V\) and \(W\) are Smith equivalent, then for all normal subgroups \(N\) of \(G\) with prime index, (1) \(V^N\) and \(W^N\) are \(G/N\)-free, (2) the unit spheres \(S(V^N)\) and \(S(W^N)\) are \(G/N\)-homotopy equivalent, and (3) if \(\dim V^N\neq 0\) or \(\dim W^N \neq 0\), then \(\operatorname {res}_N^G V \cong \operatorname {res}_N^G W\). The second one is as follows. Suppose that \(G\) is an Oliver group, \(G_0\) a proper normal subgroup of \(G\) such that \(G/G_0\) is a cyclic group, and \(N\) a proper normal subgroup of \(G\) such that \(G_0 < N < G\), \((| G/N|, | N/G_0|) = 1\), and \(G/N\) is a cyclic group of odd order, \(N/G_0\) is not of prime power order, and \(| G/N| = 3\) or \((| G/N|, 3) = 1\). Let \(A\) and \(B\) be \(G/N\)-free real \(G/N\)-modules of dimension \(4\) such that \(S(A)\) and \(S(B)\) are \(G/N\)-homotopy equivalent. Then there exist Smith equivalent real \(G\)-modules \(V\) and \(W\) such that \(V^N \cong A\), \(W^N \cong B\), and \(\operatorname {res}_P^G V \cong \operatorname {res}_P^G W\) for all Sylow subgroups \(P\) of \(G\).NEWLINENEWLINEThe Smith set of \(G\) is the subset of the real representation ring \(\operatorname {RO}(G)\) consisting of all elements \(x = [V]-[W]\) such that \(V\) and \(W\) are Smith equivalent real \(G\)-modules, and the primary Smith set of \(G\) is the subset of the Smith set of \(G\) consisting of all elements \(x = [V]-[W]\) such that \(\operatorname {res}_P^G V\cong \operatorname {res}_P^G W\) for all Sylow subgroups \(P\) of \(G\). It is well known that the Smith set of \(G\) is not additively closed if \(G\) has a quotient cyclic group of order \(8\) and that there are many Oliver groups \(G\) such that the primary Smith set of \(G\) is a subgroup of \(\operatorname {RO}(G)\). The author show that there are many finite Oliver groups \(G\) such that the primary Smith set of \(G\) is not additively closed by combining the above theorems.