The mod 2 equivariant cohomology algebras of configuration spaces (Q5902802)

The symmetric group \(\Sigma_ m\) acts freely on the configuration space \(F({\mathbb{R}}^ q,m)=\{(x_ 1,...,x_ m)\); \(x_ i\in {\mathbb{R}}^ q\), \(x_ i\neq x_ j\) if \(i\neq j\), \(1\leq i,j\leq m\}\). \(H^*(F({\mathbb{R}}^ q,\infty)/\Sigma_{\infty};{\mathbb{Z}}_ p)=\lim_{\overset \leftarrow m}H^*(F({\mathbb{R}}^ q,m)/\Sigma_ m;{\mathbb{Z}}_ p)\) is equipped with a Hopf algebra structure for \(1\leq q\leq \infty\), where \({\mathbb{Z}}_ p\) denotes the prime field of p elements. The Hopf algebra structure of \(H^*(F({\mathbb{R}}^ q,\infty)/\Sigma_{\infty};{\mathbb{Z}}_ 2)\) and \(H^*(F({\mathbb{R}}^ q,m)/\Sigma_ m;{\mathbb{Z}}_ 2)\) by means of the Dickson generator system for \(H^*(\Sigma_{\infty};{\mathbb{Z}}_ 2)\) [cf. the author, ibid. 6, No.2, 41-48 (1981; Zbl 0518.20044)] and the epimorphism i(F,q): \(H^*(\Sigma_{\infty};{\mathbb{Z}}_ 2)\to H^*(F({\mathbb{R}}^ q,\infty)/\Sigma_{\infty};{\mathbb{Z}}_ 2)\) is determined. It generalizes the author's [op. cit.], \textit{M. Nakaoka}'s [J. Math. Osaka City Univ. 13, 45-55 (1962; Zbl 0134.267)] and \textit{D. B. Funk}'s [Funkts. Anal. Prilozh. 4, No.2, 62-73 (1970; Zbl 0222.57031)] results. For \(p>2\), these algebras will be also studied by the author in a subsequent paper.