\(k\)-rational homotopy fixed points, \(k\in \mathbb{N}\) (Q6546428)

This paper considers a CW-complex \(X\) satisfying the following conditions.\N\N1. Simply-connected.\N\N2. Rational (so \(\pi_*(X)=\pi_*(X)\otimes\mathbb{Q}\) and \(H_*(X;\mathbb{Z}) = H_*(X;\mathbb{Q})\)).\N\N3. Rationally elliptic (so both \(\pi_*(X)\) and \(H_*(X;\mathbb{Z})\) are finite dimensional).\N\NLet \(X^k\) denote the \(k\)-skeleton of \(X\), and \(\mathcal{E}_*(X)\) denote the group of self-homotopy equivalences of \(X\) that induce the identity on \(H_*(X;\mathbb{Z})\). If there exists a non-identity element \([\alpha]\in\mathcal{E}_*(X)\) and a non-zero \(x\in\pi_k(X^k)\) such that \(x\) is invariant under the action of \(\pi_k(\alpha)\), then \(X\) is said to have a \(k\)-rational homotopy fixed point.\N\NSuppose that \(X\) is of formal dimension \(n\), i.e. \(n\) is the largest integer \(i\) such that \(H^i(X;\mathbb{Z})\neq 0\). Using the Quillen model of differential graded Lie algebras, the author constructs a group homomorphism \(\psi:\pi_n(X^n) \to \mathcal{E}_*(X)\), and shows that \(X\) has an \((n-1)\)-rational homotopy fixed point if \(\psi\) is not surjective.