Homotopy groups of the space of self maps of the projective 3-space (Q638739)

For based spaces \(X\) and \(Y\), let \(\text{map}_*(X,Y)\) denote the space of based maps from \(X\) to \(Y\), and let \(\mathbf{P}^k\) denote the \(k\) dimensional real projective space. Let \(\text{aut}(X)\) denote the space of self-homotopy equivalences over \(X\). In this paper, the author computes the homotopy groups \(\pi_k(\text{map}_*(\mathbf{P}^3,\text\textbf{P}^3))\) and he shows that there is an isomorphism \(\pi_k(\text{map}_*(\mathbf{P}^3,\text\textbf{P}^3))\cong \Gamma_n\oplus \pi_{n+3}(S^3)\) for \(n\geq 0\), where \(\Sigma^n\) is the \(n\)-fold suspension and \(\Gamma_n\) denotes the based homotopy set \(\Gamma_n=[\Sigma^n\mathbf{P}^2,S^3]\). Moreover, he proves that there is an isomorphsim \(\pi_n(\text{aut(\textbf{P}}^3))\cong \Gamma_n\oplus \pi_{n+3}(S^3)\oplus \pi_n(\mathbf{P}^3)\) for \(n\geq 1\), and he computes the group \(\Gamma_n\) explicitly for \(n\leq 21\). The method of the proof is based on elementary computations of the Puppe sequence and Toda's composition method.