Finite-dimensional spaces in resolving classes (Q2887094)

The author uses his theory of resolving classes developed in [\textit{J. Strom}, Fundam. Math. 178, No. 2, 97--108 (2003; Zbl 1052.55015)] to give a new, more elementary proof of the celebrated theorem of Haynes Miller that the space of pointed maps from \(B \mathbb{Z}/p\) to any finite CW complex \(K\) is weakly contractible [\textit{H. Miller}, Ann. Math. (2) 120, 39--87 (1984; Zbl 0552.55014)]. Let \(X\) be a CW complex of finite type. The main result gives the contractibility of \(\mathrm{map}_*(X, K)\) for all simply connected, finite CW complexes \(K\) assuming only the contractibility of \(\mathrm{map}_*(X, S^{n})\) for \(n \equiv 1 \mod k\) for some \(k\) and all \(n\) sufficiently large. The restriction to simply connected \(K\) is removed provided \(\pi_1(X)\) has no nontrivial perfect quotients. Taking \(X = B \mathbb{Z}/p\) reduces the proof of the Sullivan conjecture to proving that \(\mathrm{map}_*(B \mathbb{Z}/p, S^{2n+1}) \sim *\) for all \( n \geq 1\). This easier result was proved in [\textit{H. Miller}, Algebraic topology, Proc. Conf., Aarhus 1982, Lect. Notes Math. 1051, 401--417 (1984; Zbl 0542.55016)] and is outlined in an appendix here.