The topology of spaces of maps between real projective spaces (Q1890290)

Denoting by Map(X,Y) the space of continuous maps from \(X\) to \(Y\), with the compact open topology, the author writes Map\(_1(\mathbb RP^n,\mathbb RP^n)\) for the component of the inclusion \(i_{m,n}:\mathbb RP^m\to \mathbb RP^n\), when \(m\leq n\). The orthogonal group \(O(n+1)\) maps to Map\((\mathbb RP^m,\mathbb RP^n)\) via matrix multiplication. Denoting by \(PV_{n+1,m+1}\) the homogeneous space (Stiefel manifold) of cosets of \(\Delta_{m+1}\times O(n-m)\) in \(O(n+1)\), where \(\Delta_{m+1}\) is the center, there is an obvious map \(PV_{n+1,m+1}\to \text{Map}_1(\mathbb RP^m,\mathbb RP^n)\). The principal result here says that this map is an \(N\)-equivalence (isomorphisms on homotopy groups through \(N=2(n-m)-2\), and surjective in the next dimension). The methods use fibrations of Map\(_1\) spaces, as well as the specific, geometric identification of certain maps. A similar result, over the complex numbers, is due to \textit{S. Sasao} [J. London Math. Soc. 8, 193--197 (1974; Zbl 0284.55020)]. A partial result, in the more difficult non-commutative case of quaternions, was obtained by the present author in [Kodai Math. J. 6, 279--288 (1983; Zbl 0527.55022)].