Mapping spaces from projective spaces (Q303921)

The paper is best described by quoting from the author's abstract:NEWLINENEWLINE``We denote the \(n\)-th projective space of a topological monoid \(G\) by \(B_nG\) and the classifying space by \(BG\). Let \(G\) be a well-pointed topological monoid having the homotopy type of a CW complex and \(G'\) a well-pointed grouplike topological monoid. We prove that there is a natural weak equivalence between the pointed mapping space \(\text{Map}_0(B_nG,BG')\) and the space \(\mathcal{A}_n(G,G')\) of all \(\mathcal{A}_n\)-maps from \(G\) to \(G'\). Moreover, if we suppose \(G=G'\), then an appropriate union of path-components of \(\text{Map}_0(B_nG,BG)\) is delooped.''NEWLINENEWLINEThe paper is by nature rather technical, but it is well-written and comprehensively referenced. As an application of the theory developed it is shown that the evaluation fiber sequence \(\text{Map}_0(B_nG,BG) \to \text{Map}(B_nG,BG) \to BG\) extends to the right. In other applications higher homotopy commutativity and \(\mathcal{A}_n\)-types of gauge groups are investigated.