Mapping spaces from projective spaces

We denote the n-th projective space of a topological monoid G by BnG 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 Map0(BnG,BG′) and the space An(G,G′) of all An-maps from G to G′. Moreover, if we suppose G=G′, then an appropriate union of path-components of Map0(BnG,BG) is delooped.

This fact has several applications. As the first application, we show that the evaluation fiber sequence Map0(BnG,BG)→Map(BnG,BG)→BG extends to the right. As other applications, we investigate higher homotopy commutativity, An-types of gauge groups, Tfk-spaces and homotopy pullback of An-maps. The concepts of Tfk-space and Cfk-space were introduced by Iwase–Mimura–Oda–Yoon, which is a generalization of Tk-spaces by Aguadé. In particular, we show that the Tfk-space and the Cfk-space are exactly the same concept and give some new examples of Tfk-spaces.