We denote the n-th projective space of a topological monoid G by BnG and the classifying space by BG. Let G be a wellpointed 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, Tk f-spaces and homotopy pullback of An-maps. The concepts of Tk f -space and Cf k -space were introduced by Iwase-Mimura-Oda-Yoon, which is a generalization of Tk-spaces by Aguadé. In particular, we show that the Tk f- space and the Ck f -space are exactly the same concept and give some new examples of Tk f-spaces.
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)