### Abstract

We denote the n-th projective space of a topological monoid G by B_{n}G 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 Map_{0}(B_{n}G, BG') and the space A_{n}(G, G') of all A_{n}-maps from G to G'. Moreover, if we suppose G = G', then an appropriate union of path-components of Map_{0}(B_{n}G, BG) is delooped. This fact has several applications. As the first application, we show that the evaluation fiber sequence Map_{0}(B_{n}G, BG) → Map(B_{n}G, BG) → BG extends to the right. As other applications, we investigate higher homotopy commutativity, A_{n}- types of gauge groups, T_{k} ^{f}-spaces and homotopy pullback of A_{n}-maps. The concepts of T_{k} ^{f} -space and Cf k -space were introduced by Iwase-Mimura-Oda-Yoon, which is a generalization of T_{k}-spaces by Aguadé. In particular, we show that the T_{k} ^{f}- space and the C_{k} ^{f} -space are exactly the same concept and give some new examples of T_{k} ^{f}-spaces.

Original language | English |
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Pages (from-to) | 173-203 |

Number of pages | 31 |

Journal | Homology, Homotopy and Applications |

Volume | 18 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)