Mapping spaces from projective spaces

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Abstract

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.

Original languageEnglish
Pages (from-to)173-203
Number of pages31
JournalHomology, Homotopy and Applications
Volume18
Issue number1
DOIs
Publication statusPublished - Jan 1 2016

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All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)

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