### Abstract

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.

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

Number of pages | 31 |

Journal | Homology, Homotopy and Applications |

Publication status | Published - 2016 |

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### Cite this

*Homology, Homotopy and Applications*, 173-203.

**Mapping spaces from projective spaces.** / Tsutaya, Mitsunobu.

Research output: Contribution to journal › Article

*Homology, Homotopy and Applications*, pp. 173-203.

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TY - JOUR

T1 - Mapping spaces from projective spaces

AU - Tsutaya, Mitsunobu

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

M3 - Article

SP - 173

EP - 203

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

SN - 1532-0073

ER -