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Homotopy commutativity in Hermitian symmetric spaces

Part of: Homotopy theory Homotopy groups

Published online by Cambridge University Press:  18 April 2022

Daisuke Kishimoto Open the ORCID record for Daisuke Kishimoto [Opens in a new window] ,
Masahiro Takeda  and
Yichen Tong
Daisuke Kishimoto
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan e-mails: kishimoto@math.kyushu-u.ac.jp, takeda.masahiro.87u@st.kyoto-u.ac.jp, tong.yichen.25m@st.kyoto-u.ac.jp
Masahiro Takeda
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan e-mails: kishimoto@math.kyushu-u.ac.jp, takeda.masahiro.87u@st.kyoto-u.ac.jp, tong.yichen.25m@st.kyoto-u.ac.jp
Yichen Tong
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan e-mails: kishimoto@math.kyushu-u.ac.jp, takeda.masahiro.87u@st.kyoto-u.ac.jp, tong.yichen.25m@st.kyoto-u.ac.jp
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Abstract

Ganea proved that the loop space of $\mathbb{C} P^n$ is homotopy commutative if and only if $n=3$. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C} P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds $G/T$ for a maximal torus T of a compact, connected Lie group G.

MSC classification

Primary: 55P35: Loop spaces 55Q15: Whitehead products and generalizations

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 746 - 752
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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