TY - JOUR

T1 - Lusternik-Schnirelmann category of non-simply connected compact simple Lie groups

AU - Iwase, Norio

AU - Mimura, Mamoru

AU - Nishimoto, Tetsu

N1 - Funding Information:
* Corresponding author. E-mail addresses: iwase@math.kyushu-u.ac.jp (N. Iwase), mimura@math.okayama-u.ac.jp (M. Mimura), nishimoto@kinwu.ac.jp (T. Nishimoto). 1 The first named author is supported by the Grant-in-Aid for Scientific Research #14654016 from Japan Society for the Promotion of Science.

PY - 2005/5/14

Y1 - 2005/5/14

N2 - Let F → X → B be a fibre bundle with structure group G, where B is (d - 1)-connected and of finite dimension, d ≥ 1. We prove that the strong L-S category of X is less than or equal to m+ dim B/d, if F has a cone decomposition of length m under a compatibility condition with the action of G on F. This gives a consistent prospect to determine the L-S category of non-simply connected Lie groups. For example, we obtain cat (PU(n)) ≤ 3(n - 1) for all n ≥ 1, which might be best possible, since we have cat (PU(pr)) = 3(pr - 1) for any prime p and r ≥ 1. Similarly, we obtain the L-S category of SO (n) for n ≤ 9 and PO(8). We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category.

AB - Let F → X → B be a fibre bundle with structure group G, where B is (d - 1)-connected and of finite dimension, d ≥ 1. We prove that the strong L-S category of X is less than or equal to m+ dim B/d, if F has a cone decomposition of length m under a compatibility condition with the action of G on F. This gives a consistent prospect to determine the L-S category of non-simply connected Lie groups. For example, we obtain cat (PU(n)) ≤ 3(n - 1) for all n ≥ 1, which might be best possible, since we have cat (PU(pr)) = 3(pr - 1) for any prime p and r ≥ 1. Similarly, we obtain the L-S category of SO (n) for n ≤ 9 and PO(8). We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category.

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U2 - 10.1016/j.topol.2004.11.006

DO - 10.1016/j.topol.2004.11.006

M3 - Article

AN - SCOPUS:17444373099

SN - 0166-8641

VL - 150

SP - 111

EP - 123

JO - Topology and its Applications

JF - Topology and its Applications

IS - 1-3

ER -