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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lusternik-Schnirelmann category of $ \mathbf{Spin}{(9)}$


Authors: Norio Iwase and Akira Kono
Journal: Trans. Amer. Math. Soc. 359 (2007), 1517-1526
MSC (2000): Primary 55M30; Secondary 55N20, 57T30
Published electronically: October 17, 2006
MathSciNet review: 2272137
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Abstract: First we give an upper bound of $ \mathrm{cat}{(E)}$, the L-S category of a principal $ G$-bundle $ E$ for a connected compact group $ G$ with a characteristic map $ \alpha: {\Sigma}V \to G$. Assume that there is a cone-decomposition $ \{F_{i}\,\vert\,0 \leq i\leq m\}$ of $ G$ in the sense of Ganea that is compatible with multiplication. Then we have $ \mathrm{cat}{(E)} \leq\mathrm{Max}(m{+}n,m{+}2)$ for $ n \geq 1$, if $ \alpha$ is compressible into $ F_{n} \subseteq F_{m}\simeq G$ with trivial higher Hopf invariant $ H_n(\alpha)$. Second, we introduce a new computable lower bound, $ \mathrm{Mwgt} {(X; {\mathbb{F}_2}})$ for $ \mathrm{cat}({X})$. The two new estimates imply $ \mathrm{cat}({\mathbf{Spin}{(9))}}=\mathrm{Mwgt} ({\mathbf{Spin}{(9)};{\mathbb{F}_2}}) = 8 > 6 =\mathrm{wgt}({\mathbf{Spin}{(9)};{\mathbb{F}_2}})$, where $ (\mathrm{wgt}{-;R})$ is a category weight due to Rudyak and Strom.


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Additional Information

Norio Iwase
Affiliation: Faculty of Mathematics, Kyushu University, Fukuoka 810-8560, Japan
Email: iwase@math.kyushu-u.ac.jp

Akira Kono
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 607-8502, Japan
Email: kono@kusm.kyoto-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04120-1
PII: S 0002-9947(06)04120-1
Keywords: Lusternik-Schnirelmann category, spinor groups, partial products.
Received by editor(s): January 7, 2005
Published electronically: October 17, 2006
Additional Notes: The first author was supported by the Grant-in-Aid for Scientific Research #15340025 from the Japan Society for the Promotion of Science.
Article copyright: © Copyright 2006 American Mathematical Society