Foliation preserving Lie group actions and characteristic classes

Author:
Haruo Suzuki

Journal:
Proc. Amer. Math. Soc. **85** (1982), 633-637

MSC:
Primary 57R30; Secondary 57R20

DOI:
https://doi.org/10.1090/S0002-9939-1982-0660619-9

MathSciNet review:
660619

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\tilde {\mathcal {F}}$ be a codimension $k$ foliation of a manifold $M$ and $\mathcal {F}$ a subfoliation of $\tilde {\mathcal {F}}$ with codimension $q$. Let a Lie group $G$ of dimension $k$ act on $M$ transversally locally freely to $\tilde {\mathcal {F}}$ and preserving $\mathcal {F}$. Let $\mathcal {F}’$ be the foliation determined by $\mathcal {F}$ and the $G$-action. Then we have the following relations between exotic classes of $\mathcal {F}$ and $\mathcal {F}’:{\alpha _\mathcal {F}}([{\hat c_I}{c_J}]) = {\alpha _{\mathcal {F}’}}([{\hat c_I}{c_J}])$ for $I = ({i_1}, \ldots ,{i_\lambda })$, $J = ({j_1}, \ldots ,{j_\mu })$, $1 \leqslant {j_\gamma },{j_l} \leqslant q - k$, and ${\alpha _\mathcal {F}}([{\hat c_I}{c_J}]) = 0$ otherwise.

- Raoul Bott,
*Lectures on characteristic classes and foliations*, Lectures on algebraic and differential topology (Second Latin American School in Math., Mexico City, 1971) Springer, Berlin, 1972, pp. 1–94. Lecture Notes in Math., Vol. 279. Notes by Lawrence Conlon, with two appendices by J. Stasheff. MR**0362335** - Shiing Shen Chern,
*Geometry of characteristic classes*, Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress (Dalhousie Univ., Halifax, N.S., 1971) Canad. Math. Congr., Montreal, Que., 1972, pp. 1–40. MR**0370613** - Michael-Robert Herman,
*The Godbillion-Vey invariant of foliations by planes of $T^{3}$*, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Springer, Berlin, 1977, pp. 294–307. Lecture Notes in Math., Vol. 597. MR**0451261** - Daniel Lehmann,
*Classes caractéristiques exotiques et ${\cal I}$-connexité des espaces de connexions*, Ann. Inst. Fourier (Grenoble)**24**(1974), no. 3, xiv, 267–306 (French, with English summary). Avec une appendice par B. Callenaere et D. Lehmann. MR**362342** - Connor Lazarov and Herbert Shulman,
*Obstructions to foliation preserving Lie group actions*, Topology**18**(1979), no. 3, 255–256. MR**546795**, DOI https://doi.org/10.1016/0040-9383%2879%2990008-9

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57R30,
57R20

Retrieve articles in all journals with MSC: 57R30, 57R20

Additional Information

Keywords:
Characteristic classes,
foliations,
Lie group actions

Article copyright:
© Copyright 1982
American Mathematical Society