THE TOPOLOGY OF S2-FIBER BUNDLES

Title & Authors
THE TOPOLOGY OF S2-FIBER BUNDLES
Cho, Yong-Seung; Joe, Do-Sang;

Abstract
Let$\small{P{\rightarrow}M}$ be an oriented $\small{S^2-fiber}$ bundle over a closed manifold M and let Q be its associated SO(3)-bundle, then we investigate the ring structure of the cohomology of the total space P by constructing the coupling form TA induced from an SO(3) connection A. We show that the cohomology ring of total space splits into those of the base space and the fiber space if and only if the Pontrjangin class $\small{p_1(Q)\;{\in}\;H^4(M;\mathbb{Z})}$ vanishes. We apply this result to the twistor spaces of 4-manifolds.
Keywords
$\small{S^2}$-fiber bundle;coupling 2-form;twistor space;
Language
English
Cited by
References
1.
R. Bott and L. Tu, Differential forms in algebraic topology, Springer-Verlag, 1986

2.
Y. Cho and D. Joe, Anti-symplectic involutions with Lagrangian fixed loci and their quotients, Proc. Amer. Math. Soc. 130 (2002), 2797-2801

3.
A. Dold and H. Whitney, Classification of oriented sphere bundles over a 4-complex, Ann. of Math. vol. 69, 667-677

4.
V. Guillemin, E. Lerman, and S. Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge University Press, 1996

5.
H. Lawson, Jr. and M. Michelsohn, Spin geometry, Princeton Math. Ser. vol 38, Princeton University Press, Princeton, NJ, 1989

6.
J. W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Math. Notes vol. 44, Princeton University Press, Princeton, NJ, 1996

7.
J. Milnor and J. Stasheff, Characteristic classes, Princeton University Press, 1974

8.
D. McDuff and D. Salamon, Introduction to symplectic topology, Oxford, 1995

9.
A.Weinstein, A universal phase space for particle in Yang-Mills fields, Lett. Math. Phys. 2 (1978), 417-420