ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

6. Topology

1991 MS Classification: 57,58 Hashiguchi, Norikazu, Nihon University, Japan The surgery formula for the discrete Godbillon-Vey invariant Ghys and Sergiescu define the discrete Godbillon-Vey class GSGV in H^{2}(P L^{+}(S^{1}); R) where P L^{+}(S^{1}) is the group of orientation preserving piecewise linear(=PL) homeomorphisms of the circle. We can generalize GSGV to an invariant of codimension one transversely PL foliations in oriented closed 3-dimensional manifolds. These foliations are deformed by the surgery along a simple closed curve with a holonomy in a leaf. This surgery is topologically a Dehn surgery. Theorem. Let F be a codimension one transversely PL foliation and C, a simple closed curve in a leaf whose holonomy is linear, x ↦ λx (λ > 0). If F′ is made from F by the surgery along C, then

GSGV (F′) = GSGV (F) − (log λ)^{2}.