November 18, 1997

S.~S.~Anisov

Topological complexity of $T^2$-bundles over $S^1$ and continued fractions

This (joint with S.~K.~Lando) work is an attempt to extend Kazarian's method
of computing the singularities'' to fiber bundles of dimension greater than
one.

A torus can be represented as a $CW$-complex with two vertices and three
edges. These edges form a {\it net}. If the length of one of its edges
vanishes, the {\it flip\/} occurs, and the corresponding fiber of the torus
bundle over $S^1$ is said to be {\it degenerate}. We show that the algebraic
number of degenerate fibers is an invariant of the bundle, i.e., does not
depend on the choice of an embedding of the net. We also give an expression
for this number in terms of the denominators of a certain continued fraction
related to the bundle.