November 18, 1997

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

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.