  Let $X$ be a closed Riemann surface. When $X$ is embedded into a projective
space, the first rational cohomology group can be concretely obtained from the
monodromy in the family of its smooth hyperplane sections by C. Schnell's tube
mapping. We generalize this result to the first integral homology group by
relating the tube mapping with the topological Abel--Jacobi mapping. By making
use of the mapping class group action, we prove that all tube classes
constructed from the elementary vanishing cycles form a cofinite subgroup of
the first integral homology group of $X$.
