  By Torelli topology the author understands aspects of the topology of
surfaces (potentially) relevant to the study of Torelli groups. The present
paper is devoted to a new approach to the results of W. Vautaw about Dehn
multi-twists in Torelli groups and abelian subgroups of Torelli groups. The new
proofs are more transparent and lead to stronger estimates of the rank of an
abelian subgroup when some additional information is available. An unexpected
application of our methods is a new proof of the algebraic characterization of
the Dehn twist about separating circles and the Dehn-Johnson twists about
bounding pairs of circles. This proof is much shorter than the original one and
bypasses one of the main difficulties, which may be called the extension
problem, specific to Torelli groups as opposed to the Teichm\"uller modular
groups. The extension problem is discussed in details in the second paper of
this series.
