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By Torelli topology the author understands aspects of the topology ofsurfaces (potentially) relevant to the study of Torelli groups. The presentpaper is devoted to a new approach to the results of W. Vautaw about Dehnmulti-twists in Torelli groups and abelian subgroups of Torelli groups. The newproofs are more transparent and lead to stronger estimates of the rank of anabelian subgroup when some additional information is available. An unexpectedapplication of our methods is a new proof of the algebraic characterization ofthe Dehn twist about separating circles and the Dehn-Johnson twists aboutbounding pairs of circles. This proof is much shorter than the original one andbypasses one of the main difficulties, which may be called the extensionproblem, specific to Torelli groups as opposed to the Teichm\"uller modulargroups. The extension problem is discussed in details in the second paper ofthis series.

Elements of Torelli topology: I. The rank of abelian subgroups