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We give new, elementary proofs of the key properties of two families of maps:the Birman--Craggs maps of the Torelli group, and Sato's maps of the level 2congruence subgroup of the mapping class group. We use 3-manifold techniquesthat enable us to avoid referring to deep results in 4-manifold theory as inthe original constructions. Our constructive method provides a unifiedframework for studying both families of maps and allows us to relate Sato'smaps directly to the Birman--Craggs maps when restricted to the Torelli group.We obtain straightforward calculations of Sato's maps on elements given interms of Dehn twists, and we give an alternative description of theabelianization of the level 2 mapping class group.

A surgery approach to abelian quotients of the level 2 congruence group and the Torelli group