  When can one 3-manifold be transformed to another by a finite sequence of
Dehn surgeries which are restricted to preserve the first homology of the
manifolds ? What is the resulting equivalence relation on 3-manifolds ? What if
the surgery circle is further restricted to lie more deeply in the lower
central series of the fundamental group ? We answer these questions. It is
shown that many of these questions have answers in terms of classical
toplogical invariants. Relations with Heegard splittings and the Torelli group
are discussed. This is also related to whether or not one 3-manifold may be
obtained from another by Dehn surgery on a link of restricted type. These
equivalence relations form the philosophical basis of the authors joint work
with Paul Melvin on a theory of finite type invariants for arbitrary
3-manifolds.
