There are many situations in topology where the homology type of a space is fixed or is dependent only on coarse
combinatorial data whereas the fundamental group is a rich source of complexity. For example, in knot theory or in the
study of the algebraic plane curves complements. Furthermore, in studying deformations of such embeddings, typically
the homology groups of the exteriors do not vary, or are controlled by the combinatorics of the allowable
singularities, whereas the fundamental group varies with few obvious constraints. Therefore to define interesting
topological invariants of such embeddings, or of certain deformation classes of embeddings, it is vital to understand
to what extent the homology of a space constrains its fundamental group. These issues are often profitably studied in
purely group-theoretic terms. What aspects of a group are unchanged, or stable, under homology equivalences? The model
theorem in this regard is the landmark 1963 result of J. Stallings (below) that each quotient, A/A_n, of a group by
any term of its lower central series is preserved under any homological equivalence. Bill Dwyer later improved on
Stallings' results. More recently, Shelly Harvey invented a superseries of the derived series and, together with me,
proved analaogues of the Stallings and Dwyer theorems. In this talk we introduce a new notion, that of the stability
of a series of groups under a class of maps, which gives a common framework in which all of these theorems can be
understood. The work is joint with Shelly Harvey.