If A is a graded connected algebra then we define anew invariant, poly depth A, which is finite if Ex_A~*(M, A) ≠ 0 for some A-module M of at most polynomial growth. THEOREM 1 : If f: X → Y is a continuous map of finite category, and if the orbits of H_*(ΩY) acting in the homology of the homotopy fibre grow at most polynomially, then H_*(ΩY) has finite polydepth. THEOREM 5: If L is a graded Lie algebra and polydepth UL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, Σ_(i=k+1)~(k+d) dim L_i ≥ k~r, k ≥ some k(r).
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