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We consider the classification up to homotopy of homotopy equivalences of compact 3-manifolds. Given two compact 3-manifolds (with base point and CTV-decomposition) (M, m) and (N, n), in [3] we found an algebraic criterion for the existence of a degree 1 map ƒ : (M, bM) —* (N, bN) extending a given map f defined on the relative 1-skeleton (M, bM). Here we consider the space H(M, m) of degree 1 homotopy equivalences ƒ: (M, m) —» (M, m) such that ƒ \bM = Id and ƒ is homotopic rel bM U {m} to a map coinciding with the identity on (M, bM) . If bM = 0, it is equivalent to say that ƒ induces the identity automorphism of n^M, m). (If bM ¥=0, we assume that m E bM.) Important results are the following.

Obstruction theory in 3-dimensional topology: classification theorems