THREE-MANIFOLD TOPOLOGY 7 and for some j, { g | J!] f N. . J X j The "essence" of the proofs of the preceding theorems (1.1, 1.6, 1.9, 1.12 and 1.13) is discussed on pages 40-41 of [He ]. Basically my approach is the same. Given a singular map f : F — M, I construct a factorization M P f * F M where the corresponding problem for f : F — M has a solution. Then by using techniques of equivariant surgery in covering spaces, I am able to arrive at a solution to the problem for f : F — M. This method works well for 1.1, 1.6 and their generalizations (1.12 and 1.13) mentioned above however, I have had no success using this method to prove the Sphere Theorem (1.9). It is easy enough to find a factorization where f : F — M has a solution (see Exercise 1.29). The problem lies in the limited methods of equivariant surgery. This matter is discussed further in Chapter III. THE TOWER CONSTRUCTION The commutative diagram L^-^ VCL- M 1.14 K-L L 0- v ^ M is called £ level over f : K — L if K, M and M are all simplicial

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