8 WILLIAM JACO complexes, both of the maps f : K M, £ : K M are simplicial maps, L = f(K), L = ?(K), V = U(L), 7= U(L) are relative regular neighborhoods of L in M, L in M, respectively, and M is a covering space of V with simplicial covering projection p. The level is said to be trivial, two-sheeted, regular or universal depending on the covering space (M, p) of M being trivial, two-sheeted, regular or universal, respectively. The commutative diagram 1.15 K ^-^ V ( ^ M n n ^ n K n £- V . £- M , n-1 n-1 n-1 Pn-1 Kx c= - Vx (= - M- K o C ^vo °- H o is called a tower (over f^ : K M^) £^ f height n if each subdiagram 1.16 K i + l^ V i + l^ M l + l P i+1 K. C -^ V. C -^ M. i l l is a level over f. : K l M. for each i, 0 i n. i ' The fundamental lemma for our purposes is: 1.17. LEMMA: A tower of two-sheeted levels having height n can be extended to a ^ tower of two-sheeted levels having height n + 1 provided that V has £ two-sheeted covering M , with projection map p -
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