THREE-MANIFOLD TOPOLOGY 9 1.18. REMARK: If T T (K) = 1, then a tower of two-sheeted levels of height n can be extended to a tower of two-sheeted levels of height n + 1 provided that V has a two-sheeted covering. n 1.19. REMARK: If V is a compact 3-manifold, BV 4 0 and some component of dV is not a 2-sphere, then V has a two-sheeted covering. 1.20. EXERCISE: If V is a compact, orientable 3-manifold and some component of dV is not a 2-sphere, then ImCH-OV $) c ^ &AV $)) is not trivial. 1.21. EXERCISE: There exists non-orientable 3-manifolds, V , with g closed, orientable surfaces in dV having genus g (for arbitrary g) yet, HX(V ) is finite. 1.22. LEMMA: l£ K is a finite simplicial complex, Mn is a triangulated 3-manifold and fft : K — Mft i£ simplicial, then there exists £ non- negative integer £ such that any tower over f0 : K — MQ o£ height greater than £ must have a ^ trivial level. Proof: For any simplicial map f of K into a simplicial complex M, define the complexity of f, written C(f), to be the cardinality of the set of pairs of simplicies £(f) = {(?, T) € K X K : o o - _ O 4 T and f(CT) = f(T)j. Since K is finite, £(fn) * is a nonnegative integer. Set Q = £(fQ). Then Lemma 1.22 follows from the observation that if V. =-» M. i. l

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