Boundary Preserving Maps of 3-manifolds
نویسنده
چکیده
We prove an extension of Waldhausen's theorem [5] conjectured by Hempel in [3]. We prove the following extension of Waldhausen's theorem [5]: Theorem 1. Let M, N be P2-irreducible 3-manifolds. Suppose that M is compact, sufficiently large and f: (M, dM) -^ (N, dN) is a continuous map inducing an injection fM: trx(M)^>irx(N). Then, there is a proper homotopy f: (M, 3M) —» (N, dN) such that f0 = fand either (i)/,: M —> N is a covering map, or (ii) M is an I-bundle over a closed surface, andfx(M) c dN, or (iii) N (hence also M) is a solid torus or a solid Klein bottle and /,: M —»• N is a branched covering with branch set a circle, or (iv) M is a cube with handles andfx(M) c dN. If f\B: B -^ C is already a covering map, we may assume f\B — f\B, for all t (where B is any component of dM and C the component of dN containing f(B)). This theorem is conjectured in [3], where it is proved under additional restrictions. When M, N are orientable and /„ is an isomorphism, a variant of this result is proved by Evans in [2]. Our argument yields a simple proof of his result too. We refer to [1] and [4] for the concepts 'geometric degree', 'absolute degree', 'orientation-true', etc. The term 'degree of a map F will be used for the twisted degree of / and will be denoted by deg/ as in [4]. Lemma 2. Let M be a compact irreducible 3-manifold and let f: (M, dM) —» (Af, dN) be a map into any aspherical 3-manifold such that /„: 7r,(M)-» trx(N) is injective. Let S be a component of dM, S' the component dN containing f(S). If the geometric degree of (f\S): S —> S' is zero, then M is a cube with handles and f is properly homotopic to a map into dN. Remark. If (f\S)+(mx(S)) is free subgroup of wx(S'), then the geometric degree of /| S is zero. Proof. Since the geometric degree of (f\S) is zero, (f\S) is homotopic to a map of S into S'-P for anyp G S'. Hence (/|S)^(trx(S)) = H is a finitely generated free subgroup of wx(S'-p). Representing H by the fundamental group of a wedge X of circles, we can write (/|S), up to homotopy, as a composite S-» A"-» S" p. Received by the editors September 22, 1978 and, in revised form, February 12, 1979. AMS (A/05) subject classifications (1970). Primary 57A10.
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تاریخ انتشار 2010