of the commutator subgroup of a knot group
of the commutator subgroup of a knot group
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结群的换向子群的
DOI：
10.1090/s0002-9939-1971-0275416-9
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发表时间：
1971
期刊：
影响因子：
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通讯作者：
D. Sumners
中科院分区：
文献类型：
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作者：
D. Sumners
给出了著名定理的简短拓扑证明：如果 G 是结群，G' 是其交换子群，则 H2(G'; Z) =0。本注释的目的是给出以下著名定理 [1]、[2]、[6]、[7] 的简短拓扑证明：定理。如果 G 是结群，G' 是其交换子群，则 H2(G'; Z) = 0。证明。令 S 表示 S3 中驯服嵌入的 S' 的有界补集。 S 是一个紧致 3 流形有边界，同伦等价于有限二维单纯复形 K。设 G=w7r(K)。众所周知[5]，K 是非球面的（ri(K) =0, i> 2），因此 K 是 Eilenberg-MacLane 空间 K(G, 1)。令K表示K的无限循环覆盖空间；也就是说，7r(K) =G'（G 的交换子群）和 HI(K; Z) = J(t)（由 t 生成的无限循环乘法群）作为单纯覆盖平移群作用于 K。 K 也是非球面，是 G' 的 EilenbergMacLane 空间。令r 表示J(t) 的有理群环。根据[3]、[4]，我们对于所有 q 而言，单纯链群 Cq(k; Q) 是有限生成的自由 r 模，生成器与 K 的 q-单纯形 1-1 对应。由于 r 是主理想域，因此 Hq(k; Q) 是一个 f.g.所有 q 的 r 模块。现在折叠 k 上的覆盖平移的无限循环群产生轨道空间 K。按照 Milnor [4 ]，这可以通过链复合体的短精确序列（作为 r 模） (t 1) wihil C*(K; Q) l ) C*(s ; Q) o ec C*f(K; Q) hOmolo 进行代数表达，从而产生同源的长精确序列 编辑收到24、 1970 年和修订版，1970 年 8 月 14 日。AMS 1970 主题分类。主要 55A25，18H10。
A short topological proof is given for the well-known theorem that if G is a knot group and G' its commutator subgroup, then H2(G'; Z) =0. The purpose of this note is to give a short topological proof of the following well-known theorem [1], [2 ], [6 ], [7 ]: THEOREM. If G is a knot group and G' is its commutator subgroup, then H2(G'; Z) = 0. PROOF. Let S denote the bounded complement of a tamely embedded S' in S3. S is a compact 3-manifold-with-boundary, and is homotopy equivalent to a finite 2-dimensional simplicial complex K. Let G=w7r(K). As is well known [5], K is aspherical (ri(K) =0, i> 2), hence K is the Eilenberg-MacLane space K(G, 1). Let K denote the infinite cyclic covering space of K; that is, 7r(K) =G' (the commutator subgroup of G), and HI(K; Z) = J(t) (the infinite cyclic multiplicative group generated by t) acts on K as the group of simplicial covering translations. K is also aspherical, and is the EilenbergMacLane space for G'. Let r denote the rational group ring of J(t). Following [3], [4] we have for all q that the simplicial chain groups Cq(k; Q) are finitely generated free r-modules, with generators in 1-1 correspondence with the q-simplexes of K. Since r is a principal ideal domain, then Hq(k; Q) is a f.g. r-module for all q. Now collapsing out the infinite cyclic group of covering translations on k yields the orbit space K. Following Milnor [4 ], this is expressed algebraically by the short exact sequence of chain complexes (as r-modules) (t 1) wihil C*(K; Q) l ) C*(s ; Q) o ec C*f(K; Q) hOmolo which yields the long exact sequence of homology Received by the editors May 24, 1970 and, in revised form, August 14, 1970. AMS 1970 subject classifications. Primary 55A25, 18H10.