On a Conjecture of C. T. C. Wall
On a Conjecture of C. T. C. Wall
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论C. T. C. Wall的猜想
DOI：
10.1112/jlms/s2-14.2.331
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发表时间：
1976
影响因子：
1.2
通讯作者：
F. Johnson
中科院分区：
文献类型：
--
作者：
L. Auslander;F. Johnson
定理1肯定地解决了WALL在文献[6]第231页提出的一个猜想。从拓扑学上讲，这一结果很有趣，至少有两个原因。首先，如果F是无挠多{循环或有限}群，则F是Poincare*对偶群[4]；因此，由于Bieri和Eck mann[2]的结果，X(F，1)的同伦被有限复控制。定理1证明了不仅X(F，1)的壁有限障碍[5]消失了，而且它的Spivak纤维至少有一个手术障碍为零的光滑法向不变量。除了比耶里-埃克曼定理之外，对于任意的庞加莱对偶群来说，这一切都是未知的。另外，回想在[6；第15章]中，WALL对每个多Z群F产生了一个K(T，1)型的闭拓扑流形，直到同胚为止都是唯一的(可能在3维和4维上除外)。他无法证明它们是可组合三角的，而且可能存在障碍T(F)EH4(F；Z2)，这可能阻止了这一点。然而，我们得到了
Theorem 1 settles in the affirmative a conjecture made by Wall on page 231 of [6]. Topologically, the result is interesting for at least two reasons. Firstly, if F is a torsionfree poly {cyclic or finite} group, then F is a Poincare* Duality group [4]; hence, by a result of Bieri and Eckmann [2], X (F, 1) is dominated in homotopy by a finite complex. Theorem 1 shows that not only does the Wall finiteness obstruction [5] of X (F, 1) vanish, but that also its Spivak fibration has at least one smooth normal invariant whose surgery obstruction is zero. Beyond the Bieri-Eckmann theorem, none of this is known for an arbitrary Poincare'Duality group. In addition, recall that in [6; Chapter 15], Wall produced for each poly-Z group F, a closed topological manifold Mr of type K (T, 1), unique (except perhaps in dimensions 3 and 4) up to homeomorphism. He was unable to prove they were combinatorially triangulable and there was, potentially, an obstruction T (F) eH4 (F; Z2) which perhaps prevented this. However, we get