Centerless groups—an algebraic formulation of Gottlieb’s theorem
Centerless groups—an algebraic formulation of Gottlieb’s theorem
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DOI：
10.1016/0040-9383(65)90060-1
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发表时间：
1965-10
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影响因子：
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通讯作者：
J. Stallings
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文献类型：
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作者：
J. Stallings
D. GOTTLIEB [I]证明了有限非球面多面体的基本群的中心是平凡的，其欧拉特征不为零。他的证明使用了Nielsen [2]和Wecken [3]的思想，他们根据某些同伦性质对映射K+ K的不动点进行分类。Gottlieb的这个证明可以转化为代数，这就是这里要做的。我们讨论的莱夫谢茨不动点定理在代数的背景下，我们定义“迹”的方式，同伦自同态的链复杂的有相同的“莱夫谢茨数”。我们定义了复形生成投射模的“秩”和复形的“欧拉特征线”，并证明了Gottlieb定理。
D. GOTTLIEB [I] has proved that the center of the fundamental group of a finite, aspherical polyhedron, whose Euler characteristic is non-zero, is trivial. His proof uses an idea of Nielsen [2] and Wecken [3], who classify the fixed points of a mapping K+ K according to certain homotopy properties.This proof of Gottlieb’s can be transformed into algebra, and that is what will be done here. We discuss the Lefschetz fixed point theorem in an algebraic context; we define “trace” in such a way that homotopic endomorphisms of a chain complex have the same “Lefschetz number”. We are able to define “rank” of a finitely generated projective module and the “Euler characteristic” of a complex; and then we prove Gottlieb’s theorem.