The lower bound conjecture for 3- and 4-manifolds
The lower bound conjecture for 3- and 4-manifolds
复制标题
3-和4-流形的下界猜想
DOI：
10.1007/bf02392331
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发表时间：
1970
期刊：
影响因子：
3.7
通讯作者：
D. Walkup
中科院分区：
文献类型：
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作者：
D. Walkup
对于任意闭连通d-流形M，设[M]（M）表示向量集f（K）=[]0（K）.. fd（K）），其中K在M的所有三角剖分上的范围内，]k（K）表示K的k-单形的个数。本文的主要结果是下面的定理1到定理5，它们与第2章中讨论的Dehn-Sommerville方程一起，给出了某些较简单的3-和4-流形的[M]（M）的一个特征.定理1和定理5给出的关于3和4球的结果对单纯多面体有直接和明显的意义，即，闭有界凸多面体，其所有的固有面都是单形。特别是他们提供了一个强有力的肯定决议在4和5维的所谓下界猜想单纯多面体。对于讨论这一猜想，这在4维至少可以追溯到一份文件Briiclmer在1909年，和一些有限的结果在更高的维度读者是指第10.3节Griinbaum的书多面体[2]。定理3涉及射影三维空间的三角剖分，它对中心对称单纯多面体的一个特殊子类也有直接的含义。这个结果被称为定理6。某些特殊的抽象单纯复形类~/d（n），d >~ 1，n/>0，都出现在这些定理的陈述和证明中。当d >~ 2时，每个类~，td（n）由一类闭d-流形的某些特别简单的三角剖分组成，这些闭d-流形可以被描述为具有n个可定向或不可定向柄的d-球面。可以如下归纳地定义类~/d（n）：
For any closed connected d-manifold M let ](M) denote the set of vectors / (K)= (]0(K) ..... fd(K)), where K ranges over all triangulations of M and ]k(K) denotes the number of k-simplices of K. The principal results of this paper are Theorems 1 through 5 below, which, together with the Dehn-Sommerville equations reviewed in w 2, yield a characterization of ](M) for some of the simpler 3and 4-manifolds. The results for the 3and 4spheres given in Theorems 1 and 5 have immediate and obvious implications for simplicial polytopes, i.e., closed bounded convex polyhedra all of whose proper faces are simplices. In particular they provide a strong affirmative resolution in dimensions 4 and 5 of the socalled lower bound conjecture for simplicial polytopes. For a discussion of this conjecture, which in dimension 4 goes back at least to a paper by Briiclmer in 1909, and some limited results in higher dimensions the reader is referred to Section 10.3 of Griinbaum's book on polytopes [2]. Theorem 3, which is concerned with triangulations of projective 3-space, also has an immediate implication for a special subclass of the centrally symmetric simplicial polytopes. This result is stated as Theorem 6. Some special classes ~/d(n), d >~ 1, n/>0, of abstract simplicial complexes figure in the statement and proof of these theorems. For d >~ 2 each class ~,td(n) consists of certain especially simple triangulations of a class of closed d-manifolds which might be described as d-spheres with n orientable or nonorientable handles. The classes ~/d(n) may be defined inductively as follows: