On simply connected, 4-dimensional polyhedra
On simply connected, 4-dimensional polyhedra
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简单连通的 4 维多面体
DOI：
10.1007/bf02568048
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发表时间：
1949
影响因子：
0.9
通讯作者：
J. Whitehead
中科院分区：
文献类型：
--
作者：
J. Whitehead
1.导论.本文的主要目的是证明一个单连通的四维多面体的同伦型完全由它的关联上同调环mod. m（m-0，1，2.）决定，以及一个附加的结构元件。后者是用乘积来定义的，这是由L. Pontrjaginl），并且最近由NE Steenrod（S）进行了更广泛的研究。这里我们需要的是庞特亚金的方法，它把一个2n维的上同调类px，mod.4r，和每一个n维的上同调类x，mod.2r联系起来。我们称px为x的Pontrjagin平方。若f是上同调类x中的模2 r余圈，则px用上链表示，用Steenrod记法记为fUl + lU，~ f.作为系数，可以通过M的方法组合成单个环。Bockstein a）.我们给这个环额外的代数结构，通过引入一定的运营商A，也Pontrjagin平方。我们将结果描述为P的上同调环，并证明：
1. Introduction. Our main purpose is to show that the homotopy type of a simply connected, 4-dimensional polyhedron is completely determined by its inter-re] ated co-homology rings, mod. m (m-----0, 1, 2....), together with one additional element of structure. The latter is defined in terms of a product, which was introduced by L. Pontrjaginl), and which has recently been studied in greater generality by NE SteenrodS). What we want here is Pontrjagin's method of associating a 2ndimensional co-homology class, px, mod. 4r, with every n-dimensional co-homology class, x, mod. 2r. We shall call px the Pontrjagin square of x. If f is a co-cycle, mod. 2r, in the co-homology class x, then px is represented by the co-chain which, in Steenrod's notation, is written as fU l+ lU,~ f.The co-homology rings of a polyhedron, P, with integers reduced mod. m (m--0, 1, 2....) as coefficients, may be combined into a single ring by a method due to M. Bockstein a). We give this ring additional algebraic structure by introducing a certain operator A and also the Pontrjagin squares. We describe the result as the co-homology ring of P and prove that: