More Torsion in the Homology of the Matching Complex
More Torsion in the Homology of the Matching Complex
复制标题
匹配复合体同源性中的更多扭转
作者：
J. Jonsson
集合 X 上的匹配是大小为 2 的 X 的成对不相交子集的集合。使用计算机，我们分析匹配复形 M n 的积分同源性，它是集合 {1,…, n} 上匹配的单纯复形。主要结果是检测 p ∊ {5, 7, 11, 13} 同源性中的 p 阶元素。具体来说，我们证明，对于 n ≥ 18 和 n ∊ {14, 16}，M n 的同源性中存在 5 阶元素。之前唯一已知的值是 n = 14，在这种特殊情况下，我们有一个新的无计算机证明。此外，我们表明，对于 23 到 41 之间的所有奇数 n 以及 n = 30，M n 的同源性中存在 7 阶元素。此外，M47 同源性中存在 11 阶元素，M62 同源性中存在 13 阶元素。最后，我们计算 d (M n ; ) 的扭转部分的 Sylow 3 和 5 子群的秩，其中 13 ≤ n ≤ 16；对于 n ≤ 12，同源性的完整描述已经存在。为了证明结果，我们使用表示论方法，检查通过让某些基团作用于链复合体而获得的 M n 链复合体的子复合体。
A matching on a set X is a collection of pairwise disjoint subsets of X of size two. Using computers, we analyze the integral homology of the matching complex M n , which is the simplicial complex of matchings on the set {1,…, n}. The main result is the detection of elements of order p in the homology for p ∊ {5, 7, 11, 13}. Specifically, we show that there are elements of order 5 in the homology of M n for n ≥ 18 and for n ∊ {14, 16}. The only previously known value was n = 14, and in this particular case we have a new computer-free proof. Moreover, we show that there are elements of order 7 in the homology of M n for all odd n between 23 and 41 and for n = 30. In addition, there are elements of order 11 in the homology of M47 and elements of order 13 in the homology of M62. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of d (M n ; ) for 13 ≤ n ≤ 16; a complete description of the homology already exists for n ≤ 12. To prove the results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of M n obtained by letting certain groups act on the chain complex.