To every directed graph G one can associate a complex #DELTA#(G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph G_n, where it leads to the study of some interesting representations of the symmetric group and corresponds (via the Stanley-Reisner correspondence) to an interesting quotient ring. Our main result states that #DELTA#(G_n) is shellable, in particular, Cohen-Macaulay, which can be further translated to say that the Stanley-Reisner ring of #DELTA#(G_n) is Cohen-Macaulay. Besides that, by computing the homology groups of #DELTA#(G) for the cases when G is essentially a tree and when G is a double directed cycle, we touch upon the general question of the interaction of the combinatorial properties of a graph and the topological properties of the associated complex.
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机译：每个有向图G都可以关联由有向子森林组成的复数＃DELTA＃（G）。 R. Stanley向我们建议的这种构造在完整的双向图G_n的情况下尤其重要，在这种情况下，它导致研究对称组的一些有趣表示，并（通过Stanley-Reisner对应）对应于一个有趣的商环。我们的主要结果表明，＃DELTA＃（G_n）是可剥壳的，尤其是科恩-马考雷（Cohen-Macaulay），可以进一步解释为＃DELTA＃（G_n）的Stanley-Reisner环是科恩-马考雷。除此之外，对于G本质上是一棵树而G是双重有向环的情况，通过计算＃DELTA＃（G）的同源性组，我们触及了图和图的组合特性相互作用的一般问题相关复合体的拓扑特性。
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