Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Independence complexes and Edge covering complexes via Alexander duality. | Independence complexes and Edge covering complexes via Alexander duality Kazuhiro Kawamura Institute of Mathematics University of Tsukuba Ibaraki Tsukuba 305-8571 Japan kawamura@ Submitted Jun 8 2010 Accepted Feb 6 2011 Published Feb 14 2011 Mathematics Subject Classiciation 05C10 55P10 05C05 05C69 05C99 Abstract The combinatorial Alexander dual of the independence complex Ind G and that of the edge covering complex EC G are shown to have isomorphic homology groups for each non-null graph G. This yields isomorphisms of homology groups of Ind G and EC G with homology dimensions being appropriately shifted and restricted. The results exhibits the complementary nature of homology groups of Ind G and EC G which had been proved by Ehrenborg-Hetyei 10 Engstrom 11 and Marietti-Testa 16 for forests at homotopy level. 1 Introduction and Preliminaries All graphs are assumed to be finite and simple. Topology of independence complexes has recently drawn much attention of various authors. See for example 2 5 6 7 9 10 11 12 14 16 15 etc. Ehrenborg and Hetyei 10 proved that the independence complex of a forest is either contractible or is homotopy equivalent to a sphere. Also Engstrom 11 and Marietti-Testa 15 independently gave algorithms to determine the dimension of the associated sphere see 13 for another approach . Marietti and Testa 16 have shown that the homotopy types of the independence complex Ind F and the edge covering complex EC F of a forest F are closely related they are either both homotopy equivalent to spheres or both contractible. Furthermore the dimensions of the associated spheres are both related to the domination number and differ by the number of components of F 16 Theorem . The referee of the first manuscript kindly pointed out that the method of Engstrom 11 can be applied to obtain these homotopy equivalences. The result of the present paper shows that this complementary phenomenon is observed to certain extent for every non-null .
