tailieunhanh - Báo cáo toán học: " Subdivision yields Alexander duality on independence complexes"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Subdivision yields Alexander duality on independence complexes. | Subdivision yields Alexander duality on independence complexes Peter Csorba Department of Mathematics and Computer Science Eindhoven University of Technology 513 5600 MB Eindhoven The Netherlands pcsorba@ Submitted Dec 1 2008 Accepted Apr 24 2009 Published May 12 2009 Mathematics Subject Classification 55P10 05C69 05E25 Dedicated to Anders Bjorner on the occasion of his 60th birthday. Abstract We study how the homotopy type of the independence complex of a graph changes if we subdivide edges. We show that the independence complex becomes the Alexander dual if we place one new vertex on each edge of a graph. If we place two new vertices on each edge then the independence complex is the wedge of two spheres. Placing three new vertices on an edge yields the suspension of the independence complex. 1 Introduction Independence complexes of various graph classes . trees cycles 2D grids were studied in numerous papers 2 4 5 6 9 10 11 12 . We study how edge subdivision definition 1 changes the homotopy type of the independence complex. This is motivated by the homology calculation 7 of Ind G3 . Schoutens 15 observed and proved that Hí Ind G R Hn_i_2 Ind G2 R using the double complex and the tic-tac-toe lemma. This explains that the reduced Euler characteristic sometimes changes the sign if we place one new vertex on each edge of a graph Ỹ Ind G 1 ly G l x Ind G2 . Alexander duality explains this on the homotopy level. Ind G is a subcomplex of a simplex with n V G vertices. If G is connected then Ind G is a subcomplex of Sn_2 the boundary of a simplex with n vertices. We can consider this Sn-2 as the equator of Sn_1. We will show that the complement of Ind G Sn-1 Ind G is homotopy equivalent to Ind G2 . In section 2 we review some definitions and collect the necessary tools for the proofs. In This research has been supported by DIAMANT an NWO mathematics cluster . THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R11 1 section 3 we will show that Ind G2