Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Discrete Morse inequalities on infinite graphs. | Discrete Morse inequalities on infinite graphs Rafael Ayala Luis M. Fernandez and Jose A. Vilches Departamento de Geometria y Topologia Facultad de Matematicas Universidad de Sevilla Sevilla SPAIN rdayala@ lmfer@ vilches@ Submitted Jul 19 2007 Accepted Mar 9 2009 Published Mar 20 2009 Mathematics Subject Classification 05C10 57M15 Abstract The goal of this paper is to extend to infinite graphs the known Morse inequalities for discrete Morse functions proved by R. Forman in the finite case. In order to get this result we shall use a special kind of infinite subgraphs on which a discrete Morse function is monotonous namely decreasing rays. In addition we shall use this result to characterize infinite graphs by the number of critical elements of discrete Morse functions defined on them. Introduction. Under a classical point of view Morse theory looks for links between global properties of a smooth manifold and critical points of a function defined on it see 5 . For instance the so-called Morse inequalities relate the Betti numbers of the manifold and the numbers of such critical points. R. Forman 1 introduced the notion of discrete Morse function defined on a finite cw-complex and in this combinatorial context he developed a discrete Morse theory as a tool for studying the homotopy type and homology groups of these complexes. He also proved the corresponding Morse inequalities analogous to the classical ones obtained in the smooth case. This theory has shown to have many applications for instance see 2 . Following Forman s suggestions stated in 1 we have begun the study of discrete Morse functions on infinite simplicial complexes by establishing the extension of Morse inequalities which allow us to understand the relationships between the topology of the complex critical simplices of a discrete Morse function defined on it and the monotonous behaviour of this function at the ends of the complex. The authors made this work for The authors are partially .
