SUMMARY: 
The book represents the author’s PhD thesis entitled ”The study of simplicial complexes of nonpositive curvature” written under the guidance of Prof. Dr. Dorin Andrica at the Faculty of Mathematics and Computer Science, ”BabesBolyai” University, ClujNapoca, Romania. The PhD thesis was sustained in public on the 11th of December 2009. The book is naturally divided into two parts. In the first part we introduce the notion of discrete MorseSmale characteristic of a finite simplical complex and we define exact and Fperfect discrete Morse functions on a finite simplicial complex, F being any coefficient field. Further we give a few examples
of Fperfect discrete Morse functions on finite simplicial complexes (see [4], [30]). In the second part of the book we investigate metric and combinatorial conditions which guarantee the collapsibility of a finite cell complex of dimension 2 and 3. 
