arXiv:math/0404310v1 [math.GT] 17 Apr 2004 POSITIVE DEHN TWIST EXPRESSIONS FOR SOME NEW INVOLUTIONS IN MAPPING CLASS GROUP YUSUF Z. GURTAS Abstract. The well-known fact that any genus g symplectic Lefschetz fibra- tion X 4 → S 2 is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manifolds that are realized as symplectic Lefschetz fibra- tions. In this article we provide new words in the mapping class group, hence new symplectic Lefschetz fibrations. We also compute the signatures of those symplectic Lefschetz fibrations. Introduction The last decade has experienced a resurgence of interest in the mapping class groups of 2-dimensional closed oriented surfaces. This is primarily due to Donald- son’s theorem [3] that any closed oriented symplectic 4−manifold has the structure of a Lefschetz pencil and Gompf’s theorem [5] that any Lefschetz pencil supports a symplectic structure. These theorems, together with the fact that any genus g symplectic Lefschetz fibration is given by a word that is equal to the identity ele- ment in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4−dimensional symplectic topology. Therefore the elements of finite order in the mapping class group are of special importance. There are very few examples of elements of finite order for which explicit positive Dehn twist products are known and a great deal is known about the structure of the 4−manifolds that they describe. In this article we give an algorithm for positive Dehn twist products for a set of new involutions in the mapping class group that are non-hyperelliptic. These involutions are obtained by combining the positive Dehn twist expressions for two well-known involutions of the mapping class group. An application of the words in the mapping class group that we produced is to determine the homeomorphism types of the 4−manifolds that they describe. This calculation involves computing two invariants of those manifolds. The first one, which is very easy to compute, is the Euler characteristic, and the second one is the signature. Using the algorithm described in [17] we wrote a Matlab program that computes the signatures of some of these manifolds. The computations that we have done using this program point to a closed formula for the signature, which is mentioned in the last part of this article. This article will be followed by two other articles, the first one of which will contain the same computations for the multiple case. Namely, the explicit positive 1991 Mathematics Subject Classification. Primary 57M07; Secondary 57R17, 20F38. Key words and phrases. low dimensional topology, symplectic topology, mapping class group, Lefschetz fibration . 1