A generalization of the Morita-Mumford classes to extended mapping class groups for surfaces

Let Σ _g,1 be an oriented compact surface of genus g with 1 boundary component, and Γ _g,1 the mapping class group of Σ _g,1 . We define a bigraded series of cohomology classes m _i,j ∈H ^2i+j−2 (Γ _g,1 ;⋀ ^j H ₁(Σ _g,1 ;ℤ)), 2i+j−2≥1,i,j≥0. When j=0, the class m _i+1,0 is the i-th Morita- Mumford class Mo]Mu]. It is proved that H ^r (Γ _g,1 ;⋀ ^s H ₁(Σ _g,1 ;ℚ)) is generated by m _i,j 's for the case r+s=2 and the case g≥5 and (r,s)=(1,3). Especially the Johnson homomorphism extended to the whole mapping class group by Morita Mo3] has an implicit representation by the classes m _0,3 and m _0,2 m _1,1 over ℚ. Oblatum 28-IV-1995 & 8-II-1997