Tautological characteristic classes II: the Witt class

Let K be an arbitrary infinite field. The cohomology group H² (SL (2, K), H₂\, SL (2, K) ) contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in SL (2, K) it is useful to have classes stable under deformations (Fenchel--Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovar. The Milnor--Wood inequality asserts that an SL (2, R) -bundle over a surface of genus g admits a flat structure if and only if its Euler number is (g-1). We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general.