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)) H 2 ( S L ( 2 , K ) ; H 2 S L ( 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 Nekovàř. The Milnor-Wood inequality asserts that an SL(2,R) S L ( 2 , R ) -bundle over a surface of genus g admits a flat structure if and only if its Euler number is ≤ (g - 1) ≤ ( 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.