Sharp 4(g−1) bound for the Witt class

Establish whether, for every infinite field K and every flat SL(2, K)-bundle P over a closed oriented surface of genus g, the Witt class w(P) in I2(K) always satisfies |w(P)| ≤ 4(g−1), where |·| denotes the standard Witt norm given by the anisotropic dimension of the representing quadratic form.

Background

The paper proves an “easy” general bound for the Witt class: for any infinite field K and genus g surface, the norm of the Witt class of a flat SL(2, K)-bundle is at most 4g−2 (Theorem B(a)).

In the classical real case, the Euler class satisfies the stronger Milnor–Wood inequality, which corresponds to a 4(g−1) bound for the Witt class’s real component. This motivates asking whether the same stronger 4(g−1) bound might hold for the full Witt class over general fields.

The authors explicitly state that it is unclear whether this improvement holds in general and report no counterexamples, framing it as an open question.