Higher-genus counterexamples to the strong Milnor–Wood inequality for the PSL(2,K) Witt class

Determine whether there exist oriented closed surfaces of genus g > 1 and infinite fields K of characteristic not equal to 2 with Stufe s(K) = 2 for which a flat PSL(2,K)-bundle has Witt class in I₂(K) of norm exceeding 4(g − 1), thereby providing counterexamples to the strong Milnor–Wood inequality for the Witt class beyond genus 1.

Background

Section 6 studies Milnor–Wood type bounds for the Witt class associated to flat PSL(2,K)-bundles over closed oriented surfaces of genus g. The authors establish a weak upper bound on the norm by 4(g−1)+2 and show that, constructively, all elements in I₂(K) of norm at most 4(g−1) can be realized (the strong Milnor–Wood inequality).

They then analyze specific fields and genera: for K = Q, the realized Witt classes coincide exactly with those of norm ≤ 4(g−1). For fields of Stufe 2, they exhibit a genus 1 counterexample where elements of norm 2 occur, exceeding the strong inequality’s threshold at g = 1, and raise the question of whether similar counterexamples exist for higher genus surfaces.