  We investigate some relations between the duality and the topological
filtration in algebraic K-theory. As a result, we obtain a construction of the
first Steenrod square for Chow groups modulo two of varieties over a field of
arbitrary characteristic. This improves previously obtained results, in the
sense that it is not anymore needed to mod out the image modulo two of torsion
integral cycles. Along the way we construct a lifting of the first Steenrod
square to algebraic connective K-theory with integral coefficients, and
homological Adams operations in this theory. Finally we provide some
applications to the Chow groups of quadrics.
