  Using the Gordon-Litherland pairing, one can define invariants (signature,
nullity, determinant) for ${\mathbb Z}/2$ null-homologous links in thickened
surfaces. In this paper, we study the concordance properties of these
invariants. For example, if $K \subset \Sigma \times I$ is ${\mathbb Z}/2$
null-homologous and slice, we show that its signatures vanish and its
determinants are perfect squares. These statements are derived from a cobordism
result for closed unoriented surfaces in certain 4-manifolds.
  The Brown invariants are defined for ${\mathbb Z}/2$ null-homologous links in
thickened surfaces. They take values in ${\mathbb Z}/8 \cup \{\infty\}$ and
depend on a choice of spanning surface. We present two equivalent methods to
defining and computing them, and we prove a chromatic duality result relating
the two. We study their concordance properties, and we show how to interpret
them as Arf invariants for null-homologous links. The Brown invariants and knot
signatures are shown to be invariant under concordance of spanning surfaces.
