  We construct a nonlinear sigma (NL$\sigma$) model description of 2+1d spin
systems, by coupling together antiferromagnetic spin chains via interchain
exchange terms. Our mapping incorporates methods developed recently by
ourselves and by Senthil and Fisher, which aim at describing competition
between antiferromagnetic and valence-bond-solid orders in quantum magnets. The
resulting 2+1d $O$(4) NL$\sigma$ model contains a topological $\theta$-term
whose vacuum angle $\theta$ varies continuously with $\delta$, the
bond-alternation strength of the interchain interaction. This implies that the
$\theta$-vacua structure for this NL$\sigma$ model can be explored by tuning
$\delta$ in a suitable 2d spin system, which is strongly reminiscent of the
situation for 1+1 AF spin chains with bond-alternation.
