  We study the rational homotopy groups of open books in terms of those of
their pages and bindings. Under homotopy conditions on the monodromy we prove a
rational dichotomy theorem for open books, as an extension of the classical
dichotomy in rational homotopy theory. As a direct application, we show that
for Milnor's open book decomposition of an odd sphere with monodromy of finite
order the induced action of the monodromy on the homology groups of its page
cannot be nilpotent. We also calculate the integral homotopy groups of open
books in terms of those of their pages and bindings when their monodromies are
homotopic to the identity map.
