  We develop a "local theory" of multidimensional quasiperiodic $\SL(2,\R)$
cocycles which are not homotopic to a constant. It describes a $C^1$-open
neighborhood of cocycles of rotations and applies irrespective of arithmetic
conditions on the frequency, being much more robust than the local theory of
$\SL(2,\R)$ cocycles homotopic to a constant. Our analysis is centered around
the notion of monotonicity with respect to some dynamical variable. For such
{\it monotonic cocycles}, we obtain a sharp rigidity result, minimality of the
projective action, typical nonuniform hyperbolicity, and a surprising result of
smoothness of the Lyapunov exponent (while no better than H\"older can be
obtained in the case of cocycles homotopic to a constant, and only under
arithmetic restrictions). Our work is based on complexification ideas, extended
"\`a la Lyubich" to the smooth setting (through the use of asymptotically
holomorphic extensions). We also develop a counterpart of this theory centered
around the notion of monotonicity with respect to a parameter variable, which
applies to the analysis of $\SL(2,\R)$ cocycles over more general dynamical
systems and generalizes key aspects of Kotani Theory. We conclude with a more
detailed discussion of one-dimensional monotonic cocycles, for which results
about rigidity and typical nonuniform hyperbolicity can be globalized using a
new result about convergence of renormalization.
