  The Hamiltonian reduction of SU(2) Yang-Mills theory for an arbitrary \theta
angle to an unconstrained nonlocal theory of a self-interacting positive
definite symmetric 3 \times 3 matrix field S(x) is performed. It is shown that,
after exact projection to a reduced phase space, the density of the Pontryagin
index remains a pure divergence, proving the \theta independence of the
unconstrained theory obtained. An expansion of the nonlocal kinetic part of the
Hamiltonian in powers of the inverse coupling constant and truncation to lowest
order, however, lead to violation of the \theta independence of the theory. In
order to maintain this property on the level of the local approximate theory, a
modified expansion in the inverse coupling constant is suggested, which for a
vanishing \theta angle coincides with the original expansion. The corresponding
approximate Lagrangian up to second order in derivatives is obtained, and the
explicit form of the unconstrained analogue of the Chern-Simons current linear
in derivatives is given. Finally, for the case of degenerate field
configurations S(x) with rank|S| = 1, a nonlinear \sigma-type model is
obtained, with the Pontryagin topological term reducing to the Hopf invariant
of the mapping from the three-sphere S^3 to the unit two-sphere S^2 in the
Whitehead form.
