Title: Reduced Wannier Representation of Topologically Obstructed Bands

We present a method for constructing a maximally localized Wannier basis spanning a subspace of a manifold with a non-zero Chern number. Topological obstructions generally prevent such manifolds from being fully represented by exponentially localized Wannier states. Here, we avoid the obstruction through a modification of the standard projection method for constructing Wannier functions, enabling the construction of a set of exponentially localized Wannier functions that span a subspace of the topological bands. We refer to these functions as "reduced" Wannier states, reflecting that they only partially cover the topological manifold. The fraction of the topological manifold that they span can be systematically incresed, though at the expense of reduced localization and increased computational cost. We demonstrate this approach using the Haldane tight-binding model of a Chern insulator.