Rigidity of Fibrations over Nonpositively Curved Manifolds?

THE THEORY of manifold approximate fibrations is the correct bundle theory for topological manifolds and singular spaces. This theory plays the same role in the topological category as fiber bundle theory plays in the differentiable category and as block bundle theory plays in the piecewise linear category. For example, neighborhoods in topologically stratified spaces can be characterized and classified within the theory of manifold approximate fibrations (see [l 1,3]). This paper is concerned with manifold approximate fibrations over closed manifolds of nonpositive curvature. The main result is that this curvature assumption on the base space implies that such manifold approximate fibrations are rigid in the sense that if two manifold approximate fibrations over a manifold of nonpositive curvature are homotopy-theoretically equivalent, then they are equivalent in a much stronger, geometric way appropriate in this theory (they are controlled homeomorphic). Although we work within the setting of manifold approximate fibrations, our results also hold, and are new, for the more classical theories of fibrations and fiber bundle projections between manifolds. These results are established by working with the full moduli space of all manifold approximate fibrations over a closed manifold B of nonpositive curvature. We study a forgetful map from this moduli space which assigns to a manifold approximate fibratidn over B, the underlying map of a closed manifold to B. We call this map the forget control map for manifold approximate fibrations and show that it is homotopy-split injective. This paper is the second in a series dealing with controlled topology over manifolds of nonpositive curvature. In the first paper [8] we establish that the theory of controlled homeomorphisms coincides with the theory of bounded homeomorphisms over Hadamard manifolds. This point of view is extended in the present paper by showing that the theory of manifold approximate fibrations coincides with the theory of manifold bounded fibrations over Hadamard manifolds. We combine this finding with our classifying differential [6] to prove the homotopy injectivity of the forget control map mentioned above. In the future papers in this series [9, lo], we will show that the splitting of the forget control map for manifold approximate fibrations is compatible with various other constructions, such as taking the underlying Hurewicz fibration or taking the surgery-theoretic normal invariant. This will allow us to prove homotopy split injectivity for the forgetful