In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X (more geometrically a Poincaré space), a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information. If the dimension of X is 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manif