A fundamental problem of geometric topology is to describe all possible manifolds which are homotopy (simple homotopy) equivalent to a given manifold X. The set of equivalence classes form a structure set S(X) which fits into the surgery exact sequence. To describe the set S(X) we must know the set of normal invariants [x,G/TOP], the surgery obstruction groups L_*(\pi), and the assembly map $[X,G/TOP]\to L_*(\pi)$. The image of assembly map consists of elements that can be realized by normal maps of closed manifolds. In this talk, we describe relations between assembly maps, manifolds with filtration, Browder-Quinn surgery obstruction groups for stratified manifolds, Browder-Livesay invariants, and surgery spectral sequence. In particular, we give geometrical description of forbidden invariants in closed manifold surgery problem and consider several examples.