Surgery Exact Sequence - Classification of Manifolds

Classification of Manifolds

The answer to the organizing questions of the surgery theory can be formulated in terms of the surgery exact sequence. In both cases the answer is given in the form of a two-stage obstruction theory.

The existence question. Let be a finite Poincaré complex. It is homotopy equivalent to a manifold if and only if the following two conditions are satisfied. Firstly, must have a vector bundle reduction of its Spivak normal fibration. This condition can be also formulated as saying that the set of normal invariants is non-empty. Secondly, there must be a normal invariant such that . Equivalently, the surgery obstruction map hits .

The uniqueness question. Let and represent two elements in the surgery structure set . The question whether they represent the same element can be answered in two stages as follows. First there must be a normal cobordism between the degree one normal maps induced by and, this means in . Denote the normal cobordism . If the surgery obstruction in to make this normal cobordism to an h-cobordism (or s-cobordism) relative to the boundary vanishes then and in fact represent the same element in the surgery structure set.