The main result of the Browder-Novikov-Sullivan-Wall surgery theory (1962-1969) is that for n>4 a space X is homotopy equivalent to a compact n-dimensional topological (resp. differentiable) manifold if and only if X is homotopy equivalent to a finite CW complex M with n-dimensional Poincaré duality, and there is a topological (resp. vector) bundle over M for which the corresponding normal map (f,b):N--> M from an n-dimensional manifold N has zero surgery obstruction in the Wall L-group of quadratic forms over Z[π₁(X)]. Thus there are two obstructions, a primary topological K-theory obstruction to the existence of a bundle, and (depending on the vanishing of the primary one, and a choice of reason) a secondary obstruction in algebraic L-theory. The original theory was for differentiable manifolds: the extension to topological manifolds due to Kirby and Siebenmann (1970) remains a major success of surgery theory. All this is explained (at some length) in Wall's own book Surgery on compact manifolds (1970/1998) and also in my own books Algebraic L-theory and topological manifolds (1992) and Algebraic and geometric surgery (2002), as well as many other references (such as Wolfgang Lück's notes listed in a previous post). I have made available a large number of surgery-related resources on my website.

The main result of the Browder-Novikov-Sullivan-Wall surgery theory (1962-1969) is that for $n>4$ a space $X$ is homotopy equivalent to a compact n-dimensional topological (resp. differentiable) manifold if and only if $X$ is homotopy equivalent to a finite $CW$ complex $M$ with $n$-dimensional Poincaré duality, and there is a topological (resp. vector) bundle over $M$ for which the corresponding normal map $(f,b):N\to M$ from an $n$-dimensional manifold $N$ has zero surgery obstruction in the Wall $L$-group of quadratic forms over $Z[\pi_1(X)]$. Thus there are two obstructions, a primary topological $K$-theory obstruction to the existence of a bundle, and (depending on the vanishing of the primary one, and a choice of reason) a secondary obstruction in algebraic $L$-theory. The original theory was for differentiable manifolds: the extension to topological manifolds due to Kirby and Siebenmann (1970) remains a major success of surgery theory. All this is explained (at some length) in Wall's own book Surgery on compact manifolds (1970/1998) and also in my own books Algebraic L-theory and topological manifolds (1992) and Algebraic and geometric surgery (2002), as well as many other references (such as Wolfgang Lück's notes listed in a previous post). I have made available a large number of surgery-related resources on my website.