How can you tell if a space is homotopy equivalent to a manifold?

In surgery theory (which is basically a whole field of mathematics which tries to answer questions as the above), the next obstruction to the existence of a manifold in the homotopy type is that every finite complex with Poincaré duality is the base space of a certain distinguished fibration (Spivak normal fibration) whose fibre is homotopy equivalent to a sphere. (In order to get a unique such fibration, identify two fibrations if they are fiber homotopy equivalent or if one is obtained from the other by fiberwise suspension.)

For manifolds, this fibration is the spherization of the normal bundle, so the Spivak normal fibration comes from a vector bundle. This is invariant under homotopy equivalence. Thus the next obstruction is: the Spivak normal fibration must come from a vector bundle.

If I remember right, then it was Novikov who first proved that for simply-connected spaces of odd dimension at least 5, this is the only further obstruction.

In general, there is a further obstruction with values in a group $L_n(\pi_1,w)$ which depends on the fundamental group, first Stiefel-Whitney class and the dimension. See Lück's notes on surgery theory at http://wwwmath.uni-muenster.de/u/lueck/publ/lueck/ictp.pdf.

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.

Sean: this gives Poincare space which is not homotopy equivalent to a closed manifold. the idea is that the Spivak fibration of the $5$ dimensional Poincare space doesn't lift to a stable vector bundle. One can prove this as follows: let $X^5$ be as in Madsen and Milgram. Then $X$ fibers over $S^3$ with fiber $S^2$. Call this fibration $\xi$, and let the projection $X \to S^3$ be $p$. Note that $p$ has a section, call it $s$.

It's not hard to prove that the Spivak fibration of $X$ in this case is just $p^*\xi$. One then needs to verify that $p^*\xi$ doesn't lift to a stable vector bundle. But if it did, then so would $$\xi = (p \circ s)^*\xi = s^*p^* \xi\ .$$ But it's very easy to see that $\xi$ doesn't lift ($\xi$ is given by the nontrivial element of $\pi_2(F) = \mathbb{Z}_2$, where $F$ classifies spherical fibrations with section, whereas $\pi_2(O)$ is trivial).