I have this question, if every 3-dimensional Poincaré complex is a 3-dimensional topological manifold?
Definition (Poincaré complex)
$X$ is a n-dimensional Poincaré complex if $X$ have the same homotopy type of a finite CW complex with a homomorphism $$ w: \pi_1 (X) \rightarrow \mathbb{Z}/2 $$ and $M \in H_n (X;\mathbb{Z}[w])$, such that
$$H^\ast (X;\mathbb{Z}[w]) \rightarrow ^{\frown [M]} H_{n-s}(X;\mathbb{Z}[w])$$ is a isomorphism.
First of all, you can always modify an $n$-dimensional closed (oriented if you prefer) connected $n$-dimensional manifold to make it into a finite CW $n$-dimensional Poincare complex which is not a manifold: Just attach a 1-cell to it along a vertex of this cell. This does not change the homotopy type, hence, keeps the space $X$ a Poincare complex, but $X$ is clearly not a manifold. Thus, a meaningful question to ask is:
In dimension 2, it is a nontrivial theorem due to Bieri, Linnell and Muller that the question has positive answer. In dimension 3 the answer is negative, an example is due to C.B. Thomas, it is somewhere in his book:
Elliptic Structures on 3-Manifolds, Cambridge University Press, 1986.
In fact, he had this example already in 1977 modulo the Smale Conjecture which was proven by Hatcher in 1983.
It is a famous open problem (due to C.T.C. Wall) if every finite aspherical n-dimensional Poincare complex is homotopy equivalent to an n-dimensional manifold, $n\ge 3$. From what I know, the question is expected to have positive answer in dimension 3 and negative answer in (sufficiently) higher dimensions. Already in dimension 3 this problem is notoriously difficult.
Two more papers to read:
C.T.C Wall, Poincare duality in dimension 3, 2004.
J. Klein, Poincare duality spaces, In: Surveys on surgery theory, Vol. 1, 135–165, Ann. of Math. Stud., 145, Princeton Univ. Press, Princeton, NJ, 2000.
See also my answer here.