5 problems

Let $F^{2m} \to E \to B^{2n}$ be a fibration of oriented Poincaré complexes, and suppose that the action of $\pi_1(B)$ on $H^{m}(F;\mathbb Z_2)$ is trivial. Klaus–Teichner's conjec…

Free-fundamental-group conjecture. Then $X^4$ is (simple) homotopy equivalent to an ANR homology $4$--manifold.

Quinn's conjecture. If $s(X^4)=0$, then $X^4$ is (simple) homotopy equivalent to an ANR homology $4$--manifold.

Let $X$ be a finitely dominated aspherical Poincaré complex. The manifold-structure conjecture. The complex $X$ is homotopy equivalent to a closed topological manifold. This is one…

A finite Poincaré complex is a finite complex satisfying Poincaré duality. Aspherical Poincaré complexes conjecture. Every finite Poincaré complex is homotopy equivalent to a close…