Two homotopy equivalent closed $n$-manifolds are homeomorphic

Question

I heard somewhere that there is a wrong conjecture that states:

Two homotopy equivalent closed $n$-manifolds ($n\geq 3$) are homeomorphic.

and the lens spaces are the first counterexample to this conjecture. I want to know the author of this wrong conjecture, and a reference for it. I am also curious about other counterexamples? (Here is a related question)

Answer 1

The right conjecture (due to Armand Borel, not Horowitz) is that any two homotopy-equivalent closed aspherical manifolds are homeomorphic, see here. (A manifold is aspherical if it is connected and has contractible universal covering space.) This conjecture is wide-open (in dimensions $\ge 4$) but is expected to be false. The conjecture is known to be true for Riemannian manifolds of negative curvature (very hard work of Farrel and Jones). It is also known to be true for 3-dimensional (aspherical) manifolds, which is primarily due to Perelman's solution of the Geometrization Conjecture.

Answer 2

I mentioned in the comments one can find families of counterexamples parametrized by a K-theoretic invariant that is normally nontrivial for a nontrivial fundamental group. Computing these invariants is generally difficult. Here is something on the opposite end of the spectrum; something rather easy to produce nontrivial elements of that only works for simply-connected manifolds.

Let $M$ be a simply-connected manifold of dimension $2k-1$ and $2n$ be such that $2k+2n-1>4$. If $Top$ is the colimit of the $Top(n)$, the homeomorphism group of $\mathbb{R}^n$, and $G$ is the colimit of the $G(n)$, the monoid of self homotopy equivalences of $S^{n-1}$, then it is a result of surgery theory (dependent on our conditions) that $M \times S^{2n}$ has homotopy equivalences $N \rightarrow M \times S^{2n}$, $N$ a manifold, in bijective correspondence to $[M \times S^{2n}, G/Top]$.

Another very deep result of surgery theory is that $\pi_{4n}(Top/O)=\mathbb{Z}$, $\pi_{4n+2}(Top/O)=\mathbb{Z}/2$ and trivial otherwise. So if we take the composition $M \times S^{2n} \rightarrow S^{2n} \rightarrow Top/O$, with the latter a nontrivial map as above, then this gives a nontrivial element of $[M \times S^{2n}, Top/O]$.

So we conclude that at least in every odd dimension greater than four, we have many counterexamples.