One approach to a homeomorphism classification of a closed manifolds simply homotopy equivalent to a closed manifold $X$ of dimension $>4$ is to compute the topological structure set $\mathcal S^s_\text{TOP}(X)$ and the group of homotopy classes of simple homotopy equivalence $\text{Aut}_s(X)$. Then $\text{Aut}_s(X)$ acts on $\mathcal S^s_\text{TOP}(X)$ be composition and the quotient is the desired set of homeomorphism classes closed manifolds simply homotopy equivalent to $X$.

For $X=\mathbb RP^n$ the topological structure set is known, and the structure set always has at least 4 elements, and it is finite unless $n-3$ is divisible by $4$.

The group $\text{Aut}_s(\mathbb RP^n)$ is trivial if $n$ is even and has order $2$ if $n$ is odd. See Corollary 6 in "Coverings of fibrations" by Becker and Gottlieb.

Thus there are lots of fake even-dimensional real projective spaces in dimensions $>4$.

By Freedman's work this can be extended to dimension $4$, see Invariant knots of free involutions on $S^4$ by Ruberman for examples of fake $\mathbb RP^4$.

One approach to a homeomorphism classification of closed manifolds simply homotopy equivalent to a closed manifold $X$ of dimension $>4$ is to compute the topological structure set $\mathcal S^s_\text{TOP}(X)$ and the group of homotopy classes of simple homotopy equivalences $\text{Aut}_s(X)$. Then $\text{Aut}_s(X)$ acts on $\mathcal S^s_\text{TOP}(X)$ by composition and the quotient is the desired set of homeomorphism classes of closed manifolds simply homotopy equivalent to $X$.

For $X=\mathbb RP^n$ the topological structure set is known, and the structure set always has at least 4 elements, and it is finite unless $n-3$ is divisible by $4$.

The group $\text{Aut}_s(\mathbb RP^n)$ is trivial if $n$ is even and has order $2$ if $n$ is odd. See Corollary 6 in "Coverings of fibrations" by Becker and Gottlieb.

Thus there are lots of fake even-dimensional real projective spaces in dimensions $>4$.

By Freedman's work this can be extended to dimension $4$, see Invariant knots of free involutions on $S^4$ by Ruberman for examples of fake $\mathbb RP^4$.