Let $X$ be a homology sphere which is not homotopic to a sphere. (For example, the Poincare $3$-sphere.) Denote by $M$ the suspension of $X$. Then I believe that $f^!\mathbb{R}=\mathbb{R}[4]$, yet $M$ is not even a topological manifold. (Homologically one cannot distinguish $M$ form a $4$-manifold.)

Let $X$ be a homology sphere which is not homotopic to a sphere. (For example, the Poincare $3$-sphere.) Denote by $M$ the suspension of $X$. Then I believe that $f^!\mathbb{R}=\mathbb{R}[4]$, yet $M$ is not even a topological manifold. (Homologically one cannot distinguish $M$ from a $4$-manifold.)

Let $X$ be a homology sphere which is not homotopic to a sphere. (For example, the Poincare $3$-sphere. Denote by $M$ the suspension of $X$. Then I believe that $f^!\mathbb{R}=\mathbb{R}[4]$, yet $M$ is not even a topological manifold. (Homologically one cannot distinguish $M$ form a $4$-manifold.)

Let $X$ be a homology sphere which is not homotopic to a sphere. (For example, the Poincare $3$-sphere.) Denote by $M$ the suspension of $X$. Then I believe that $f^!\mathbb{R}=\mathbb{R}[4]$, yet $M$ is not even a topological manifold. (Homologically one cannot distinguish $M$ form a $4$-manifold.)