Let $M$ be a connected manifold of finite type. We denote $Ch_{\mathbb{Q}}(M),$ $Ch_{\mathbb{Z}}(M)$ and $Ch_{\mathbb{\pm}\mathbb{Z}}(M)$ by cohomological dimensions of $M$ over $\mathbb{Q},$ $\mathbb{Z}$ and $\pm\mathbb{Z}$ (local coefficients) respectively. Is it always true that $Ch_{\mathbb{Q}}(M)\leq Ch_{\mathbb{Z}}(M)\leq Ch_{\mathbb{\pm}\mathbb{Z}}(M)?$ I know that if $M$ is orientable then $Ch_{\mathbb{Z}}(M)= Ch_{\mathbb{\pm}\mathbb{Z}}(M).$ Can we say that $Ch_{\mathbb{Z}}(M)< Ch_{\mathbb{\pm}\mathbb{Z}}(M)$ if $M$ is non-orientable?


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