Relation between cohomological dimensions of manifolds

Question

$\DeclareMathOperator\Ch{Ch}$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 $\pm\mathbb{Z}$ (coefficients in the orientation sheaf $\mathbb{\pm}\mathbb{Z}$) 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? $H^{*}(M,\mathbb{\pm}\mathbb{Z})$ is the homology of $Hom_{\mathbb{Z}[\pi_{1}(X)]}(C_{*}(M^{c}),\mathbb{Z}),$ where $C_{*}(M^{c})$ is the singular chain complex of universal cover $M^{c}$ of $M,$ and where the action of (the class of) a loop on the $\mathbb{Z}$ is multiplication by $\pm1$ according to whether this loop preserves or reverses orientation. We denote by $Ch_{\pm\mathbb{Z}}(M)$ the smallest integer with property that $$H^{i}(M,\pm\mathbb{Z})=0,\quad\mbox{ for all $i>Ch_{\pm\mathbb{Z}}(M)$}.$$

Answer 1

By Bredon's Sheaf Theory, Proposition II.16.15, if $X$ is locally paracompact then $\dim_L X\leq \dim_{\mathbb Z}X$ for any ring $L$ with unit. So that should answer the question about the relation between $\dim_{\mathbb Q}$ and $\dim_{\mathbb Z}$.

For the other part, I'm not completely sure what you mean by $\mathbb Z_\pm$, but if it's something to do with twisted coefficients then my guess is that for nice spaces like this it would be possible to prove the $\mathbb Z$ and $\pm\mathbb Z$ dimensions are the same using covering spaces. I'll think some more about the details.

Update: looking more carefully at the definition of dimension in Bredon, for a family of supports $\Phi$ and ring with unit $L$, he defines $\dim_{\Phi,L} X$ to be the least integer $n$ such that $H^k_{\Phi}(X; A)=0$ for all sheaves $A$ of $L$-modules and all $k>n$. Since we're using manifolds, the system of supports can be taken to be the usual closed supports. So, assuming this is the same notion of cohomological dimension that you mean, I'm not sure then what it means to talk about dimension over something that's not a fixed ring. Can you say more about your definition?