I have several related questions regarding homological vs. cohomological dimension of a space/group (this is not a duplicate of this).
The standard definition of the cohomological dimension $cd(X)$ of a space $X$ is
definition 1: $cd(X)=$ The minimum over $n\in \mathbb{N}$, such that $H^i(X,M)=0$ for all $i>n$ and all local coefficient systems $M$.
One can similarly define the homological dimension $hd(X)$ (I am not sure it is a standard terminology)
definition 2: $hd(X)=$ The minimum over $n\in \mathbb{N}$, such that $H_i(X,M)=0$ for all $i>n$ and all local coefficient systems $M$.
My main question is
Question1: Do we always have $cd(X)=hd(X)?$ What if one or both assumed to be finite? what if $X$ is a finite CW complex?
For $X = BG$, i.e a classifying space of a group $G$, the definitions reduce to the standard ones in group cohomology. I believe that the $cd(BG)$ coincides with the minimal length of a projective resolution of the trivial $G$-module $\mathbb{Z}$ over the group ring $\mathbb{Z}[G]$ (Namely, the projective dimension of $\mathbb{Z}$).
Question 2: Is it true that the $hd(BG)$ coincides with the flat dimension of $\mathbb{Z}$ as a $\mathbb{Z}[G]$-module?
Finally, if $X$ is a connected CW complex with $cd(X)=0$, then $X$ is contractible. One can similarly ask
Question 3: If $X$ is a CW complex with $hd(X)=0$, is it true that $X$ is contractible?
(Edit: The following argument has a gap is wrong. It is not clear true that if $hd(X)=0$ then $X$ is a filtered colimit of finite subcomplexes $X'$ with $hd(X')=0$. An exmaple is given here)
Regardless of the answer to question 1, I think the answer to this is yes by the following argument. Since the universal cover is acyclic and simply connected, it is contractible, so this reduces to a question about group cohomology. If $G$ is finitely generated (if $X$ is a finite CW for example), then $\mathbb{Z}$ is a finitely presented $\mathbb{Z}[G]$-module so if it is flat then it is projective and hence the cohomological dimension is zero as well. Can one reduce the general case to this one since every CW complex is a filtered colimit of its finite subcomplexes. Is there a more direct argument?
Best Answer
Q2) You mean $X=BG$, right? Then the answer is yes by standard homological algebra.
Q1) The answer to the first question is no, e.g. $X=M(\mathbb Q,n)=S^n_{\mathbb Q}$ the rational $n$-sphere has homological dimension $n$ but cohomological dimension $n+1$ (since $\mathbb Q$ is flat but not projective as an abelian group). For $X$ finite I don't know the answer in general, but if $X$ is in addition simply connected then yes, since the chain complex of $X$ is quasi-isomorphic to its homology and the projective and flat dimensions coincide on finitely generated abelian groups.
Q3) This is true if $X$ is simply connected (Whitehead's theorem)
but false otherwise, e.g. $X=BG$ with $G$ acyclic.It is also positively answered in the question for $X=BG$ with $G$ finitely presented.