Cohomology of a space with local coefficients and singular cohomological dimension

Question

If $X$ is a space with (singular) cohomological dimension n, i.e. $H^i(X;\mathbb{Z})=0$ for all $i>n$, (may be cohomology with rational coefficients). Is it true that $H^i(X;\mathcal{L})=0$ for all $i>n$, where $\mathcal{L}$ is a system of local coefficients on $X$? Can someone give some reference?

Answer 1

The answer is no. It's easiest to give examples arising from group cohomology (so the spaces are $K(\pi,1)$'s). The reason for this is that if $G$ is a nontrivial group, then there must exist some local coefficient system $M$ such that $H^i(G;M) \neq 0$ for some $i$. Indeed, let $H < G$ be a nontrivial cyclic subgroup, and let $M$ be the coinduction of the trivial $\mathbb{Z}$-module from $H$ to $G$. Then Shapiro's lemma (see here) says that

$$H^i(G;M) = H^i(H;\mathbb{Z}),$$

which has to be nonzero for some $i$.

Dramatic examples of groups with the property you seek are so-called "acyclic groups". These are groups $G$ such that $H^i(G;\mathbb{Z})=0$ for all $i \geq 1$. For a nice survey of such groups, see the last section of

MR1967745 (2004c:20001) Berrick, A. J.(SGP-SING) A topologist's view of perfect and acyclic groups. Invitations to geometry and topology, 1–28, Oxf. Grad. Texts Math., 7, Oxford Univ. Press, Oxford, 2002.

Answer 2

How about $X=\Bbb RP^2\times L^2_3$, where $L^2_3$ is the cone of the $3$-fold cover $S^1\to S^1$. Here $H^4(X;\Bbb Z)\simeq\Bbb Z/2\otimes\Bbb Z/3=0$, but $H^4(X;\mathcal L)\simeq\Bbb Z\otimes\Bbb Z/3$, where $\mathcal L$ is the pullback of the orientation sheaf of $\Bbb RP^2$.

As a side remark, "singular cohomological dimension" is not something that people normally do, perhaps because singular cohomology is not Brown representable, or because of the Barratt-Milnor example of a compact subset of $\Bbb R^3$ (in fact it is just the one-point compactification of $\Bbb R^2\times\Bbb Z$) which has infinite "singular cohomological dimension". If you do care about spaces not homotopy equivalent to CW-complexes, you can look up some books on traditional dimension theory, which deal with usual (that is, Cech) cohomological dimension. For spaces homotopy equivalent to CW-complexes, there's no distinction because all ordinary cohomology theories coincide.