I am confused about this "homology with local coefficients" business. First, I am confused by its name. What is "local" about it?


Let $X$ be a topological space with a universal cover. Let $G=\pi_1(X)$. Let $M$ be a $G$-module, that is, a left $\mathbb{Z}[G]$-module: an abelian group with a left action of $G$. Then we can do two things:

1) Take the homology with local coefficients in $M$, which I take it as defined in Hatcher, through the action of $G$ by deck transformations on the universal cover of $X$,

2) Do the following procedure (which occurred to me and I don't know if it is relevant). If $R$ is a ring, define $C_n^R(X)$ as the free $R$-module with basis the singular $n$-chains, and a boundary map defined as for the usual $C_n(X)$. Then you can take homology, which gives you "$R$-modules of homology". Just as you do when you define homology with coefficients in an abelian group, you can define homology with coefficients in an $R$-module. Now, take $R=\mathbb{Z}[G]$. Consider this "homology with coefficients in $M$".

Do these two procedures yield isomorphic modules?


2 Answers 2


To answer your first question, one way to define (co)homology with local coefficients is the following.

Let $X$ be a space, let $\Pi(X)$ be the fundamental groupoid of $X$, ie. a category with objects points of $X$ and morphisms $x \rightarrow y$ given by homotopy classes of paths. A local coefficient system $M$ on $X$ is a functor $M: \Pi(X) \rightarrow \mathcal{A}b$ from the fundamental groupoid to the category of abelian groups. In particular, $M$ associates a "group of coefficients" $M(x)$ to every point of $x \in X$.

Associated to $M$ is the singular complex given by

$C _{n}(X, M)= \bigoplus _{\sigma \in Sing_{n}(X)} M(\sigma(1,0,\ldots,0))$,

where $Sing _{n}(X)$ is the set of all maps $\Delta ^{n} \rightarrow X$. The differential can be defined using the fact that the groups $M(x)$ are functorial with respect to paths in $X$. Observe that this is a very similar to the usual definition of singular complex with coefficients in an abelian group $A$, which would be

$C _{n}(X, A) = \bigoplus _{\sigma \in Sing_{n}(X)} A$,

except in the "non-local" case, we count the occurances of any $\sigma: \Delta^{n} \rightarrow X$ in a given chain using the same group $A$ and in the local case, we use $M(\Delta^{n}(1, 0, \ldots, ))$, which might be different for different $\sigma$.

This is the locality (localness?) in the name, which should be contrasted with globality of usual homology with coefficients, where the choice of the group $A$ is global and the same for all points.

The definition I have given above is enlightening but perhaps not suitable for computations. Luckily under rather weak assumptions one can use the definition you allude to. Let me explain. Let $X$ be path-connected, let $x \in X$ and consider $\pi = \pi_{1}(X, x)$ as a category with one object and morphisms the elements of the group.

The obvious inclusion $\pi \hookrightarrow \Pi(X)$ is an equivalence of categories and so the functor categories $[\pi, \mathcal{A}b], [\Pi(X), \mathcal{A}b]$ are equivalent, too. But the left functor category is exactly the category of $Z[\pi]$-modules! In particular, we have a bijection between isomorphism classes of local coefficient systems on $X$ and $Z[\pi]$-modules.

If $X$ is nice enough to admit a universal cover $\tilde{X}$, the above allows us to give another definiton of homology with local coefficients, the one you know. Let $M$ be a local coefficient system and let $M^\prime$ be the associated $\mathbb{Z}[\pi]$-module under the above equivalence (which is unique up to a unique isomorphism). Since $\pi$ acts on $\tilde{X}$, it also acts on $C _{\bullet}(X, \mathbb{Z})$ and so the latter is a chain complex of $\mathbb{Z}[\pi]$-modules. We the can define homology with local coefficients to be homology of the complex

$C_{n}(X, M ^\prime) = C_{n}(\tilde{X}, \mathbb{Z}) \otimes _{\mathbb{Z}[\pi]} M^\prime$

One can show that this two definitions I gave agree, that is, for $X$ like above we have an isomorphism $H_{n}(X, M) \simeq H_{n}(X, M^\prime)$.

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    $\begingroup$ Do you know a reference where I can find that this definitions agree? $\endgroup$
    – Bill
    Aug 24, 2014 at 8:46
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    $\begingroup$ I think I got it. Thanks anyway! But is there a good basic refence for this subject, where all the different points of view are treated? $\endgroup$
    – Bill
    Aug 24, 2014 at 22:26
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    $\begingroup$ I don't have it. $\endgroup$
    – Bill
    Aug 25, 2014 at 2:10
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    $\begingroup$ Can you do this for constructible sheaves. $\endgroup$ Nov 19, 2015 at 0:32

The two homology groups you get will be different (in general), since for the second approach you did not encode the information that $G$ is the fundamental group of $X$ (you could as well have taken any group!). As the practical matter, try to compute these in the case of $X$ equal the circle and $M={\mathbb Z}$ with nontrivial ${\mathbb Z}G$-module structure (generator of $G$ acts by $-1$). Already for the 0-th homology you will notice the difference!

Lastly, the terminology "local coefficients" might be coming from the notion of "local system", which (in the simplest case) is another name for the sheaf of sections of a flat vector bundle. If you think of a sheaf or a bundle given by local gluing equations, the terminology kind of makes sense.


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