Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated vector bundle. The connection $A$ gives rise to a twisted deRham complex with cohomology $H^i(M,\text{Ad} P)$. Using the holonomy map to identify (gauge equivalence classes of) flat connections with (conjugacy classes of) representations of $\pi_1(M)$, let $\mathfrak{g}_A$ denote the $\pi_1(M)$-module with action given by the composition of the representation with the adjoint representation of $G$. Let $H^i(\pi_1(M),\mathfrak{g}_A)$ be the corresponding group cohomology groups (cf. e.g. Brown).

Now, I would like to have isomorphisms $H^i(M, \text{Ad} P) \to H^i(\pi_1(M), \mathfrak{g}_A)$ for $i = 0, 1$.

In the case where $M$ is a surface of genus at least 2, rather than a 3-manifold, this is worked out in papers by Goldman, but what can we say in this 3-dimensional setup?


In general, a twisted coefficient system on a manifold $M$ (also called a local system by more algero-geometrically minded people) is given by a representation of $\pi_1(M)$ (the holonomy representation, also called the monodromy representation by more algebrao-geometrically minded people). Conversely, any representation of $\pi_1(M)$ gives a twisted coefficient system on $M$.

(There is no need for $M$ to be a manifold here; any space for which the usual theory of $\pi_1$ and covering spaces goes through would be fine.)

If $V$ is the representation of $\pi_1(M)$, giving rise to the twisted coefficient system $\mathcal L$, then there will be map $H^i(\pi_1(M), V) \to H^i(M,\mathcal L)$. However, these will not be isomorphisms in general unless $M$ is aspherical, i.e. if its universal cover $\tilde{M}$ is contractible, i.e. if $M$ is a $K(\pi,1)$ (for $\pi = \pi_1(M)$). (Here I am recalling the basic topological interpretation of group cohomology, which you can find in many places.)

What happens in general is that there is a spectral sequence (a special case of the Hochschild--Serre spectral sequence) $$H^i(\pi_1(M), H^j(\tilde{M},V) ) \implies H^{i+j}(M,\mathcal L).$$

Note that if $M$ is hyperbolic (or, more generally, negatively curved), then its universal cover is contractible, and so one does get your desired isomorphism.

Added: To get a feeling for what can happen if $\tilde{M}$ is not contractible, you can consider the case when $M = \mathbb RP^2$, so that $\tilde{M}$ is $S^2$ and $\pi_1(M)$ is cyclic of order $2$, acting on $S^2$ by the antipodal map. Take $V$ to be the trivial representation (over $\mathbb Z$, or $\mathbb Z/2\mathbb Z$, say). Then the preceding spectral sequence gives a way to compute the cohomology (with $\mathbb Z$-coefficients, or $\mathbb Z/2\mathbb Z$-coefficients) in terms of the group cohomology of the cyclic group of order two acting on the cohomology of the sphere. (So it acts trivially on $H^0$, and by $-1$ on $H^2$.)

Of course, you could do the analogous computation for $\mathbb R P^3$ as well (which is more directly relevant to your question).

  • $\begingroup$ Do you happen to know of a good reference for this stuff (e.g. the map $H^i(\pi_1(M), V) \to H^i(M,\mathcal L)$ and the spectral sequence)? $\endgroup$ – Eric O. Korman Mar 3 '14 at 3:55
  • $\begingroup$ Dear Eric, Not off the top of my head; it is all standard, but it is stuff I learnt at a stage in my career when I wasn't really using textbooks anymore, so I don't know what references are good. You could try googling Hochschild--Serre, with perhaps another key word so that you get links that discuss the topological version of this spectral sequence (and not just the one that involves only group cohomology). By the way, I'm sure if you ask for references as an actual question, many people here will be able to give you references. Regards, $\endgroup$ – Matt E Mar 3 '14 at 4:26

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