Disclaimer: This is a cross-listing of a [math.stackexchange](https://math.stackexchange.com/questions/1143589/exact-sequence-of-groups-to-exact-sequence-sheaves) post. While not research level, after a week of no response, I figured I would ask it here.


For a topological group $G$ and a topological space $X$, denote by $\underline{G^X}$ the sheaf of continuous functions from $X$ into $G$.


Suppose we have an exact sequence of groups
$$
1\rightarrow F\rightarrow G\rightarrow H\rightarrow 1.
$$
What are sufficient conditions for the corresponding sequence of sheaves
$$
1\rightarrow \underline{F^X}\rightarrow \underline{G^X}\rightarrow \underline{H^X}\rightarrow 1
$$
to be exact?



Exact sequence of groups to exact sequence of sheaves
The following is perhaps more of an extended comment than an answer. The sequence of sheaves is exact iff the quotient map $G\to H$ has a section over a neighborhood of every point (in fact, because of the group structure, it suffices to have a section over any single nonempty open set). In particular, for instance, this means the sequence of sheaves is always exact if $G$ is a Lie group. In general, however, the sequence need not be exact, as the following example shows.


Let $G$ be the free topological group on the Cantor set $K$; this is just the free group on the underlying set equipped with the obvious colimit topology (in general that topology might fail to be a group topology, but everything works fine because $K$ is compact). Similarly, let $H$ be the free topological group on $[0,1]$. Let $p:K\to[0,1]$ be any continuous surjection, and consider the induced map $q:G\to H$. It is not hard to see this is a quotient map of topological groups. But $q$ cannot have a section over any open set, since $H$ is locally connected and $G$ is totally disconnected.


