Exact sequence of groups to exact sequence of sheaves

Question

Disclaimer: This is a cross-listing of a math.stackexchange 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?

Answer 1

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.

Answer 2

Here is a positive answer when $G$ is the universal covering group of $H$ (so then $F=\pi_1H$ which is discrete, and $\pi_1G=0$), such as $\mathbb{Z}\to\mathbb{R}\to S^1$.

Denote by $M(U,G)$ the $G$-valued continuous functions on open $U\subset X$. Then covering space theory says the following induced sequence is exact: $$0\to M(U,F)\to M(U,G)\to M(U,H)\to[U,H]\to 0$$ where $[\cdot,\cdot]$ is homotopy classes of maps. But $Sheaf([\cdot,H])=0$, because given any $f:U\to H$ and $x\in U$ there is a neighborhood $x\in V\subset U$ with $f|_V$ nullhomotopic. Thus your desired sheafification is exact.

In general, for my argument I need a lifting criterion, which means I need the surjection $G\to H$ to be a fibration, moreover with the fiber $F$ discrete (so that I can talk about unique path lifting). Thus this also works for the case of discrete groups.