Let
$$ 1\to G'\to G\to G''\to 1 $$
be a short exact sequence of real Lie groups.
I assume that $G',\,G$, and $G''$ have finitely many connected components, but are not necessarily connected.
I am looking for a reference for the following extremely well-known fact: the above short exact sequence gives rise to an exact sequence
$$
\pi_1 G''\to \pi_0 G'\to\pi_0 G\to\pi_0 G''\to 1.
$$
Most of the references and links (say,
Wikipedia
and Hatcher's book "Algebraic topology")
give the required sequence for a fibration $E\to B$ with fiber $F$, but only
under the assumption that the base $B$ is path-connected, which is not what I need.
This link contains the desired assertion (Theorem 5.4 on page 4).
However, I would prefer to have a reference to a book, or to a published paper, or at least to an arXiv preprint, rather than to a web page that can disappear in a year or two…
I stress that I am looking for a reference rather than for a proof. I need only the above 4-term exact sequence, and I can construct the connecting map and to prove the exactness by hand. Still I would like to have a reference….
Motivation: I am going to use this exact sequence in order to compute $\pi_0G$.
Best Answer
A statement can be found on p. 123 of M. Arkowitz, Introduction to Homotopy Theory, Springer, New York, 2011.
In his notation $[W,X]$ is the set of basepoint preserving homotopy classes of maps $W\rightarrow X$. The result you seek is obtained from Corollary 4.2.19 by taking $W=S^0$. Below the initial statement the details are spelled out further. Arkowitz considers homotopy groups with coefficients in an abelian group $G$, and these reduce to ordinary homotopy groups when $G=\mathbb{Z}$, which is the statement on the last line of p. 123.
Note, since you are working with Lie groups and homomorphisms, $\pi_0G,\pi_0G'$ and $\pi_0G''$ are groups, the maps in the sequence $\pi_0G'\rightarrow\pi_0G\rightarrow\pi_0G'$ are homomorphisms, and indeed the sequence is exact as a sequence of groups. I couldn't pin down an exact reference for this, but it is covered by Proposition 2.2.6 on p.40 and the definition of exactness for pointed sets on p. 116.
Finally its worth pointing out that Arkowitz uses pointed Hurewicz fibrations rather than Serre fibrations. This is structure you have. Any pointed map $p:E\rightarrow B$ between well-pointed spaces which is an unpointed fibration is a pointed fibration. Moreover, any locally-trivial map with paracompact base is a Hurewicz fibration. Since the projection $G\rightarrow G/H$ of a Lie group onto its factor space by a closed subgroup is a surjective submersion, it has local sections, and so is locally-trivial.