Just a comment on the role of the Higher Homotopy Seifert-van Kampen Theorems:

they should be regarded as an extra tool in algebraic topology. There are quite severe conditions on their applicability but when they apply they compute quite a lot. Just as the 1-dimensional theorem, in its groupoid formulation, is about calculating 1-types, so the 2-dim theorem is about computing 2-types, in the form of crossed modules (over groupoids). However computing the second homotopy group from this 2-type may not be straightforward. But then the situation is the same for the 1-dim theorem, as is evidenced by the complications of theorems such as the Kurosh subgroup theorem, which can be seen to be about the fundamental group(s) of a cover of a wedge of $K(G_i,1)$'s.

As a taster, based on the 2-d theorem, work with Chris Wensley enabled the computation of the crossed module representing the 2-type of the mapping cone of a map $Bf: BG \to BH$ induced by a morphism $f: G \to H$. Of course. the second homotopy group, even as a module over the fundamental group, is but a pale shadow of the 2-type. You can see some of this in our book (pdf available from my web page on the book).

R. Brown, P.J. Higgins, R. Sivera, *Nonabelian algebraic
topology: filtered spaces, crossed complexes, cubical homotopy
groupoids*, EMS Tracts in Mathematics Vol. 15, 703 pages. (August
2011).

November 8, 2013: As a taster, let $X$ be the homotopy pushout of the classifying spaces of the two maps of groups $P \to P/M, P \to P/N$ where $M,N$ are normal subgroups of the group $P$. The the homotopy 2-type of $X$ is determined by the crossed module $M \circ N \to P$, the coproduct of the two crossed $P$-modules, which is given by the pushout of crossed modules

$$\begin{matrix} (1 \to P) & \to & (N \to P) \cr
\downarrow && \downarrow \cr
(M \to P)& \to & (M \circ N \to P)
\end{matrix} $$. It follows that
$$ \pi_2(X) \cong (M \cap N)/ [M,N]. $$
(Of course we know $\pi_1 X$ by the 1-dimensional van Kampen Theorem.) This result is applied in Bardakov, Valery G; Mikhailov, Roman; Vershinin, Vladimir V.; Wu, Jie,
"Brunnian braids on surfaces". *Algebr. Geom. Topol.* 12 (2012), no. 3, 1607–1648.

morphism$\kappa:G \otimes G \to G$ whose kernel is $\pi_3 S(K(G,1))$. Abstract??!! $\endgroup$ – Ronnie Brown Aug 9 '13 at 13:51