There are a number of questions arising from this question.

Firstly, what can one really explicitly compute or calculate by the methods given by E.H. Brown?

Secondly, we should also try to compute $\pi_n, n \geq 2$ as a module over $\pi_1$, as the example of $S^n \vee S^1$ shows. However even seeing $\pi_2 X$ as a $\pi_1(X)$-module still gives only a pale shadow of the $2$-type.

Thirdly, the emphasis on computing in EH Brown's paper is via the Postnikov system, which has problems. Saunders Mac Lane remarked to me in 1972 that it would seem impractical to try to compute the homotopy $2$-type of a union since you would first have to compute $\pi_2$ as a module over $\pi_1$; and then somehow describe the $k$-invariant of the union in terms of the $k$-invariants of the individual pieces. Each step, apart from the computation of $\pi_1$ by the theorem of Seifert-van Kampen, seems pretty difficult. Also one can presumably specify a cocycle, but how does one specify a cohomology class?

A different approach was taken by Higgins and me in a paper published in the Proc. LMS (1978), available here as [31], and which follows the lead of J.H.C. Whitehead in his paper Combinatorial Homotopy II, and his paper with Mac Lane ``On the $3$-type of a complex" , Proc. Nat.
Acad. Sci. U.S.A. 36 (1950) 41--48, which we now call $2$-type. They describe the $2$-type of a connected complex $K$ (with base point) in terms of the crossed module $(\pi_2(K, K^1) \to \pi_1(K^1))$, which we write as $\Pi_2(K)$.

Now suppose that $K$ is the union of subcomplexes $L,M$ with intersection $N$. Our generalisation of the Seifert-van Kampen Theorem to dimension $2$ implies that the following diagram:

$$ \matrix{\Pi_2(N)&\to & \Pi_2(M) \cr
\downarrow && \downarrow\cr
\Pi_2(L) & \to & \Pi_2(K) }$$
is a pushout of crossed modules.

This gives a complete determination of $\Pi_2(K)$ which, using also the notion of free crossed module, can be translated into terms of presentations of crossed modules. In some cases this also yields finite calculations of the pushout.

Notice that this result gives ``in principle" a determination of $$\pi_2(K)= Ker (\pi_2(K,K^1) \to \pi_1(K^1))$$ as a $\pi_1(K)$-module, but it not so obvious how to give explicit computations. Thus this method is the reverse of the traditional method, which proposes to compute first $\pi_2$ and then the $2$-type!

The properties of the pairs of complexes $(X,X^1)$ for $X=L,M,N$ that are required for this pushout are that the individual spaces are connected and non empty, and that $\pi_1(X^1) \to \pi_1(X)$ is surjective: we then say that $(X,X^1)$ is {\it connected}. Part of the theorem is that then the union $(K,K^1)$ is also connected.

Note that the result involves in general nonabelian groups, and so is not available through traditional tools of algebraic topology; current proofs involve some form of $2$-dimensional homotopy groupoids.

These results would seem to be relevant to work in geometric group theory. Full details of the above results and their applications are given in Part I of the book titled in part Nonabelian algebraic topology, which also shows how parts of these methods extend to higher dimensions. Algebraic models of homotopy type in higher dimensions do allow some explicit computations, involving so called cat$^n$-groups, and crossed $n$-cubes of groups, but do not easily give a handle on the general computability of homotopy groups.

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