This is my attempt at an answer. I think I might be off in justifying the second diagram. It's based on the proof in Bredon, but I think it might be a little simpler. Corrections are welcome, of course!
For convenience, we introduce some notation and conventions. We define subsets of $$S^n= \{ (c,t):c\in S^{n-1},t\in I \} /\sim$$ by $X=\{(c,t):t\leq 1/2\}$, $Y=\{(c,t):t\geq 1/2\}$, and $Z=\{(c,1/2)\}$. We define subsets of $$S^{2n-1}=\{ (a,t,b):a,b\in S^{n-1},t\in I\}$$ by $A=\{(a,t,b)\in S^{2n-1}:t\leq 1/2\}$ and $B=\{(a,t,b)\in S^{2n-1}:t\geq 1/2\}$. We define subsets of $A\cap B=\{(a,1/2,b)\}$ by $S^{n-1}_A = \{(a,1/2,b_0)\}$ and $S^{n-1}_B = \{(a_0,1/2,b)\}$ for fixed $a_0,b_0\in S^{n-1}$. We consider $S^{2n-1}$ as the boundary of $e^{2n}$; in this cell, $S^{n-1}_A$ is the boundary of a cell $e^n_A$ and $S^{n-1}_B$ is the boundary of a cell $e^n_B$. We write $K=S^n\cup_{h(g)}e^{2n}$ for the complex in which we must compute $H(h(g))$. Throughout, all cohomology is integral.
To compute cup products in $K$, we use the commutative diagram
H^n(K) x H^n(K) ---------> H^{2n}(K)
^ cup ^
| |
| cup |
H^n(K,X) x H^n(K,Y) -----> H^{2n}(K,S^n)
which follows from the map $(K,\emptyset,\emptyset)\rightarrow (K,X,Y)$ and naturality; the vertical maps are isomorphisms by the long exact sequence in cohomology for pairs. Let the generator $x$ of $H^n(K)$ correspond to the generator $x_Y$ of $H^n(K,X)$ and the generator $x_X$ of $H^n(K,Y)$, so that $x_Y\cup x_X$ in $H^{2n}(K,S^n)$ corresponds to $x^2$ in $H^{2n}(K)$.
To understand the image of the cup product, we use the evident map $j:(e^{2n},A,B)\rightarrow (K,X,Y)$. We have the diagram
H^n(K,X) = H^n(S^n,X) = H^n(Y,Z) = H^{n-1}(Z)
| |
| j^* | j^*
| |
V V
H^n(e^{2n},A) = H^n(e^n_A,S^{n-1}_A) = H^{n-1}(S^{n-1}_A)
which commutes because of the naturality of the isomorphisms involved (namely, along the top: cellular cohomology, excision and homotopy invariance, lexseq for pairs; along the bottom: homotopy invariance, lexseq for pairs). By definition of degree, the map $j^\ast:H^{n-1}(Z)\rightarrow H^{n-1}(S^{n-1}_A)$ is multiplication by $\alpha$. Hence, so is the map $j^\ast : H^n(K,X)\rightarrow H^n(e^{2n},A)$. Similarly, the map $j^\ast:H^n(K,Y)\rightarrow H^n(e^{2n},B)$ is multiplication by $\beta$. Now, $j^\ast:H^{2n}(K,S^n)\rightarrow H^{2n}(e^{2n},S^{2n-1})$ is an isomorphism, and $j^\ast(x_Y\cup x_X)$ is $\alpha\beta$ times a generator of $H^{2n}(e^{2n},S^{2n-1})$. Thus $x_Y\cup x_X$ is $\alpha\beta$ times a generator of $H^{2n}(K,S^n)$. So by the first diagram, $H(h(g))=\alpha\beta$.
Here are some answers on the HHSvKT - I have been persuaded by a referee that we ought also to honour Seifert.
These theorems are about homotopy invariants of structured spaces, more particularly filtered spaces or n-cubes of spaces. For example the first theorem of this type
Brown, R. and Higgins, P.~J. On the connection between the second relative homotopy
groups of some related spaces. Proc. London Math. Soc. (3) \textbf{36}~(2) (1978) 193--212.
said that the fundamental crossed module functor from pairs of pointed spaces to crossed modules preserves certain colimits. This allows some calculations of homotopy 2-types and then you need further work to compute the 1st and 2nd homotopy group; of course these two homotopy groups are pale shadows of the 2-type.
As example calculations I mention
R. Brown ``Coproducts of crossed $P$-modules: applications to second homotopy groups and to the homology of groups'', {\em Topology} 23 (1984) 337-345.
(with C.D.WENSLEY), `Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72.
In the second paper some computational group theory is used to compute the 2-type, and so 2nd homotopy groups as modules, for some mapping cones of maps $ Bf: BG \to BH$ where $f:G \to H$ is a morphism of groups.
For applications of the work with Loday I refer you for example to the bibliography on the nonabelian tensor product
http://groupoids.org.uk/nonabtens.html
which has 144 items (Dec. 2015: the topic has been taken up by group theorists, because of the relation to commutators) and also
Ellis, G.~J. and Mikhailov, R. A colimit of classifying spaces.
{Advances in Math.} (2010) arXiv: [math.GR] 0804.3581v1 1--16.
So in the tensor product work, we determine $\pi_3 S(K(G,1))$ as the kernel of a morphism $\kappa: G \otimes G \to G$ (the commutator morphism!). In fact we compute the 3-type of $SK(G,1)$ so you can work out the Whitehead product $\pi_2 \times \pi_2 \to \pi_3$ and composition with the Hopf map $\pi_2 \to \pi_3$.
These theorems have connectivity conditions which means they are restricted in their applications, and don not solve all problems! There is still some interest in computing homotopy types of some complexes which cannot otherwise be computed. It is also of interest that the calculations are generally nonabelian ones.
So the aim is to make some aspects of higher homotopy theory more like the theory of the fundamental group(oid); this is why I coined the term `higher dimensional group theory' as indicating new structures underlying homotopy theory.
Even the 2-dimensional theorem on crossed modules seems little known or referred to! The proof is not so hard, but requires the notion of the homotopy double groupoid of a pair of pointed spaces. See also some recent presentations available on my preprint page.
Further comment: 11:12 24 Sept.
The HHSvKT's have two roles. One is to allow some calculations and understanding not previously possible. People concentrate on the homotopy groups of spheres but what about the homotopy types of more general complexes? One aim is to give another weapon in the armory of algebraic topology.
The crossed complex work applies nicely to filtered spaces. The new book (pdf downloadable) gives an account of algebraic topology on the border between homotopy and homology without using singular or simplicial homology, and allows for some calculations for example of homotopy classes of maps in the non simply connected case. It gets some homotopy groups as modules over the fundamental group.
I like the fact that the Relative Hurewicz Theorem is a consequence of a HHSvKT, and this suggested a triadic Hurewicz Theorem which is one consequence of the work with Loday. Another is determination of the critical group in the Barratt-Whitehead n-ad connectivity theorem - to get the result needs the apparatus of cat^n-groups and crossed n-cubes of groups (Ellis/Steiner).
The hope (expectation?) is also that these techniques will allow new developments in related fields - see for example work of Faria Martins and Picken in differential geometry.
Developments in algebraic topology have had over the decades wide implications, eventually in algebraic number theory. People could start by trying to understand and apply the 2-dim HHSvKT!
Edit Jan 11, 2014 Further to my last point, consider my answer on excision for relative $\pi_2$: https://math.stackexchange.com/questions/617018/failure-of-excision-for-pi-2/621723#621723
See also the relevance to the Blakers-Massey Theorem on this nlab link.
December 28, 2015 I mention also a presentation at CT2015 Aveiro on A philosophy of modelling and computing homotopy types. Note that homotopy groups are but a pale "shadow on a wall" of a homotopy type. Note also that homotopy groups are defined only for a space with base point, i.e. a space with some structure. My work with Higgins and with Loday involves spaces with much more structure; that with Loday involves $n$-cubes of pointed spaces. As with any method, it is important to be aware of what it does, and what it does not, do. One aspect is that the work with Loday deals with nonabelian algebraic models, and obtains, when it applies, precise colimit results in higher homotopy. One inspiration was a 1949 Theorem of JHC Whitehead in "Combinatorial Homotopy II" on free crossed modules.
Best Answer
OK, after having made so many stupid comments, I felt obligated to remember what I knew about unstable homotopy theory in order to try to say something meaningful.
Recall that $\pi_7(S^4\vee S^4)\cong\pi_7S^4\oplus\pi_7 S_4\oplus \mathbb{Z}$, where the third summand is the kernel of the homomorphism $$\pi_7(S^4\vee S^4)\longrightarrow \pi_7(S^4\times S^4)$$ induced by the inclusion of the coproduct in the product, generated by the map representing the Whitehead product operation.
Moreover, for any $\alpha\in\pi_7S^4$, the failure in the commutativity of the following diagram $$\begin{array}{ccc} S^7&\stackrel{\alpha}\rightarrow&S^4\\\ \downarrow&&\downarrow\\\ S^7\vee S^7&\stackrel{\alpha\vee\alpha}\rightarrow&S^4\vee S^4 \end{array}$$ where the vertical arrows are the coproducts, is $(0,0,H(\alpha))\in\pi_7(S^4\vee S^4)$, where $H$ denotes the Hopf invariant (see G. Whitehead's book).
The map $\alpha+(-1)\alpha$ is the composite $$S^7\stackrel{\rm coprod.}\longrightarrow S^7\vee S^7\stackrel{\alpha\vee\alpha}\longrightarrow S^4\vee S^4\stackrel{(1,-1)}\longrightarrow S^4.$$ Moreover, the composite $$S^4\stackrel{\rm coprod.}\longrightarrow S^4\vee S^4\stackrel{(1,-1)}\longrightarrow S^4$$ vanishes. Therefore $\alpha+(-1)\alpha$ it coincides with the image of $(0,0,H(\alpha))\in\pi_7(S^4\vee S^4)$ by the homomorphism $$\pi_7(S^4\vee S^4)\longrightarrow \pi_7S^4$$ induced by $(1,-1)$, which is $H(\alpha)[1_{S^4},1_{S^4}]\in \pi_7S^4$, where the bracket denotes the Whitehead product operation. In particular $$\nu+(-1)\nu=[1_{S^4},1_{S^4}]\in \pi_7S^4$$ which generates the kernel of the suspension homomorphism (by the work of Blakers and Massey on homotopy groups of triads) $$\pi_7S^4\longrightarrow\pi_8S^5$$ described in Lennart Meier's comment above.