Degree-four characteristic evaluations add under the clutching product

Choose two disjoint oriented closed balls $B_1,B_2$ in $S^3$, avoiding the basepoint. Let $c_i:S^3\to B_i\mathrm{/}\mathrm{\partial }{B}_i\cong S^3$ collapse the complement of the interior of $B_i$, using an orientation-preserving identification with the target sphere. Each $c_i$ has degree one and is based homotopic to the identity by [L6]. Consequently $g_i=g\circ c_1$ and $g_i^{\mathrm{\prime }}=g^{\mathrm{\prime }}\circ c_2$ are based homotopic to $g,g^{\mathrm{\prime }}$; the map $g_i{g}_i^{\mathrm{\prime }}$ is homotopic to $g{g}^{\mathrm{\prime }}$, is identity outside $B_1\cup B_2$, and equals the appropriate factor on each ball. Thus pointwise multiplication represents the oriented pinch sum of the two clutching classes. Only continuous representatives are needed for clutching classification.

Suspend the pinch $S^3\to S^3\vee S^3$ to obtain the oriented pinch $p:S^4\to S^4\vee S^4$. Identify the two fibres over the wedge point by their based trivializations, giving a bundle $E_g\vee E_{g^{\mathrm{\prime }}}$ on the wedge. Over the suspended pinch each cone has a product trivialization, and the equatorial transition on $B_1$ is $g\circ c_1$, on $B_2$ is $g^{\mathrm{\prime }}\circ c_2$, and elsewhere is identity. Its clutching class is therefore that of $g{g}^{\mathrm{\prime }}$ by step 1.1; [L2] gives $E_{g{g}^{\mathrm{\prime }}}\cong p^{\ast }(E_g\vee E_{g^{\mathrm{\prime }}})$. This uses the suspended pinch, not a claim that a nontrivial hemispherical quotient pullback is trivial on its source hemisphere.

In degree four the wedge cohomology is the direct sum of its two summand cohomologies. By naturality [L3], [L4], the characteristic class $c=e$ or $p_1$ on the wedge has restrictions $c(E_g),c(E_{g^{\mathrm{\prime }}})$, and $c(E_{g{g}^{\mathrm{\prime }}})=p^{\ast }c$. The oriented pinch sends $[S^4]$ to the sum of the two fundamental classes, since its two quotient sphere maps have local degree $+1$. Naturality and additivity of evaluation [L5] give $⟨c(E_{g{g}^{\mathrm{\prime }}}),[S^4]⟩=⟨c(E_g),[S^4]⟩+⟨c(E_{g^{\mathrm{\prime }}}),[S^4]⟩$.

The constant identity clutching gives a trivial bundle, with zero Euler evaluation (a constant nowhere-zero section) and zero first Pontryagin class. Applying step 3.1 to $g,g^{-1}$, whose pointwise product is identity, shows that inversion negates both evaluations. This proves all assertions.