If $K$ is an oriented simplicial complex and $K=K_1 \cup K_2$, and $K_1= K -\{\text{$s=$ simplex of highest dimension}\}$, $K_2=\{\text{s, all proper faces of $s$\}}$ then there is a commutative diagram
such that the horizontal maps are the connecting homomorphisms of the Mayer-Vietoris sequences.
If $a : K \rightarrow |K|$ and $b: K_1 \cap K_2 \rightarrow |K_1 \cap K_2|$ then I can see that I would have to show that $b_*D^k = D^{|K|}a_*$.
Therefore I would have to show that $$bD^Kz_q + B_{q-1}(|K_1 \cap K_2|) = D^{|K|}az_q + B_{q-1}(|K_1 \cap K_2|)$$
but I'm having trouble understanding the relationship between $D^K, b_*, D^{|K|}$ and $a_*$.
Anyone have any ideas?