# Homology braid I (Ex)

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Consider the following commutative braid of exact sequences:

$\displaystyle \xymatrix{ A \ar[dr] \ar@/^2pc/[rr]^{\alpha} && B \ar[dr] \ar@/^2pc/[rr]^{\beta} && C \\ & D \ar[dr] \ar[ur] && E \ar[dr] \ar[ur] & \\ F \ar[ur] \ar@/_2pc/[rr]_{\gamma} && G \ar[ur] \ar@/_2pc/[rr]_{\delta} && H \\ }$
1) Show that there is a rudimentary Mayer-Vietoris exact sequence
$\displaystyle D \to B\oplus G \to E.$
2) Show that there is defined an isomorphism
$\displaystyle \mathrm{Ker}(\beta)/\mathrm{Im}(\alpha) \cong \mathrm{Ker}(\delta)/\mathrm{Im}(\gamma).$