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 \\ }](/images/math/3/5/f/35f824aa183b9bef2de6a02f21b3ee52.png)
![\displaystyle D \to B\oplus G \to E.](/images/math/d/2/2/d22ced9238be1be740eb708ab285b6f7.png)
![\displaystyle \mathrm{Ker}(\beta)/\mathrm{Im}(\alpha) \cong \mathrm{Ker}(\delta)/\mathrm{Im}(\gamma).](/images/math/5/c/d/5cd63c1c69115316aefbe4c70b8de159.png)
Consider the following commutative braid of exact sequences: