Homology braid I (Ex)
From Manifold Atlas
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| − | .. | + | Consider the following commutative braid of exact sequences: |
| + | $$ | ||
| + | \xymatrix{ | ||
| + | A \ar[dr] \ar@/^2pc/[rr]^{\alpha} && B \ar[dr] \ar@/^2pc/[rr]^{\beta} && C \\ | ||
| + | & D \ar[dr] \ar[ur] && E \ar[dr] \ar[ur] & \\ | ||
| + | F \ar[ur] \ar@/_2pc/[rr]_{\gamma} && G \ar[ur] \ar@/_2pc/[rr]_{\delta} && H \\ | ||
| + | } | ||
| + | $$ | ||
| + | '''1)''' Show that there is a rudimentary Mayer-Vietoris exact sequence $$D \to B\oplus G \to E.$$ | ||
| + | |||
| + | '''2)''' Show that there is defined an isomorphism $$\mathrm{Ker}(\beta)/\mathrm{Im}(\alpha) \cong \mathrm{Ker}(\delta)/\mathrm{Im}(\gamma).$$ | ||
| + | |||
</wikitex> | </wikitex> | ||
== References == | == References == | ||
Latest revision as of 21:10, 25 August 2013
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)
