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> | ||
+ | == References == | ||
+ | {{#RefList:}} | ||
+ | [[Category:Exercises]] | ||
+ | [[Category:Exercises without solution]] |
Latest revision as of 21:10, 25 August 2013
Consider the following commutative braid of exact sequences:
1) Show that there is a rudimentary Mayer-Vietoris exact sequence
2) Show that there is defined an isomorphism