Timeline for Getting the most general form of Mayer-Vietoris from the Eilenberg-Steenrod axioms
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 18, 2018 at 0:36 | history | bounty ended | Johannes Hahn | ||
| Dec 18, 2018 at 0:36 | vote | accept | Johannes Hahn | ||
| Dec 18, 2018 at 0:36 | comment | added | Johannes Hahn | I'm quite happy with a proof that works with mapping cones. Do you know of any proof that relies solely on the long exact sequences for pairs? | |
| Dec 17, 2018 at 20:12 | comment | added | John Rognes | You could take the second lemma to be the definition of what it means for a triple to be excisive. The third lemma is simpler than the five-lemma; it just uses the long exact sequence of (D',D). Regarding the first lemma(ta) you may object that forming mapping cones might take you out of an "admissible category for homology theory", as on page 5 of Eilenberg-Steenrod. Are you looking for a proof that refers to only the pairs of spaces you can form from A_{12}, A_1, A_2, A, X_{12}, X_1, X_2, X and the empty space? | |
| Dec 17, 2018 at 19:00 | comment | added | Johannes Hahn | So let me get this straight: First new (to me) lemma is "The two possible iterated mapping cones of a commutative square are homoemorphic" and similarly, "the various iterated mapping cones of a commutative cube are homoemorphic". The second lemma is a special case of the five-lemma: "$(Y,Y_1,Y_2)$ is excisive iff the double mapping cone of the square $\begin{smallmatrix}Y_1&\to&Y\\\uparrow&&\uparrow\\Y_{12}&\to&Y_2\end{smallmatrix}$ is acyclic." The third lemma is also an application of the five-lemma: If $D\subseteq D'$ are two acyclic spaces, the mapping cone $C_D^{D'}$ is also acyclic." | |
| Dec 17, 2018 at 1:51 | comment | added | John Rognes | PS: This is basically the archaic formulation of Dylan Wilson's comments from October 2017. | |
| Dec 17, 2018 at 1:46 | history | answered | John Rognes | CC BY-SA 4.0 |