Timeline for Reference request: Flipping the factors in the Künneth formula
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 11, 2015 at 15:24 | vote | accept | Aaron Tikuisis | ||
| Jun 9, 2015 at 8:06 | answer | added | Mark Grant | timeline score: 5 | |
| May 6, 2015 at 9:41 | comment | added | Sean Tilson | Also, all you need is the right hand square to commute and that implies the result. I believe this follows by looking at what happens in chain complexes. The map $H_*(C_*)\otimes H_*(D_*) \to H_*(C_* \otimes D_*)$ is nice enough that it commutes with the twist map (up to sign as observed above). Then there is the map $S_*(X) \otimes S_*(Y) \to S_*(X\times Y)$ which commutes with the twist (up to sign) after taking homology. This is definitely a more delicate issue than I had thought. I would recommend looking at Switzer's Algebraic Topology book, chapter 13. (Let me know if you need a copy) | |
| May 6, 2015 at 9:33 | comment | added | Sean Tilson | @AaronTikuisis: Signs are always an issue, and this is why "the usual one" is a loaded statement. In the case you are interested in, all elements of the chain complex have internal degrees that determine the sign of the twist map. On chain complexes, the map $C_p\otimes D_q \to D_q\otimes C_p$ sends a homogeneous element $a \otimes b$ to $(-1^{pq})b \otimes a$. When you are just working with abelian groups $G$ and $H$ are complexes in degree $0$ and so it can't be the second map. | |
| May 6, 2015 at 8:41 | comment | added | Aaron Tikuisis | One thing to note that makes the result non-obvious: there are (at least) two natural maps Tor$^\mathbb{Z}_1(G,H) \to$ Tor$^\mathbb{Z}_1(H,G)$, namely (1) the usual one and (2) the usual one composed with multiplication by -1. | |
| May 6, 2015 at 7:26 | comment | added | Sean Tilson | @AaronTikuisis: Quite right. I was being silly above. | |
| May 6, 2015 at 6:57 | comment | added | Aaron Tikuisis | The 5 lemma definitely doesn't do it. The thing to prove is that the diagram commutes (which is a hypothesis for the 5 lemma). Also, it doesn't follow from naturality of the maps in the Künneth formula, because this naturality is in each variable and not specifically about swapping the variables. | |
| May 5, 2015 at 23:33 | comment | added | Sean Tilson | There were votes to close? I can see why that would be alarming. I don't have enough reputation to see such things I guess. I didn't think the comment was too censorious. I just feel like I should be careful on mediums like this and I wanted to clarify that I didn't mean anything rude or condescending. Electronic communication allows people to attach their own tone to things, and I like to get rid of ambiguity. Cheers! | |
| May 5, 2015 at 23:30 | comment | added | Yemon Choi | @SeanTilson Thanks for your comments; I certainly see where you are coming from. I hope my comment was not seen as too censorious, I was just a bit alarmed by all the votes to close and the only comment at the time being your first one. | |
| May 5, 2015 at 23:26 | comment | added | Sean Tilson | @MarkGrant: Doesn't the five lemma also prove the above? Am I missing something? | |
| May 5, 2015 at 23:25 | comment | added | Sean Tilson | @YemonChoi: I don't usually make such comments, but I don't feel that my comment was meant in any rude fashion or as an insult. It certainly was not intended to be hurtful or condescending. Whether or not it was a reference request or a request for a proof, I would have made the same comment. I guess I had been assuming that question in first year graduate courses were off topic regardless of the field. Maybe I misunderstood the nuance of the question somehow? Regardless, as I said above, MO is certainly the place for people to ask questions about neighboring fields. | |
| May 5, 2015 at 23:22 | comment | added | Sean Tilson | Apologies, I thought that this question was not so difficult, that is probably my bias or misunderstanding of the question. The Künneth theorem above is deduced by the degeneration of the Künneth SS. The Künneth SS is natural in maps of spaces, the twist map induces all of the above. Or you could just as well view this as a fact in (homological) algebra. Again, my apologies. I think MO is exactly the place for people who aren't experts in one field to ask questions of experts in another field. | |
| May 5, 2015 at 20:32 | comment | added | Mark Grant | @SeanTilson: I disagree. This is exactly the kind of thing you would not want to reprove in a paper, hence seems like a totally reasonable reference request. | |
| May 5, 2015 at 20:30 | comment | added | Yemon Choi | Is this a trend on MO of people who did "alg top in grad school" deeming questions at that level off-topic even when asked by non-alg-top people? Also, the question is a reference request, not a request for a proof | |
| May 5, 2015 at 16:57 | review | Close votes | |||
| May 5, 2015 at 23:58 | |||||
| May 5, 2015 at 11:20 | comment | added | Sean Tilson | This should be migrated to MSE. | |
| May 5, 2015 at 8:38 | history | asked | Aaron Tikuisis | CC BY-SA 3.0 |