Timeline for Effect on homology of decorating vertices of a simplicial complex
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 22, 2012 at 18:58 | answer | added | Eric | timeline score: 1 | |
| Jun 21, 2012 at 15:46 | comment | added | Russ Woodroofe | Tricia: You're right, I was too hasty. They're somewhat similar, but the distinctness of the $v_j$'s doesn't have to hold in the order complex of the product of posets. (It does seem like there should be some nice interpretation of this in a poset context, but maybe not.) | |
| Jun 19, 2012 at 16:38 | vote | accept | Martin | ||
| Jun 19, 2012 at 15:57 | comment | added | Patricia Hersh | Russ, this doesn't sound right -- you may go up in a chain in P with order complex X while needing to decrease the decoration. Also, Cartesian product increases dimension. | |
| Jun 19, 2012 at 14:28 | comment | added | Russ Woodroofe | I'll point out that if $X$ is the order complex of a poset $P$, then $X[m]$ is the order complex of $P \times C_m$, where $C_m$ is the chain with $m$ elements. And there are all sorts of helpful results about product posets... | |
| Jun 19, 2012 at 12:47 | answer | added | John Shareshian | timeline score: 11 | |
| Jun 18, 2012 at 19:22 | comment | added | Patricia Hersh | paper on fixed point subposets of the partition lattice. | |
| Jun 18, 2012 at 19:22 | comment | added | Patricia Hersh | I was thinking you could label the facets as $(v,F)$ where $F$ is a facet of $X$ and $v\in \{ 1,\dots ,m\}^{dim F+1}$, using the balancing to order the coordinates in $v$. Then it seemed like you should get a shelling order on $X[m]$ from using lexicographic order on $\{ 1,\dots ,m\}^{dim F+1}$ and then breaking ties with the shelling order for $X$. A usual trick for calculating characters of a Cohen-Macaulay complex whose group action commutes with the boundary map is to look at the subcomplexes fixed under various elements of the group. A reference on this last idea is P. Hanlon's (cont) | |
| Jun 18, 2012 at 19:10 | answer | added | rvf0068 | timeline score: 10 | |
| Jun 18, 2012 at 18:45 | comment | added | Martin | @Vel Nias : It would take me pretty far afield to describe the exact setup. I'm trying to perform a computation involving the homology of certain kinds of arithmetic groups. I'm pretty sure that the answer is related to the homology of the spaces $X[m]$ for very special choices of $X$ and $m$, but I was having trouble performing the needed calculations. I abstracted away the special properties of the spaces I was using, and the above problem was the result. | |
| Jun 18, 2012 at 18:41 | comment | added | Martin | @Patricia Hersh : I'll give it a shot. Thanks! | |
| Jun 18, 2012 at 18:40 | comment | added | Martin | @John Wiltshire-Gordon : That would be interesting, and I'll send you an email. | |
| Jun 18, 2012 at 18:36 | comment | added | Vidit Nanda | This is a really pretty problem. Could you say more about how this came up in your research? | |
| Jun 18, 2012 at 18:20 | comment | added | Patricia Hersh | You might want to try to prove that $X$ shellable and balanced implies $X[m]$ is shellable, since this would give you a really good description of a (co)-homology basis and also might be helpful for calculating characters. | |
| Jun 18, 2012 at 17:48 | comment | added | John Wiltshire-Gordon | I think this sequence of S_m representations will be stable in the sense of Church and Farb: arxiv.org/abs/1008.1368 Send me an email if you'd like details. | |
| Jun 18, 2012 at 17:24 | history | asked | Martin | CC BY-SA 3.0 |