Timeline for Reference request on Leray numbers
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 8, 2011 at 16:09 | vote | accept | Benjamin Steinberg | ||
| Dec 5, 2011 at 19:02 | comment | added | Benjamin Steinberg | The simplicial vertex is needed to show the clique complex of a chordal graph is contractible. The direction you sketch is indeed the easier one. Both are pretty easy but I still am not having luck nailing down a good attribution. I would love to see if it is in the Boland and Lekkerkerker paper, which is very early. | |
| Dec 5, 2011 at 17:21 | comment | added | Russ Woodroofe | From the summary in mathreviews and Wegner's paper, it sounds like one could go either way. Indeed it follows easily from the simplicial vertex condition, but it also follows easily from first principles. ($\Delta$ is the clique complex of a graph, since otherwise it contains a simplex boundary as an induced subcomplex; this graph is chordal since otherwise it contains a 1-sphere as an induced subcomplex.) | |
| Dec 4, 2011 at 23:33 | comment | added | Ryan Budney | Welcome to MO, Mr. Woodroofe. | |
| Dec 4, 2011 at 23:07 | comment | added | Benjamin Steinberg | Wegner says that Boland and Lekkerkerker show that a Leray number 1 complex has a simplicial vertex and hence are 1-collapsible. Conversely, he shows 1-collapsible implies Leray number 1. Since chordal equals having a simplicial vertex it seems hidden here. But I cannot get the Boland and Lekkerkerker paper. | |
| Dec 4, 2011 at 22:51 | comment | added | Russ Woodroofe | The Fröberg reference is just the first one I know -- I wouldn't be at all surprised if there were an earlier one. (Please tell me if you find the paper you're thinking of.) The characterization is a pleasing theorem with a fairly easy proof, and indeed often rediscovered. <br> Fröberg's paper did seem to stir up quite a bit of activity in the commutative algebra and algebraic combinatorics literature. | |
| Dec 4, 2011 at 21:51 | comment | added | Benjamin Steinberg | Thanks, Russ. Are you sure about the Fr\"oberg reference? I'm pretty sure I saw a topological proof in a paper from the seventies, maybe about interval graphs. I have the impression this was rediscovered several times. | |
| Dec 4, 2011 at 21:09 | history | answered | Russ Woodroofe | CC BY-SA 3.0 |