Abstract
Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Björner, Lovász, Vrećica and Živaljević established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large n, the bottom nonvanishing homology of the matching complex Mn is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex Mn, n is a 3-group of exponent at most 9. When n ≡ 2 mod 3, the bottom nonvanishing homology of Mn, n is shown to be Z3. Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 525-570 |
| Number of pages | 46 |
| Journal | Advances in Mathematics |
| Volume | 212 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jul 10 2007 |
Fingerprint
Keywords
- Chessboard complex
- Homology
- Matching complex
- Torsion
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Torsion in the matching complex and chessboard complex. / Shareshian, John; Galloway, Michelle L.
In: Advances in Mathematics, Vol. 212, No. 2, 10.07.2007, p. 525-570.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Torsion in the matching complex and chessboard complex
AU - Shareshian, John
AU - Galloway, Michelle L
PY - 2007/7/10
Y1 - 2007/7/10
N2 - Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Björner, Lovász, Vrećica and Živaljević established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large n, the bottom nonvanishing homology of the matching complex Mn is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex Mn, n is a 3-group of exponent at most 9. When n ≡ 2 mod 3, the bottom nonvanishing homology of Mn, n is shown to be Z3. Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.
AB - Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Björner, Lovász, Vrećica and Živaljević established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large n, the bottom nonvanishing homology of the matching complex Mn is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex Mn, n is a 3-group of exponent at most 9. When n ≡ 2 mod 3, the bottom nonvanishing homology of Mn, n is shown to be Z3. Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.
KW - Chessboard complex
KW - Homology
KW - Matching complex
KW - Torsion
UR - http://www.scopus.com/inward/record.url?scp=34247132131&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=34247132131&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2006.10.014
DO - 10.1016/j.aim.2006.10.014
M3 - Article
VL - 212
SP - 525
EP - 570
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
IS - 2
ER -