Five-torsion in the homology of the matching complex on 14 vertices

Jönsson, Jakob

doi:10.1007/s10801-008-0123-6

Cited by 6 publications

(13 citation statements)

References 15 publications

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“…In both cases, the action of the transposition (x, y) onγ yields −γ, implying that 2γ = 0 by (7). Since |U a | ≥ 2, we obtain that |S λ | is divisible by 2.…”

Section: Properties Of the Chain Complex Induced By (λ S)mentioning

confidence: 89%

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More Torsion in the Homology of the Matching Complex

Jönsson¹

2010

Experimental Mathematics

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A matching on a set $X$ is a collection of pairwise disjoint subsets of $X$ of size two. Using computers, we analyze the integral homology of the matching complex $M_n$, which is the simplicial complex of matchings on the set $\{1, >..., n\}$. The main result is the detection of elements of order $p$ in the homology for $p \in \{5,7,11,13\}$. Specifically, we show that there are elements of order 5 in the homology of $M_n$ for $n \ge 18$ and for $n \in {14,16}$. The only previously known value was $n = 14$, and in this particular case we have a new computer-free proof. Moreover, we show that there are elements of order 7 in the homology of $M_n$ for all odd $n$ between 23 and 41 and for $n=30$. In addition, there are elements of order 11 in the homology of $M_{47}$ and elements of order 13 in the homology of $M_{62}$. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of $H_d(M_n;Z)$ for $13 \le n \le 16$; a complete description of the homology already exists for $n \le 12$. To prove the results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of $M_n$ obtained by letting certain groups act on the chain complex.Comment: 35 pages, 10 figure

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“…In both cases, the action of the transposition (x, y) onγ yields −γ, implying that 2γ = 0 by (7). Since |U a | ≥ 2, we obtain that |S λ | is divisible by 2.…”

Section: Properties Of the Chain Complex Induced By (λ S)mentioning

confidence: 89%

“…The proof [7] of Proposition 1.1 uses a definition of C(M n ; Z)/S λ that differs slightly from the one given above. As alluded to earlier, we present a computerfree proof of the proposition in Section 7.…”

Section: Introductionmentioning

confidence: 99%

More Torsion in the Homology of the Matching Complex

Jönsson¹

2010

Experimental Mathematics

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“…Theorem 4.5 (Jonsson [12]). H 4 (M 14 ; Z) is a finite nontrivial group of exponent a multiple of 15.…”

Section: 2mentioning

confidence: 99%

“…So far, all our results have been about the existence of 3-torsion and the nonexistence of other torsion. Almost nothing is known about ptorsion when p = 3, but in a previous paper [12], the author used a result due to Andersen [1] to prove thatH 4 (M 14 ; Z) is a finite nontrivial group of exponent a multiple of 15. We have not been able to detect 5-torsion in any other homology groupH d (M n ; Z), but in Section 5.3, we show that the case 5d = 2n − 8 is crucial for the general behavior: See Figure 1 for the homology of M n for n ≤ 14.…”

Section: Introductionmentioning

confidence: 99%

Exact sequences for the homology of the matching complex

Jönsson

2008

Journal of Combinatorial Theory, Series A

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Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex M n , which is the simplicial complex of matchings in the complete graph K n . Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of M n . First, we demonstrate that there is nonvanishing 3-torsion inH d (M n ; Z) whenever ν n d n−6 2 , where ν n = n−4 3 . By results due to Bouc and to Shareshian and Wachs,H νn (M n ; Z) is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of M n for all n = 2. Second, we prove thatH d (M n ; Z) is a nontrivial 3-group whenever ν n d 2n−9 5. Third, for each k 0, we show that there is a polynomial f k (r) of degree 3k such that the dimension ofH k−1+r (M 2k+1+3r ; Z 3 ), viewed as a vector space over Z 3 , is at most f k (r) for all r k + 2.

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“…For p = 2, 3 the complexes C p (n) and M p (n) are isomorphic, as for each X ⊆ [n] of size p, there is unique cyclic group of order p in S n having support X . Matching complexes and related complexes have been studied in the literature for their intrinsic combinatorial interest and in connection with applications in various fields of mathematics, see [Wa] for a survey and see [Jo1,Jo2,Jo3,SW] for more recent developments.…”

Section: Introductionmentioning

confidence: 99%

Top homology of hypergraph matching complexes, p-cycle complexes and Quillen complexes of symmetric groups

Shareshian

Wachs

2009

Journal of Algebra

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We investigate the representation of a symmetric group S n on the homology of its Quillen complex at a prime p. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of p-uniform hypergraph matching complexes. We conjecture an explicit formula for the representation of S n on the top homology group of the corresponding hypergraph matching complex when n ≡ 1 mod p. Our conjecture follows from work of Bouc when p = 2, and we prove the conjecture when p = 3.

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Five-torsion in the homology of the matching complex on 14 vertices

Cited by 6 publications

References 15 publications

More Torsion in the Homology of the Matching Complex

More Torsion in the Homology of the Matching Complex

Exact sequences for the homology of the matching complex

Top homology of hypergraph matching complexes, p-cycle complexes and Quillen complexes of symmetric groups

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