The Frobenius complex.
Given positive integers \(n_1,n_2,\ldots,n_k\) with \(\gcd(n_1,n_2,\ldots,n_k)=1\), the Frobenius problem is to find the largest integer -- called the \textit{Frobenius number} -- which cannot be obtained as a non-negative integer linear combination of the \(n_i\). In the present paper, the authors reformulate this classical problem by introducing a \textit{Frobenius poset}: considering \(\Lambda\), the sub-semigroup of \(\mathbb N\) generated by \(n_1,n_2,\ldots,n_k\), the Frobenius poset is \(P=(\mathbb Z,\leq_\Lambda)\) with \(n\leq_\Lambda m\) if \(m-n\in\Lambda\). The Frobenius number is then the largest integer which is not comparable to 0 in \(P\) and the authors study the homotopy type of order complexes of intervals \([0,n]_\Lambda:=\{i\in [0,n]:i,n-i\in\Lambda\}\). They consider two cases for which the Frobenius number is known. In the first case, \(\Lambda\) is generated by two relatively prime positive integers \(a\) and \(b\) and they prove (Theorem 4.1) that the order complex of \([0,n]_\Lambda\) is either homotopy equivalent to a sphere or contractible (depending on the value of \(n\) modulo \(ab\)). In the second case, \(\Lambda\) is generated by an arithmetic sequence \(\{a,a+d,a+2d,\ldots,a+(a-1)d\}\) where \(a\) and \(d\) are two relatively prime positive integers and they prove (Theorem 5.1) that the order complex of \([0,n]_\Lambda\) is homotopy equivalent to a wedge of spheres; moreover, they give a formula for calculating the Betti numbers. The proofs use discrete Morse theory. In each case, the authors get an acyclic matching on the face poset of the order complex of \([0,n]_\Lambda\) by applying the Patchwork Theorem [\textit{D. Kozlov}, Combinatorial algebraic topology. Berlin: Springer (2008; Zbl 1130.55001)] and the fact, in the second case, that the critical cells correspond to facets of the face poset of the order complex of \([0,n]_\Lambda\).
scientific article; zbMATH DE number 6121627
scientific article