The Frobenius complex
نویسندگان
چکیده
Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers Z, that is, for a sub-semigroup Λ of the non-negative integers (N,+), we define the order by n ≤Λ m if m− n ∈ Λ. When Λ is generated by two relatively prime integers a and b, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when Λ is generated by the integers {a, a + d, a + 2d, . . . , a + (a− 1)d}, the order complex is homotopy equivalent to a wedge of spheres. Résumé. Motivé par le problème de Frobenius classique, nous introduisons l’ensemble partiellement ordonné de Frobenius sur les entiers Z, c.à.d. que pour un sous-semigroupe Λ de les entiers non-négatifs (N,+) nous définissons l’ordre par n ≤Λ m si m − n ∈ Λ. Quand le Λ est engendré par deux nombres a et b, relativement premiers entre eux, noux montrons que le complexe des chaı̂nes d’un intervalle quelquonque dans l’ensemble partiellement ordonné de Frobenius est soit contractible soit homotopiquement équivalent à une sphère. Nous montrons aussi que dans le cas où Λ est engendré par les entiers {a, a + d, a + 2d, . . . , a + (a − 1)d}, le complexe des chaı̂nes a le type de homotopie d’un bouquet de sphères.
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