Substitutions of polytopes and of simplicial complexes, and multigraded betti numbers

Abstract

For a simplicial complex K on m vertices and simplicial complexes K 1 , . . . , K m , we introduce a new simplicial complex K(K 1 , . . . , K m ), called a substitution complex. This construction is a generalization of the iterated simplicial wedge studied by A. Bari, M. Bendersky, F. R. Cohen, and S. Gitler. In a number of cases it allows us to describe the combinatorics of generalized joins of polytopes P (P 1 , . . . , P m ), as introduced by G. Agnarsson. The substitution gives rise to an operad structure on the set of finite simplicial complexes in which a simplicial complex on m vertices is considered as an m-ary operation. We prove the following main results: (1) the complex K(K 1 , . . . , K m ) is a simplicial sphere if and only if K is a simplicial sphere and the K i are the boundaries of simplices, (2) the class of spherical nerve-complexes is closed under substitution, (3) multigraded betti numbers of K(K 1 , . . . , K m ) are expressed in terms of those of the original complexes K, K 1 , . . . , K m . We also describe connections between the obtained results and the known results of other authors.2010 Mathematics Subject Classification. Primary 05E45; Secondary 52B11, 52B05, 55U10, 13F55. download from IP 128.122.253.212.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 176 A. A. AYZENBERG question is how to describe the properties of the Stanley-Reisner algebra k[K P ] and the cohomology ring H * (Z P ; k) ∼ = Tor * , * k[m] (k[K P ], k) for arbitrary convex polytopes. In the work of A. Bahri, M. Bendersky, F. R Cohen, and S. Gitler [5] one finds a construction that allows, for a given simple polytope P with m facets and a set of natural numbers (l 1 , . . . , l m ), the building of a new simple polytope P (l 1 , . . . , l m ). The simplicial complex ∂P (l 1 , . . . , l m ) * can be described combinatorially in terms of minimal non-simplices. This approach allows us to describe the moment-angle complex Z ∂P (l 1 ,...,l m ) * (D 2 , S 1 ) as the polyhedral product Z P * ((D 2l i , S 2l i −1 )), which yields a more efficient description of the cohomology ring H * (Z P (l 1 ,...,l m ) ), as well as of the cohomology rings of some quasitoric manifolds over such polytopes.The use of arbitrary (i.e., not necessarily simple) polytopes in toric topology yields a wider class of examples and more general constructions. One such construction is known in convex geometry (see, for example, [1]) and, for a given a polytope P ⊂ R m and polytopes P 1 , . . . , P m , produces a new polytope P (P 1 , . . . , P m ). In general, the combinatorial structure of P (P 1 , . . . , P m ) depends on the chosen geometric representation of P ⊂ R m . However, under some conditions, the construction can be made combinatorial, i.e., we may assume that the face poset of P (P 1 , . . . , P m ) depends only on the face posets of the original polytopes P, P 1 , . . . , P m . In the particular case when P i = l i is a simplex on l i vertices, the polytope P (P 1 , . . . , P m ) c...

Toric origami structures on quasitoric manifolds

“…Another problem, which naturally arises from Lemma 3.3 is to find, for a given polytope P , a characteristic function Λ with the minimal possible range of values |ΛpFpP qq|, if at least one characteristic function is known to exist. This minimal number seems to be an analogue of Buchstaber invariant (see the definition in [11] or [1]), as was noted to us recently by N. Erokhovets. It may happen that an interesting theory hides beyond this subject.…”

On polyhedral product spaces over polyhedral joins

“…, L m ) whenever the set {i ∈ [m] | σ i / ∈ L i } is a simplex of K. It is noteworthy that simplicial complexes form an operad where the simplicial complex on m vertices is viewed as an m-adic operation. See [1] for more details.…”

Polyhedral products and features of their homotopy theory

“…Theorem 8.1), we conclude that the bigraded Betti numbers of the Tor module are the ranks of the contributions of H * (|K J |) to the cohomology of Z(K; (D 2 , S 1 )). Further computations and properties of the bigraded and multigraded Betti numbers may be found in [8] [97] [133] and [42]. The calculations in [42] are used to verify the Halperin-Carlsson conjecture, ([42, Remark 3]), in the case of free torus actions on moment-angle complexes.…”