TY - JOUR

T1 - Polyhedral products over finite posets

AU - Kishimoto, Daisuke

AU - Levi, Ran

N1 - Funding Information:
by [21, precise statement]. By Proposition 7.2, fi(s(P)) = fi(P)+k=i+1∑|P|fk(P)× νi,k(P). The claim follows by substituting this expression in (7.1). □ Acknowledgments. Daisuke Kishimoto was partly supported by Japan Society for the Promotion of Science KAKENHI grant 17K05248. Ran Levi was partly supported by Engineering and Physical Sciences Research Council grant EP/P025072/1. The authors are grateful to the University of Kyoto and the University of Aberdeen for their kind hospitality.
Publisher Copyright:
© 2022 by Kyoto University.

PY - 2022/9

Y1 - 2022/9

N2 - Polyhedral products were defined by Bahri, Bendersky, Cohen, and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper, we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets.We show that under certain restrictions on the posetP,which include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over P of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction - the polyhedral tensor product.We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product.We then restrict attention to polyhedral posets, a family of posets that includes face posets of simplicial complexes, and simplicial posets, as well as many others.We define the Stanley-Reisner ring of a polyhedral poset and showthat, as in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset P, we construct a simplicial poset s(P), and show that if P is a polyhedral poset, then polyhedral products over P coincide up to homotopy with the corresponding polyhedral products over s(P).

AB - Polyhedral products were defined by Bahri, Bendersky, Cohen, and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper, we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets.We show that under certain restrictions on the posetP,which include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over P of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction - the polyhedral tensor product.We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product.We then restrict attention to polyhedral posets, a family of posets that includes face posets of simplicial complexes, and simplicial posets, as well as many others.We define the Stanley-Reisner ring of a polyhedral poset and showthat, as in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset P, we construct a simplicial poset s(P), and show that if P is a polyhedral poset, then polyhedral products over P coincide up to homotopy with the corresponding polyhedral products over s(P).

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U2 - 10.1215/21562261-2022-0020

DO - 10.1215/21562261-2022-0020

M3 - Article

AN - SCOPUS:85139040338

SN - 2156-2261

VL - 62

SP - 615

EP - 654

JO - Kyoto Journal of Mathematics

JF - Kyoto Journal of Mathematics

IS - 3

ER -