TY - JOUR
T1 - Random simplicial complexes, duality and the critical dimension
AU - Farber, Michael
AU - Mead, Lewis
AU - Nowik, Tahl
N1 - Publisher Copyright:
© 2022 The Author(s).
PY - 2022/3/1
Y1 - 2022/3/1
N2 - In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.
AB - In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.
KW - Alexander duality
KW - Random simplicial complex
KW - critical dimension
UR - http://www.scopus.com/inward/record.url?scp=85095072620&partnerID=8YFLogxK
U2 - 10.1142/S1793525320500387
DO - 10.1142/S1793525320500387
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AN - SCOPUS:85095072620
SN - 1793-5253
VL - 14
SP - 1
EP - 31
JO - Journal of Topology and Analysis
JF - Journal of Topology and Analysis
IS - 1
ER -