Mathematics Subject Classifications: 16T05, 16T30, 05C65, 05E45 AbstractThe generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn-Sommerville relations for the combinatorial Hopf algebra of hypergraphs. We characterize a wide class of eulerian hypergraphs according to the combinatorics of underlying clutters. The analogous results hold for simplicial complexes by the isomorphism which is induced from the correspondence of clutters and simplicial complexes.
Following D.B. Karaguezian, V. Reiner, and M.L. Wachs (Matching Complexes, Bounded Degree Graph Complexes, and Weight Spaces of GL-Complexes, Journal of Algebra 2001) we study the connectivity degree and shellability of multiple chessboard complexes. Our central new results (Theorems 3.2 and 4.4) provide sharp connectivity bounds relevant to applications in Tverberg type problems where multiple points of the same color are permitted. These results also provide a foundational work for the new results of Tverberg-van Kampen-Flores type, as announced in the forthcoming paper . An overview and motivationChessboard complexes and their generalizations belong to the class of most studied graph complexes, with numerous applications in and outside combinatorics [A04, BLVŽ, BMZ, FH98, G79, J08, KRW, M03, SW07, VŽ94,ŽV92, Zi11,Ž04].The connectivity degree of a simplicial complex was selected in [J08, Chapter 10] as one of the five most important and useful parameters in the study of simplicial complexes of graphs. Following [KRW] we study the connectivity degree of multiple chessboard complexes (Section 1.4) and their generalizations (Section 2). Our first central result is Theorem 3.2 which improves the 2-dimensional case of [KRW, Corollary 5.2.] and reduces to the 2-dimensional case of [BLVŽ, Theorem 3.1.] in the case of standard chessboard complexes. * Supported by the Ministry for Science and Technology of Serbia, Grant 174034. 1Perhaps it is worth emphasizing that our methods allow us to obtain sharp bounds relevant to applications in Tverberg and van Kampen-Flores type problems (see ). Moreover, the focus in [KRW] is on the homology with the coefficients in a field and multidimensional chessboard complexes while our results are homotopical and apply to 2-dimensional chessboard complexes.High connectivity degree is sometimes a consequence of the shellability of the complex (or one of its skeletons), see [Zi94] for an early example in the context of chessboard complexes. Theorem 4.4 provides a sufficient condition which guarantees the shellability of multiple chessboard complexes and yields another proof of Theorem 3.2. The construction of the shelling offers a novel point of view on this problem and seems to be new and interesting already in the case of standard chessboard complexes.Among the initial applications of the new connectivity bounds established by Theorem 3.2 is a result of colored Tverberg type where multiple points of the same color are permitted (Theorem 5.1 in Section 5). After the first version of our paper was submitted to the arXiv we were kindly informed by Günter Ziegler that Theorem 5.1 is implicit in their recent work (see [BFZ], Theorem 4.4 and the remark following the proof of Lemma 4.2.).Other, possibly more far reaching applications of Theorems 3.2 and 4.4 to theorems of Tverberg-van Kampen-Flores type are announced in . This provides new evidence that the chessboard complexes and their generalizations are a natural framework for constructing configuration spaces relevant to Tverb...
We prove a new theorem of Tverberg-van Kampen-Flores type (Theorem 1.2) which confirms the conjecture of Blagojević, Frick, and Ziegler about the existence of 'balanced Tverberg partitions' (Conjecture 6.6 in, Tverberg plus constraints, Bull. London Math. Soc. 46 (2014) 953-967). The conditions in Theorem 1.2 are somewhat weaker than in the original conjecture and we show that the theorem is optimal in the sense that the new (weakened) condition is also necessary. Among the consequences is a positive answer (Theorem 7.2) to the 'balanced case' of the problem whether each admissible r-tuple is Tverberg prescribable, [BFZ, Question 6.9].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.