Symmetric multiple chessboard complexes and a new theorem of Tverberg type

Abstract

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].

Alexander r-tuples and bier complexes

“…Among the corollaries of our results are exact connectivity bounds for some classes of generalized chessboard complexes (including the main case of [8,Theorem 3.2]).…”

“…However, the construction of the discrete Morse function, described in Section 7, is sufficiently general and versatile to be applied in this case as well. This is very interesting since the existence of a perfect Morse function on this complex provides an alternative proof of the (critical case) of [8,Theorem 3.2]. Recall that this result paved the way for some new Tverberg-Van Kampen-Flores type results, including the [9, Theorem 1.2].…”

“…(1) A 1 avoids the segment [1, provides an alternative proof of the following theorem from [8]. (Two other proofs, both of them comparatively complex and nontrivial, relied respectively on a shelling construction, and the Nerve Lemma).…”

Splitting necklaces, with constraints

“…, r. In the case K 1 = • • • = K r = K this reduces to the definition of r-fold deleted join K * r ∆ , see [12]. The symmetrized deleted join [11] of K is defined as…”

“…The following theorem [11,Theorem 3.3] was originally proved by a direct shelling argument. As demonstrated in [9] it can be also deduced from Theorem 2.6.…”