arXiv subjects

Albert Haase

Publications and source records attributed to Albert Haase.

3 recordsLinked to original sources

arXiv2018

Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem

In 1960 Grünbaum asked whether for any finite mass in $\mathbb{R}^d$ there are $d$ hyperplanes that cut it into $2^d$ equal parts. This was proved by Hadwiger (1966) for $d\le3$, but disproved by Avis (1984) for $d\ge5$, while the case $d=4$ remained open. More generally, Ramos (1996) asked for the smallest dimension $Δ(j,k)$ in which for any $j$ masses there are $k$ affine hyperplanes that simultaneously cut each of the masses into $2^k$ equal parts. At present the best lower bounds on $Δ(j,k)$ are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska, Vrećica \& Živaljević (2006). The problem has been an active testing ground for advanced machinery from equivariant topology. We give a critical review of the work on the Grünbaum--Hadwiger--Ramos problem, which includes the documentation of essential gaps in the proofs for some previous claims. Furthermore, we establish that $Δ(j,2)= \frac12(3j+1)$ in the cases when $j-1$ is a power of $2$, $j\ge5$.

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arXiv2017

Tverberg-type theorems for matroids: A counterexample and a proof

Bárány, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex to d-dimensional Euclidean space, if the matroid has sufficiently many disjoint bases. They make a conjecture on the connectivity of k-fold deleted joins of a matroid with many disjoint bases, which would yield a much tighter result - but we provide a counterexample already for the case of k=2, where a tight Tverberg-type theorem would be a topological Radon theorem for matroids. Nevertheless, we prove the topological Radon theorem for the counterexample family of matroids by an index calculation, despite the failure of the connectivity-based approach.

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arXiv2016

Hyperplane mass partitions via relative equivariant obstruction theory

The Grünbaum-Hadwiger-Ramos hyperplane mass partition problem was introduced by Grünbaum (1960) in a special case and in general form by Ramos (1996). It asks for the "admissible" triples $(d,j,k)$ such that for any $j$ masses in $\mathbb{R}^d$ there are $k$ hyperplanes that cut each of the masses into $2^k$ equal parts. Ramos' conjecture is that the Avis-Ramos necessary lower bound condition $dk\ge j(2^k-1)$ is also sufficient. We develop a "join scheme" for this problem, such that non-existence of an $G_k$-equivariant map between spheres $(S^d)^{*k} \rightarrow S(W_k\oplus U_k^{\oplus j})$ that extends a test map on the subspace of $(S^d)^{*k}$ where the hyperoctahedral group $G_k$ acts non-freely, implies that $(d,j,k)$ is admissible. For the sphere $(S^d)^{*k}$ we obtain a very efficient regular cell decomposition, whose cells get a combinatorial interpretation with respect to measures on a modified moment curve. This allows us to apply relative equivariant obstruction theory successfully, even in the case when the difference of dimensions of the spheres $(S^d)^{*k}$ and $S(W_k\oplus U_k^{\oplus j})$ is greater than one. The evaluation of obstruction classes leads to counting problems for concatenated Gray codes. Thus we give a rigorous, unified treatment of the previously announced cases of the Grünbaum-Hadwiger-Ramos problem, as well as a number of new cases for Ramos' conjecture.

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