arXiv subjects

Michael C. Crabb

Publications and source records attributed to Michael C. Crabb.

4 recordsLinked to original sources

arXiv2024

Many partitions of mass assignments

In this paper, extending the recent work of authors with Calles Loperena and Dimitrijević Blagojević, we give a general and complete treatment of a problem of partition of mass assignments with prescribed arrangements of hyperplanes on Euclidean vector bundles. Using a new configuration test map scheme, as well as an alternative topological framework, we are able to reprove known results, extend them to arbitrary bundles as well as to put various types of constraints on the solutions. Moreover, the developed topological methods allow us to give new proofs and extend results of Guth and Katz, Schnider, and Soberón and Takahashi. In this way we place all these results under one ``roof''.

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arXiv2022

Topology of the Grünbaum--Hadwiger--Ramos problem for mass assignments

In this paper, motivated by recent work of Schnider and Axelrod-Freed \& Soberón, we study an extension of the classical Grünbaum--Hadwiger--Ramos mass partition problem to mass assignments. Using the Fadell--Husseini index theory we prove that for a given family of $j$ mass assignments $μ_1,\dots,μ_j$ on the Grassmann manifold $G_{\ell}(\R^d)$ and a given integer $k\geq 1$ there exist a linear subspace $L\in G_{\ell}(\R^d)$ and $k$ affine hyperplanes in $L$ that equipart the masses $μ_1^L,\dots,μ_j^L$ assigned to the subspace $L$, provided that $d\geq j + (2^{k-1}-1)2^{\lfloor\log_2j\rfloor}$.

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arXiv2021

Equivariant Cohomology of Configuration Spaces mod 2: The State of the Art

The equivariant cohomology of the classical configuration space $F(\mathbb{R}^d,n)$ has been been of great interest and has been studied intensively starting with the classical papers by Artin (1925/1947) on the theory of braids, by Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). We give a brief treatment of the subject from the beginnings to recent developments. However, we focus on the mod 2 equivariant cohomology algebras of the classical configuration space $F(\mathbb{R}^d,n)$, as described in an influential paper by Hung (1990). We show with a new, detailed proof that his main result is correct, but that the arguments that were given by Hung on the way to his result are not, as are some of the intermediate results in his paper. This invalidates a paper by three of the present authors, Blagojević, Lück \& Ziegler (2016), who used a claimed intermediate result from Hung (1990) in order to derive lower bounds for the existence of $k$-regular and $\ell$-skew embeddings. Using our new proof for Hung's main result, we get new lower bounds for existence of highly regular embeddings: Some of them agree with the previously claimed bounds, some are weaker.

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arXiv2015

$Z_2$-bordism and the Borsuk-Ulam Theorem

The purpose of this work is to classify, for given integers $m,\, n\geq 1$, the bordism class of a closed smooth $m$-manifold $X$ with a free smooth involution $τ$ with respect to the validity of the {\it Borsuk-Ulam property} that for every continuous map $ϕ: X \to R^n$ there exists a point $x\in X$ such that $ϕ(x)=ϕ(τ(x))$. We will classify a given free $Z_2$-bordism class $α$ according to the three possible cases that (a) all representatives $(X , τ)$ of $α$ satisfy the Borsuk-Ulam property; \ (b) there are representatives $(X_ 1, τ_1)$ and $(X_2, τ_2)$ of $α$ such that $(X_1, τ_1)$ satisfies the Borsuk-Ulam property but $(X_2, τ_2)$ does not; \ (c) no representative $(X , τ)$ of $α$ satisfies the Borsuk-Ulam property.

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