arXiv subjects

Nikola Sadovek

Publications and source records attributed to Nikola Sadovek.

10 recordsLinked to original sources

arXiv2026

On colorful generalizations of the Goodman--Pollack transversal problem

We establish a colorful and, more generally, matroidal solution to the problem of Goodman and Pollack on the existence of an $\mathbb{F}$-affine $k$-dimensional transversal to a family of convex sets in $\mathbb{F}^d$, where $0 \le k \le d - 1$ is an integer and $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$ is a field. Our results unify several classical and recent theorems. In the case $k=0$, we recover the colorful Helly theorem of Lovász, together with a matroidal extension due to Kalai and Meshulam. In the opposite extremal case $k=d-1$, we obtain Holmsen's colorful and matroidal generalization of the Goodman-Pollack-Wenger theorem. Additionally, we extend the recent noncolorful solution of the Goodman-Pollack problem by McGinnis and the author. As the main application, we obtain a matroidal and colorful Dol'nikov-type transversal theorem. Our methods are topological. We introduce matroidal joins, defined as homotopy colimits of diagrams over face posets of matroidal complexes, and derive estimates on their connectivity. The proof additionally relies on adaptations of nonexistence results for equivariant maps from Stiefel manifolds to spheres.

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arXiv2026

Complex line fields on almost-complex manifolds

We study linearly independent complex line fields on almost-complex manifolds, which is a topic of long-standing interest in differential topology and complex geometry. A necessary condition for the existence of such fields is the vanishing of appropriate virtual Chern classes. We prove that this condition is also sufficient for the existence of one, two, or three linearly independent complex line fields over certain manifolds. More generally, our results hold for a wider class of complex bundles over CW complexes.

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arXiv2026

Quantifying discontinuity

Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem.

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arXiv2026

A necessary and sufficient condition for $k$-transversals

We solve a long-standing open problem posed by Goodman \& Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in $\mathbb{R}^d$ to admit a $k$-transversal for any $0 \le k \le d-1$. This result is a common generalization of Helly's theorem ($k=0$) and the Goodman-Pollack-Wenger theorem ($k=d-1$). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex $k$-transversal to a family of convex sets in $\mathbb{C}^d$, extending the work of McGinnis ($k=d-1$). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case.

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arXiv2025

Complex analogues of the Tverberg--Vrećica conjecture and central transversal theorems

The Tverberg--Vrećica conjecture claims a broad generalization of Tverberg's classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoint theorem and the ham sandwich theorem. In this manuscript, we establish complex analogues of these results, where the corresponding transversals are complex affine spaces. The proofs of the complex Tverberg--Vrećica conjecture and its optimal colorful version rely on the non-vanishing of an equivariant Euler class. Furthermore, we obtain new Borsuk--Ulam-type theorems on complex Stiefel manifolds. These theorems yield complex analogues of recent extensions of the ham sandwich theorem for mass assignments by Axelrod-Freed and Soberón, and provide a direct proof of the complex central transversal theorem.

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arXiv2025

Bisections of mass assignments by parallel hyperplanes

In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting method to define the configuration space--test map scheme, which transforms the problem to a Borsuk--Ulam-type question on equivariant fiber bundles, along with a new computation of the parametrized Fadell--Husseini index. As the primary application, we show that any $d+k+m-1$ mass assignments to linear $d$-spaces in $\mathbb{R}^{d+m}$ can be bisected by $k$ parallel hyperplanes in at least one $d$-space, provided that the Stirling number of the second kind $S(d+k+m-1, k)$ is odd. This generalizes all known cases of a conjecture by Soberón and Takahashi, which asserts that any $d+k-1$ measures in $\mathbb{R}^d$ can be bisected by $k$ parallel hyperplanes.

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arXiv2025

Bisections of mass assignments by parallel hyperplanes

In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting method to define the configuration space--test map scheme, which transforms the problem to a Borsuk--Ulam-type question on equivariant fiber bundles, along with a new computation of the parametrized Fadell--Husseini index. As the primary application, we show that any $d+k+m-1$ mass assignments to linear $d$-spaces in $\mathbb{R}^{d+m}$ can be bisected by $k $ parallel hyperplanes in at least one $d$-space, provided that the Stirling number of the second kind $S(d+k+m-1, k)$ is odd. This generalizes all known cases of a conjecture by Soberón and Takahashi, which asserts that any $d+k-1$ measures in $\mathbb{R}^d$ can be bisected by $k$ parallel hyperplanes.

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arXiv2025

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the distortion of (possibly discontinuous) functions between them. The more these functions must distort the metrics, the larger the Gromov--Hausdorff distance must be. Topology has few tools to obstruct the existence of discontinuous functions. However, an arbitrary function $f\colon X\to Y$ induces a continuous map between their Vietoris--Rips simplicial complexes, where the allowable choices of scale parameters depend on how much the function $f$ distorts distances. We can then use equivariant topology to obstruct the existence of certain continuous maps between Vietoris--Rips complexes. With these ideas we bound how discontinuous an odd map between spheres $S^k\to S^n$ with $k>n$ must be, generalizing a result by Dubins and Schwarz (1981), which is the case $k=n+1$. As an application, we recover or improve upon all of the lower bounds from Lim, M{é}moli, and Smith (2022) on the Gromov--Hausdorff distances between spheres of different dimensions. We also provide new upper bounds on the Gromov--Hausdorff distance between spheres of adjacent dimensions.

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arXiv2024

Persistent equivariant cohomology

This article has two goals. First, we hope to give an accessible introduction to persistent equivariant cohomology. Given a topological group $G$ acting on a filtered space, persistent Borel equivariant cohomology measures not only the shape of the filtration, but also attributes of the group action on the filtration, including in particular its fixed points. Second, we give an explicit description of the persistent equivariant cohomology of the circle action on the Vietoris-Rips metric thickenings of the circle, using the Serre spectral sequence and the Gysin homomorphism. Indeed, if $\frac{2πk}{2k+1} \le r < \frac{2π(k+1)}{2k+3}$, then $H^*_{S^1}(\mathrm{VR}^\mathrm{m}(S^1;r))\cong \mathbb{Z}[u]/(1\cdot3\cdot5\cdot\ldots \cdot (2k+1)\, u^{k+1})$ where $\mathrm{deg}(u)=2$.

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arXiv2024

Convex Equipartitions inspired by the little cubes operad

A decade ago two groups of authors, Karasev, Hubard and Aronov, and Blagojević and Ziegler, have shown that the regular convex partitions of a Euclidean space into $n$ parts yield a solution to the generalised Nandakumar and Ramana-Rao conjecture when $n$ is a prime power. This was obtained by parametrising the space of regular equipartitions of a given convex body with the classical configuration space. Now, we repeat the process of regular convex equipartitions many times, first partitioning the Euclidean space into $n_1$ parts, then each part into $n_2$ parts, and so on. In this way we obtain iterated convex equipartions of a given convex body into $n=n_1...n_k$ parts. Such iterated partitions are parametrised by the (wreath) product of classical configuration spaces. We develop a new configuration space -- test map scheme for solving the generalised Nandakumar \& Ramana-Rao conjecture using the Hausdorff metric on the space of iterated convex equipartions. The new scheme yields a solution to the conjecture if and only if all the $n_i$'s are powers of the same prime. In particular, for the failure of the scheme outside prime power case we give three different proofs.

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