We present short proofs of Tverberg-type theorems for cell complexes by S. Hasui, D. Kishimoto, M. Takeda, and M. Tsutaya. One of them states that for any prime power r, any complex X topologically homeomorphic to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S^{(d+1)(r-1)-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, and any continuous map <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f:X\rightarrow {\mathbb R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> there are pairwise disjoint faces <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma _1,\ldots ,\sigma _r$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>σ</mi> <mi>r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of X such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(\sigma _1)\cap \ldots \cap f(\sigma _r)\ne \emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <mo>…</mo> <mo>∩</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mi>r</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation>.

Short Proofs of Tverberg-Type Theorems for Cell Complexes

期刊论文
发表日期：

2025年11月27日

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Roman Karasev,
Arkadiy Skopenkov

摘要

We present short proofs of Tverberg-type theorems for cell complexes by S. Hasui, D. Kishimoto, M. Takeda, and M. Tsutaya. One of them states that for any prime power r, any complex X topologically homeomorphic to $S^{(d+1)(r-1)-1}$ $S^{(d + 1) (r - 1) - 1}$ , and any continuous map $f:X\rightarrow {\mathbb R}^d$ $f : X \to R^{d}$ there are pairwise disjoint faces $\sigma _1,\ldots ,\sigma _r$ $σ_{1}, \dots, σ_{r}$ of X such that $f(\sigma _1)\cap \ldots \cap f(\sigma _r)\ne \emptyset $ $f (σ_{1}) \cap \dots \cap f (σ_{r}) \neq \emptyset$ .