Which 4-manifolds admit a flag-no-square (fns) triangulation? We introduce the “star-connected-sum" operation on such triangulations, which preserves the fns property, from which we derive new constructions of fns 4-manifolds. In particular, we show the following: (i) there exist non-aspherical fns 4-manifolds, answering in the negative a question by Przytycki and Swiatkowski; (ii) for every large enough integer k there exists a fns 4-manifold <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M_{2k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of Euler characteristic 2k, and further, (iii) <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$M_{2k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> admits a super-exponential number (in k) of fns triangulations - at least <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2^{\Omega (k \log k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>log</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> and at most <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$2^{O(k^{1.5} \log k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow> <mn>1.5</mn> </mrow> </msup> <mo>log</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>.

On Flag-No-Square 4-Manifolds

期刊论文
发表日期：

2025年9月18日

查看全文

Daniel Kalmanovich,
Eran Nevo,
Gangotryi Sorcar

摘要

Which 4-manifolds admit a flag-no-square (fns) triangulation? We introduce the “star-connected-sum" operation on such triangulations, which preserves the fns property, from which we derive new constructions of fns 4-manifolds. In particular, we show the following: (i) there exist non-aspherical fns 4-manifolds, answering in the negative a question by Przytycki and Swiatkowski; (ii) for every large enough integer k there exists a fns 4-manifold $M_{2k}$ $M_{2 k}$ of Euler characteristic 2k, and further, (iii) $M_{2k}$ $M_{2 k}$ admits a super-exponential number (in k) of fns triangulations - at least $2^{\Omega (k \log k)}$ $2^{Ω (k log k)}$ and at most $2^{O(k^{1.5} \log k)}$ $2^{O (k^{1.5} log k)}$ .