发表机构
University of Ottawa（渥太华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究庞加莱对偶空间上的模拓扑复杂度，证明其与有理拓扑复杂度相等，并推广到映射的截面不变量，给出精确不等式及几何应用。
AI 中文摘要
拓扑复杂度由Farber引入，度量配置空间上连续局部运动规划规则的数量。对于每个有限有理同伦型的单连通空间$X$，其有理上同调为有限维庞加莱对偶代数，我们证明$\mathrm{MTC}(X)=\mathrm{TC}_0(X)=\mathrm{TC}_0^M(X)$，其中$\mathrm{TC}_0(X)=\mathrm{TC}(X_{\mathbb Q})$，$M$表示幺半群不变量。无需形式性或椭圆性假设。该结果推广到更高拓扑复杂度，并在对偶下给出乘积可加性。对于每个单连通有限有理同伦型空间之间的映射$f$，其有理化具有同伦收缩，我们建立精确界$\mathrm{Msecat}(f_{\mathbb Q})\leq\mathrm{secat}(f_{\mathbb Q})\leq\mathrm{relcat}(f_{\mathbb Q})\leq\mathrm{Msecat}(f_{\mathbb Q})+1$。特别地，$\mathrm{MTC}(X)\leq\mathrm{TC}_0(X)\leq\mathrm{MTC}(X)+1$在对偶假设下成立。证明使用受控Sullivan分解和相对范畴的理想值判据。对于具有庞加莱对偶纤维的纤维化截面，我们证明导出自相交公式、Thom乘法在增广理想上的消失以及理想值评估是等价的。该判据得出：对于有限有理同伦型的单连通光滑基上的光滑丛的光滑截面，其闭单连通流形纤维（包括有理双曲纤维），三个有理截面不变量相等。当纤维具有有限维总有理同伦时，相等性也成立，无需基的上同调维数界，并且适用于合适的有限相对Frobenius模型。几何应用包括球丛、Grassmann丛和射影丛。
英文摘要
Topological complexity, introduced by Farber, measures the number of continuous local motion-planning rules on a configuration space. For every simply connected space $X$ of finite rational homotopy type whose rational cohomology is a finite-dimensional Poincaré duality algebra, we prove $\mathrm{MTC}(X)=\mathrm{TC}_0(X)=\mathrm{TC}_0^M(X)$, where $\mathrm{TC}_0(X)=\mathrm{TC}(X_{\mathbb Q})$ and $M$ denotes the monoidal invariant. No formality or ellipticity assumption is required. The result extends to higher topological complexity and gives product additivity under duality. For every map $f$ between simply connected spaces of finite rational homotopy type whose rationalization admits a homotopy retraction, we establish the sharp bound $\mathrm{Msecat}(f_{\mathbb Q})\leq\mathrm{secat}(f_{\mathbb Q})\leq\mathrm{relcat}(f_{\mathbb Q})\leq\mathrm{Msecat}(f_{\mathbb Q})+1$. In particular, $\mathrm{MTC}(X)\leq\mathrm{TC}_0(X)\leq\mathrm{MTC}(X)+1$ holds without a duality assumption. The proofs use controlled Sullivan resolutions and an ideal-valued criterion for relative category. For sections of fibrations with Poincaré duality fibre, we prove that a derived self-intersection formula, vanishing of Thom multiplication on the augmentation ideal, and ideal-valued evaluation are equivalent. This criterion yields equality of the three rational sectional invariants for smooth sections of smooth bundles over simply connected smooth bases of finite rational homotopy type, with closed simply connected manifold fibre, including rationally hyperbolic fibres. Equality also holds when the fibre has finite-dimensional total rational homotopy, without a cohomological dimension bound on the base, and for suitable finite relative Frobenius models. Geometric applications include sphere, Grassmann, and projective bundles.
