单纯复形与非紧2维及3维流形的Borel分类
Borel classification of simplicial complexes and non-compact $2$- and $3$-manifolds
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中文总结 AI 辅助
该研究推广了斯通对偶，获得单纯复形与非紧2、3维流形的同胚/PL同胚完全不变量，证明相关同胚关系可通过可数结构分类且与可数图同构关系Borel双可归约。
中文摘要 AI 辅助
我们推广了布尔代数上超滤子的斯通空间，并证明了适用于局部紧波兰空间的斯通对偶的推广形式。利用这一结果，我们得到了单纯复形在PL同胚下的完全不变量，以及非紧2维和3维流形在同胚下的完全不变量。我们证明：无边界非紧2维流形上的同胚关系、带或不带边界的非紧3维流形上的同胚关系、ℝ²和ℝ³的开子集上的同胚关系，以及ℝ³中康托集的共轭关系，均可通过可数结构分类。结合已知的下界结果，这意味着这些关系与可数图的同构关系是Borel双可归约的。我们还证明：海涅-博雷尔单纯复形的PL同胚，以及对任意n，PL n维流形的PL同胚，均可通过可数结构分类。
英文摘要
We generalize the Stone space of ultrafilters on Boolean algebras and prove a generalization of Stone duality which is applicable to locally compact Polish spaces. Using this, we obtain complete invariants for simplicial complexes up to PL-homeomorphism and for non-compact $2$- and $3$-manifolds up to homeomorphism. We prove that the homeomorphism relation on non-compact $2$-manifolds without boundary, the homeomorphism relation on non-compact $3$-manifolds with or without boundary, the homeomorphism relation on open subsets of $\mathbb{R}^2$ and $\mathbb{R}^3$, and conjugacy of Cantor sets in $\mathbb{R}^3$ are classifiable by countable structures. Together with known lower bounds, this implies that these relations are Borel bireducible with isomorphism of countable graphs. We also show that PL-homeomorphism of Heine-Borel simplicial complexes and PL-homeomorphism of PL $n$-manifolds, for every $n$, are classifiable by countable structures.
