关于扭曲同调的可实现性
On Realisability of Twisted Homology
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中文总结 AI 辅助
探讨流形\(X\)扭曲整数系数同调或上同调类由子流形实现的问题，引入扭曲配边类和空间，通过构造参数化波斯特尼科夫塔推导障碍，给出非可定向流形中不可实现整数同调类的例子。
中文摘要 AI 辅助
我们讨论流形\(X\)具有扭曲整数系数的同调或上同调类何时由子流形实现的问题。虽然这个问题本质上是经典的，但提供答案需要来自参数化同伦理论的相对现代的技术。具体而言，我们引入由系数系统扭曲的配边类，然后定义在\(\operatorname{BO}(1)\)上的扭曲托姆空间\(\operatorname{M^\mathrm{tw}O}(n)\)，它在扭曲庞特里亚金 - 托姆构造下作为该配边理论的分类对象。结果，扭曲同调类可实现当且仅当其庞加莱对偶是通过\(\operatorname{BO}(1)\)上的参数化映射\(X \to \operatorname{M^\mathrm{tw}O}(n)\)在\(\operatorname{M^\mathrm{tw}O}(n)\)中扭曲托姆类的像。最后，我们构造\(\operatorname{M^\mathrm{tw}O}(n)\)在\(\operatorname{BO}(1)\)上的参数化波斯特尼科夫塔以推导可实现性的障碍，并通过给出非可定向流形中不可实现整数同调类的首个已知例子得出结论。
英文摘要
We discuss the question of when a homology or cohomology class with twisted integer coefficients of a manifold $X$ is realised by a submanifold. While this question is classical in nature, providing an answer requires relatively modern techniques from parametrised homotopy theory. More specifically, we introduce cobordism classes twisted by a coefficient system and then define a twisted Thom space $\operatorname{M^\mathrm{tw}O}(n)$ over $\operatorname{BO}(1)$, which serves as the classifying object for this cobordism theory under a twisted Pontryagin-Thom construction. As a result, a twisted homology class is realisable if and only if its Poincaré dual is the image of the twisted Thom class in $\operatorname{M^\mathrm{tw}O}(n)$ under a parametrised map $X \to \operatorname{M^\mathrm{tw}O}(n)$ over $\operatorname{BO}(1)$. Finally, we construct the parametrised Postnikov tower of $\operatorname{M^\mathrm{tw}O}(n)$ over $\operatorname{BO}(1)$ to derive obstructions to realisability and conclude by giving the first known examples of non-realisable integer homology classes in non-orientable manifolds.
