AI 中文总结
该研究构造奇异同调的微分细化，定义微分卡积等结构并证明相关唯一性，推广至多种系数版本及有理偶同调理论，还定义纤维丛的微分映射，最终概述其在数学物理学中的潜在应用。
AI 中文摘要
我们首先构造奇异同调的微分细化，验证其满足与上同调对应公理对偶的公理。接着定义微分卡积，进而导出庞加莱对偶性，同时证明微分同调和卡积的本质唯一性。此外，我们构造相对、非紧及局部系数版本，从而可对任意带或不带边界的光滑流形陈述庞加莱和莱夫谢茨对偶性。之后，我们为任何有理偶同调理论开发类似的微分细化，使其具备相同性质。我们还为纤维丛定义“微分映射”，它与上同调中的积分映射对偶。最后，我们概述该理论在数学物理学中的若干潜在应用。
英文摘要
We first construct the differential refinement of singular homology, verifying that it satisfies the axioms dual to their cohomological counterparts. Then, we define the differential cap product, leading to Poincaré duality. The essential uniqueness of both differential homology and the cap product is proven. Moreover, we construct the relative, non-compact, and local coefficient versions, so that we can state Poincaré and Lefschetz dualities for every smooth manifold with our without boundary. Afterwards, we develop the analogous refinement of any rationally-even homology theory, with the same properties. We also define a "differentiation map" for fibre bundles, dual to the integration map in cohomology. We conclude by sketching some possible applications of this theory in mathematical physics.
Comments55 pages, no figures
