低维庞加莱对偶空间的约化性
代数拓扑
2021-11-09 v2 几何拓扑
摘要
我们讨论庞加莱对偶复形 X，以及在其维数不超过 4 时其 Spivak 法丛是否容许约化到向量丛的问题。我们证明在维数小于 4 时这样的约化总是存在，而在维数为 4 时只要 X 是可定向的便存在这样的约化。在非可定向情形，Hambleton--Milgram 给出了不可约化的反例。
引用
@article{arxiv.1711.08179,
title = {Reducibility of low dimensional Poincar\'e duality spaces},
author = {Markus Land},
journal= {arXiv preprint arXiv:1711.08179},
year = {2021}
}
备注
v2: 26 pages, largely rewritten. v1 has a mistake (Lemma 3.2 is wrong, this is explained in Proposition 3.9 and the warning preceding 3.9). v2 now contains a new proof of the main theorem. The approach of the previous version is still explained in section 3. In addition there is an appendix about PD complexes from an infinity-categorical point of view