AI 中文总结
本文研究非球面$PD_4$-对的实现问题，给出其可由1-柄和映射柱组装的充要条件，明确不同群类的实现情况及有限2维$K(π,1)$复形对应的实现特性。
AI 中文摘要
本文是三篇相关预印本的第三篇。我们研究了哪些群$π$和$PD_3$-复形$Y$可由满足$X$非球面且$π_1X\congπ$的$PD_4$-对$(X,Y)$实现，并证明当且仅当$H^2(π;\bZπ)=0$时，这类对可由$(D^4,S^3)$和群的$PD_4$-对通过添加1-柄以及边界分支$N$上的$\bZπ_1N$-同调等价的映射柱组装而成（这包含所有边界分支为$π_1$-单射的此类对，以及所有$π$为自由群的此类对，但不包含任何上同调维数$c.d.π=2$的对；添加1-柄包含连通和操作）。若存在有限2维$K(π,1)$-复形，则$π$可由某一此类对$(X,Y)$实现，但除$π$为$PD_2$-群的情形外，不存在类似于群的$PD_4$-对的明显构造基块。
英文摘要
This is the third of three related preprints. We consider here which groups $π$ and $PD_3$-complexes $Y$ are realised by $PD_4$-pairs $(X,Y)$ with $X$ aspherical and $π_1X\congπ$, and show that such a pair may be assembled from $(D^4,S^3)$ and $PD_4$-pairs of groups, by adding 1-handles and mapping cylinders of $\mathbb{Z}π_1N$-homology equivalences over boundary components $N$, if and only if $H^2(π;\mathbb{Z}π)=0$. (This includes all such pairs with $π_1$-injective boundary components and all with $π$ a free group, but none with $c.d.π=2$. Adding a 1-handle includes connected sum.) If there is a finite 2-dimensional $K(π,1)$-complex then $π$ is realisable by some such pair $(X,Y)$, but there are no obvious building blocks analogous to $PD_4$-pairs of groups, except for when $π$ is a $PD_2$-group.
CommentsSee also arXiv: 2501.125092 and arXiv:2501.12512
