AI 中文总结
本文在A-同伦理论中构造了具有任意指定同伦群的图，证明了相对A-同伦群与经典相对同伦群的等同，并建立了离散Blakers-Massey定理及Postnikov系统和Whitehead塔。
AI 中文摘要
本文研究无向图的离散同伦理论。更确切地说，我们考虑由Barcelo、Kramer、Laubenbacher和Weaver首先定义的$A$-同伦群。对于每个度数$n$和每个群$G$（当$n\geq 2$时$G$为阿贝尔群），我们构造一个图，其第$n$个$A$-同伦群同构于$G$，而其他所有$A$-同伦群均为平凡群。利用这些构造，我们证明：对于每个无穷群序列$G_1,G_2,\ldots$（其中$G_i$在$i\geq 2$时为阿贝尔群），存在一个图$X$，使得$X$的第$i$个$A$-同伦群对所有$i$都同构于$G_i$。在此过程中，我们将相对$A$-同伦群与相对经典同伦群等同起来，并证明了Blakers-Massey定理的离散版本。随后，我们构造了部分Postnikov系统和（完全的）Whitehead塔。
英文摘要
In this paper we study a discrete homotopy theory for undirected graphs. More precisely, we consider $A$-homotopy groups, first defined in the work of Barcelo, Kramer, Laubenbacher and Weaver. For every degree $n$ and every group $G$ (Abelian if $n\geq 2$), we construct a graph whose $n$-th $A$-homotopy group is isomorphic to $G$ and whose other $A$-homotopy groups all vanish. Using these, we show that for every infinite sequence of groups $G_1,G_2,\ldots$, with $G_i$ Abelian for $i\geq 2$, there is a graph $X$ such that the $i$-th $A$-homotopy group of $X$ is isomorphic to $G_i$ for all $i$. Along the way, we identify relative $A$-homotopy groups with relative classical homotopy groups, and prove a discrete version of the Blakers-Massey theorem. We then construct partial Postnikov systems and (full) Whitehead towers.
