AI 中文总结
该研究针对有限类型rack的分类映射，证明了其同伦群相关的零化性质，确定了特定同调群，并给出在辛与Alexander结构上的低维应用。
AI 中文摘要
我们研究有限类型rack X的分类映射$c\colon BX\to K(\As(X),1)$，设$t = \Type(X)$，证明当X连通时，对所有$n\ge2$有$t c_{n*}=0$；不要求连通性时，对挠子群$\Tor H_n^\mathbb{R}(X)$有$t^{n-1}c_{n*}=0$。对有限rack，在我们的符号约定下，n次有理化分类映射为轨道模的n重张量积到其n次外幂的典范投影乘以$(-1)^n$；对任意有限类型rack，我们确定$H_2^{\mathrm{gr}}(\As(X);\mathbb{Z}[1/t])$，还推导了其在辛结构和Alexander结构上的低维应用。
英文摘要
We study the classifying map $c\colon BX\to K(\As(X),1)$ of a rack $X$ of finite type. Let $t = \Type(X)$. We prove that $t c_{n*}=0$ for every $n\ge2$ when $X$ is connected, and that $t^{n-1}c_{n*}=0$ on the torsion subgroup $\Tor H_n^\mathbb{R}(X)$ without any connectedness assumption. For a finite rack, under our sign conventions, the rationalized classifying map in degree $n$ is given by $(-1)^n$ times the canonical projection from the $n$-fold tensor power of the orbit module to its $n$-th exterior power. For an arbitrary rack of finite type, we determine $H_2^{\mathrm{gr}}(\As(X);\mathbb{Z}[1/t])$. We also derive low-dimensional applications to symplectic and Alexander structures.
Comments20 pages, Key wards: rack, quandle, rack space, classifying map, associated group, rack homology, group homology, type of a rack
