Type Annihilation for Classifying Maps of Rack Spaces

Abstract

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.