Wiki. “Whitehead 塔” [Whitehead塔]
Wiki. “Whitehead 塔” [Whitehead塔]
陈述
对任意连通空间 $X$, 存在一列纤维化 $\cdots\to P_2\to P_1$, 以及相容的映射 $X\to P_i$, $$ \begin{array} {ccccc} &&\vdots\\ &&\downarrow\\ &&W_3&\leftarrow & K(\pi_3(X),3)\\ &\swarrow&\downarrow\\ &&W_2&\leftarrow & K(\pi_2(X),2)\\ &\swarrow&\downarrow\\ X & \leftarrow & W_1 \end{array} $$ 满足
- $W_n$ 的不超过 $n$ 阶同伦群平凡, 即 $W_n$ $n$-连通;
- 映射 $W_n\to X$ 在超过 $n$ 阶同伦群上为同构;
- 对于 $n\geq 2$, 映射 $W_n\to W_{n-1}$ 的纤维为 Eilenberg–MacLane 空间 $K(\pi_n(X),n-1)$.
nLab: The Whitehead tower of a pointed homotopy type $X$ is an interpolation of the point inclusion $* \to X$ by a sequence of homotopy types $*\to\cdots\to W_2\to W_1\to X$ that are obtained from right to left by removing homotopy groups from below.
nLab 指出 $W_n$ 是同伦纤维 $$ \begin{array} {ccc} W_n & \to & *\\ \downarrow && \downarrow\\ X & \to & P_{n+1}. \end{array} $$
例
$$ \cdots\to \text{String}(n)\to\text{Spin}(n)\to SO(n) \,(\to O(n)) $$