Thom spectrum

Definition

For $n \ge 0$, let $\gamma_n$ be the universal $n$-plane bundle over the classifying space $BO(n)$, and let $\varepsilon$ be the trivial line bundle. Let $(B_n, f_n, g_n)_{n \ge 0}$ be a sequence of spaces $B_n$ with maps $f_n \colon B_n \to BO(n)$ and $g_n \colon B_n \to B_{n+1}$ such that $f_{n+1} \circ g_n$ classifies the vector bundle $\xi_n \oplus \varepsilon$, where $\xi_n = f_n^* \gamma_n$. Write $\operatorname{Th}(\xi)$ for the Thom space of a bundle $\xi$. The Thom spectrum of this sequence is the spectrum $M$ with spaces $M_n = \operatorname{Th}(\xi_n)$. Let $\Sigma$ be the reduced suspension. The structure maps of $M$ are the composites $\Sigma \operatorname{Th}(\xi_n) \cong \operatorname{Th}(\xi_n \oplus \varepsilon) \to \operatorname{Th}(\xi_{n+1})$. Here the second map is induced by the bundle map $\xi_n \oplus \varepsilon \to \xi_{n+1}$ that covers $g_n$.