Lemma 10.2.21. Let \(X\) be a paracompact Hausdorff space of the homotopy type of a cell complex and let \(E \to X\) be a rank \(n\) vector bundle. Then there is an isomorphism \(\xi _{[E]} \cong \Sigma ^{\infty }(\xi _E)\) of stable spherical fibrations, and hence an isomorphism \(\th ([E]) \cong \th (E)\). More generally, for every \(N \geq 0\) we have \(\xi _{[E]-[\ul {\R ^N}]} \cong \Sigma ^{\infty }(\xi _E)[-N]\) and therefore \[ \th \big ([E]-[\ul {\R ^N}]\big ) \quad \cong \quad \th (E)[-N] \quad \cong \quad \Sigma ^{\infty -N}\Th (E). \]
Proof. By Construction 10.2.18, the restriction of \(J\) to the rank \(n\) component is \(\Sigma ^{\infty }J_n^{\mathrm {un}}\). The first isomorphism is therefore precisely Lemma 10.2.19, and passing to colimits gives \(\th ([E]) \cong \Sigma ^{\infty }\Th (\xi _E) = \th (E)\), since \(\Sigma ^{\infty }\) preserves colimits.
The virtual vector bundle \([\ul {\R ^N}]\) is constant with value \(\R ^N\), so \(\xi _{[\ul {\R ^N}]}\) is constant with value \(\Sigma ^{\infty }S^{\R ^N} \cong \S [N]\). By Remark 10.2.17 we thus obtain \[ \xi _{[E]-[\ul {\R ^N}]} \cong \Sigma ^{\infty }(\xi _E) \otimes \S [-N] \cong \Sigma ^{\infty }(\xi _E)[-N], \] and the remaining claims follow because shifts commute with colimits. □
Generated from the authoritative LaTeX source.