Pontryagin–Thom theorem

Theorem 10.3.6 (Pontryagin–Thom theorem). For every cell complex \(X\) and every \(n \in \Z \), the Pontryagin–Thom transformation is a natural isomorphism \[ \Omega _n^B(X) \xrightarrow {\cong } \pi _n(MB\otimes \S [X]). \] In particular, the groups \(\Omega _n^B(X)\) are the absolute groups of the generalized homology theory represented by \(MB\).