Normal B-structure

Definition 10.3.1 (Normal \(B\)-structure). Let \(M\) be a compact smooth \(n\)-manifold. Its stable normal bundle \(\ul {\R ^n}-T_M\) is a virtual vector bundle of rank \(0\), with classifying map \(\nu _M\colon M\to \bB O\). A normal \(B\)-structure on \(M\) is a lift of this map along \(\phi \):

Concretely, this consists of a map \(\sigma \colon M \to B\) together with a homotopy \(\phi \circ \sigma \simeq \nu _M\). If \(M\) has boundary, then the stable normal bundle of \(\partial M\) is identified with the restriction of the stable normal bundle of \(M\) using the outward normal direction. Consequently, a normal \(B\)-structure on \(M\) restricts to one on \(\partial M\). A diffeomorphism of manifolds with normal \(B\)-structures is understood to be equipped with an identification in the anima of such lifts.