Geometric B-bordism

Definition 10.3.3 (\(B\)-bordism). Let \(X\) be a topological space and let \(n \geq 0\). A \(B\)-bordism cycle of degree \(n\) over \(X\) is a triple \((M,f,\sigma )\) consisting of a closed smooth \(n\)-manifold \(M\), a continuous map \(f\colon M \to X\), and a normal \(B\)-structure \(\sigma \) on \(M\). Two cycles \((M_0,f_0,\sigma _0)\) and \((M_1,f_1,\sigma _1)\) are bordant if there exist compact smooth \((n+1)\)-manifolds \(W_0\) and \(W_1\) with normal \(B\)-structures, together with maps \(F_i\colon W_i \to X\), and a diffeomorphism \[ M_0 \amalg \partial W_0 \; \cong \; M_1 \amalg \partial W_1 \] of manifolds with normal \(B\)-structures over \(X\).

We write \(\Omega _n^B(X)\) for the set of bordism classes. Disjoint union defines an addition, with the empty manifold as zero. For \(n<0\), we set \(\Omega _n^B(X)=0\).