Bordism is an equivalence relation

Remark 10.3.4. We used the formulation of Switzer (1975), Chapter 12. Note it is an equivalence relation: reflexivity takes \(W_0 = W_1 = \varnothing \), symmetry is built into the statement, and transitivity follows by taking disjoint unions. It is also visibly additive, so \(\Omega _n^B(X)\) is a commutative monoid in which every class of the form \([\partial W]\) vanishes.

Inverses are obtained via the cylinder construction. Pulling back \(\sigma \) along the projection equips \(M \times [0,1]\) with a normal \(B\)-structure whose restriction to one end is \(\sigma \); write \(\sigma '\) for the structure induced on the second end. Then \[ (M,\sigma ) \amalg (M,\sigma ') \cong \partial (M \times [0,1]) \] as manifolds with normal \(B\)-structures, and hence \[ [(M,f,\sigma )] + [(M,f,\sigma ')] \; = \; 0 \] in \(\Omega _n^B(X)\). Thus \(\Omega _n^B(X)\) is an abelian group. For \(B = \bB O\) there is no additional structure, so \(\sigma ' = \sigma \) and every class is its own inverse; for \(B = \bB SO\) the structure \(\sigma '\) is the reversed orientation, and the groups are not \(2\)-torsion in general.