Oriented bordism classes over a space form an abelian group

The sum is well-defined: if $F_i:W_i\to X$ are bordisms witnessing equivalences of pairs for $i=0,1$, then $W_0\bigsqcup W_1$ with map $F_0\bigsqcup F_1$ is a bordism witnessing the equivalence of the disjoint unions. Boundary orientations are inherited componentwise.

Disjoint union is associative and commutative up to the orientation-preserving diffeomorphisms that rebracket or permute its components; composing the maps with these diffeomorphisms preserves the bordism classes. The empty manifold with its unique map is a two-sided identity.

The cylinder $M\times [0,1]$ with map $f\circ {\mathrm{pr}}_M$ has boundary $(M,f)\bigsqcup (-M,f)$ under the product orientation convention. Thus $(-M,f)$ is a two-sided inverse of $(M,f)$, so the operation is an abelian group operation.