Disjoint union makes bordism classes abelian groups

(Diffeomorphic manifolds are cobordant; the class of a diffeomorphism type.) Let $\phi :M\to N$ be a diffeomorphism of closed smooth $n$-manifolds. Then $W:=M\times [0,1]$ with the collars ${\theta }_0(s,x)=(x,s)$ and ${\theta }_1(s,y)=({\phi }^{-1}(y),1+s)$ is a bordism from $M$ to $N$, because ${\theta }_0$ and ${\theta }_1$ are smooth embeddings onto collar neighbourhoods of $M\times \{0\}$ and $M\times \{1\}$ (identified with $N$ by $\phi$) and the boundary parts cover $\mathrm{\partial }W$. If $\phi$ is orientation-preserving between $(M,o)$ and $(N,p)$, orient $W$ by $(-1{)}^n(o\otimes dt)$; by [F3] the incoming face carries $-o$ and the outgoing face carries $p$, so $W$ is an oriented bordism from $(M,o)$ to $(N,p)$. Consequently diffeomorphic closed manifolds, respectively orientation-preserving diffeomorphic closed oriented manifolds, are cobordant.

(Identity.) The empty $n$-manifold is a summand with $M\bigsqcup \mathrm{\varnothing }=M$ and $\mathrm{\varnothing }\bigsqcup M=M$ as smooth manifolds, and it is null-cobordant; hence $[M]+0=[M]=0+[M]$ in both theories.

(Unoriented inverses: exponent two.) Let $M$ be a closed smooth $n$-manifold. Consider $W=M\times [0,1]$ and view its whole boundary $\mathrm{\partial }W=(M\times \{0\})\bigsqcup (M\times \{1\})$ as the incoming part, with no outgoing part: the map $\theta :[0,1)\times (M\bigsqcup M)\to W$ given by $\theta (s,x)=(x,s\mathrm{/}2)$ on the first summand and $\theta (s,x)=(x,1-s\mathrm{/}2)$ on the second has disjoint open images $M\times [0,1\mathrm{/}2)$ and $M\times (1\mathrm{/}2,1]$ whose union is an open neighbourhood of $\mathrm{\partial }W$, is a smooth embedding onto it, and satisfies $\theta (\{0\}\times (M\bigsqcup M))=\mathrm{\partial }W$. Hence $M\bigsqcup M$ is null-cobordant, so $[M]+[M]=[M\bigsqcup M]=0$ in ${\mathrm{\Omega }}_n^O$.

(Associativity.) Let $M,N,P$ be closed smooth $n$-manifolds. The canonical bijection $a:(M\bigsqcup N)\bigsqcup P\to M\bigsqcup (N\bigsqcup P)$ which is the identity on each summand is a diffeomorphism by [F2]; in the oriented theory it preserves the disjoint-union orientations. By step 1.1 the two sides are cobordant (oriented cobordant), so $([M]+[N])+[P]=[M]+([N]+[P])$ by [F1].

(Commutativity.) The canonical bijection $M\bigsqcup N\to N\bigsqcup M$ swapping the summands is a diffeomorphism by [F2] and preserves the disjoint-union orientations; by step 1.1 it gives $[M]+[N]=[N]+[M]$ in both theories.

(Oriented inverses.) Let $(M,o)$ be a closed oriented $n$-manifold and orient $W=M\times [0,1]$ by $(-1{)}^n(o\otimes dt)$; by [F3] the induced boundary orientations are $-o$ on $M\times \{0\}$ and $o$ on $M\times \{1\}$. View the whole boundary as the incoming part with the single collar $\theta$ of step 1.3; the incoming face is $M\bigsqcup (-M)$ with the orientation of the source $o\bigsqcup (-o)$, and the required condition is that the induced orientation equal its negative, namely $(-o)\bigsqcup o$; this is exactly what the two faces carry. Hence $M\bigsqcup (-M)$ is null-cobordant and $[M,o]+[-M]=0$ in ${\mathrm{\Omega }}_n^{SO}$.

(The group axioms.) By [F5], associativity (G1) is step 2.1, the two-sided identity (G2) is step 1.2, and two-sided inverses (G3) are steps 1.3 and 2.3; commutativity is step 2.2. Therefore $({\mathrm{\Omega }}_n^O,+)$ and $({\mathrm{\Omega }}_n^{SO},+)$ are abelian groups for every $n\ge 0$. The construction uses only the supplied smooth structures and collars, so no choice principle is used.