Cartesian product makes bordism a graded ring

(The product is well defined.) Let $W_1$ be a bordism from $M_0$ to $M_1$ of dimension $m+1$ and $W_2$ a bordism from $N_0$ to $N_1$ of dimension $n+1$. By [F2] the products $W_1\times N_0$ and $M_1\times W_2$ are compact smooth manifolds with boundary, with boundary decompositions $(M_0\times N_0)\bigsqcup (M_1\times N_0)$ and $(M_1\times N_0)\bigsqcup (M_1\times N_1)$, and with the collars ${\theta }_i\times {\mathrm{id}}_{N_0}$ and ${\mathrm{id}}_{M_1}\times {\theta }_j^{\mathrm{\prime }}$ onto the corresponding parts. In the oriented case orient $W_1\times N_0$ by its product orientation and $M_1\times W_2$ by $(-1{)}^m$ times its product orientation. The first piece has incoming orientation $-o_{M_0}\otimes o_{N_0}$ and outgoing orientation $o_{M_1}\otimes o_{N_0}$. The product-boundary formula [F2] contributes a further $(-1{)}^m$ to both faces of the second piece, cancelling its selected orientation factor; thus those faces carry $-o_{M_1}\otimes o_{N_0}$ and $o_{M_1}\otimes o_{N_1}$. In particular the two induced orientations at the common component $M_1\times N_0$ are opposite, and the outer faces have the required incoming and outgoing product orientations. Gluing along the common collars by [F1] gives a bordism from $M_0\times N_0$ to $M_1\times N_1$, oriented when both given bordisms are. Hence cobordant representatives give cobordant products: the product is well defined on ${\mathrm{\Omega }}_m\times {\mathrm{\Omega }}_n$ in both theories.

(Diffeomorphisms give cobordisms.) If $\phi :M\to M^{\mathrm{\prime }}$ is a diffeomorphism of closed smooth $m$-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 $M^{\mathrm{\prime }}$; if $\phi$ is orientation-preserving between $(M,o)$ and $(M^{\mathrm{\prime }},o^{\mathrm{\prime }})$, the orientation $(-1{)}^m(o\otimes dt)$ of [F3] makes it an oriented bordism. Thus diffeomorphic closed manifolds represent the same class, and orientation-preserving diffeomorphic closed oriented manifolds represent the same oriented class.

(Biadditivity, associativity, distributivity, unit.) Let $M,M^{\mathrm{\prime }}$ be closed $m$-manifolds and $N$ a closed $n$-manifold. The canonical diffeomorphism $(M\bigsqcup M^{\mathrm{\prime }})\times N\to (M\times N)\bigsqcup (M^{\mathrm{\prime }}\times N)$ is orientation-preserving in the oriented theory, so by step 1.2 and [F4] $([M]+[M^{\mathrm{\prime }}])\cdot [N]=[M][N]+[M^{\mathrm{\prime }}][N]$; the same argument in the second variable gives biadditivity (distributivity over disjoint union). The canonical diffeomorphism $(M\times N)\times P\to M\times (N\times P)$ is orientation-preserving for the iterated product orientations and gives $([M][N])[P]=[M]([N][P])$. Let $\mathrm{p}\mathrm{t}$ carry the positive sign in the oriented theory. Then $\mathrm{p}\mathrm{t}\times M=M$ and $M\times \mathrm{p}\mathrm{t}=M$ as smooth (oriented) manifolds, so $[\mathrm{p}\mathrm{t}][M]=[M]=[M][\mathrm{p}\mathrm{t}]$; the product is graded in the sense $[M][N]\in {\mathrm{\Omega }}_{m+n}$.

(Graded commutativity; the unoriented commutative case.) Let $\tau :M\times N\to N\times M$, $\tau (x,y)=(y,x)$, be the transposition. Its differential interchanges the $m$ tangent directions of the first block with the $n$ of the second, so it multiplies the ordered determinant by the sign of that permutation, which is $(-1{)}^{mn}$: $\tau$ is orientation-preserving from $(M\times N,(-1{)}^{mn}(o_M\otimes o_N))$ to $(N\times M,o_N\otimes o_M)$. By step 1.2 the two oriented classes agree, so $[M][N]=[(M\times N,o_M\otimes o_N)]=(-1{)}^{mn}[(N\times M,o_N\otimes o_M)]=(-1{)}^{mn}[N][M]$ in ${\mathrm{\Omega }}_{m+n}^{SO}$. In the unoriented theory $\tau$ is a diffeomorphism, so $[M\times N]=[N\times M]$ and the product is commutative on the nose.

(Ring structure and the forgetful map.) By steps 2.1 and 2.2 the direct sum ${\mathrm{\Omega }}_{\ast }^O$ is a nonnegatively graded commutative ring with unit $[\mathrm{p}\mathrm{t}]$ in the sense of [F5], and ${\mathrm{\Omega }}_{\ast }^{SO}$ is a graded-commutative ring with the same unit and the Koszul sign rule; associativity, biadditivity, distributivity over the group operation, the grading and the unit are the assertions proved in step 2.1, and the commutativity statements are step 2.2. The forgetful map sends $[M,o]+[N,p^{\mathrm{\prime }}]$ to $[M\bigsqcup N]$ and $[M,o]\cdot [N,p^{\mathrm{\prime }}]$ to the underlying class of $(M\times N,o\otimes p^{\mathrm{\prime }})$, which is $[M\times N]=[M]\cdot [N]$, and it preserves the unit; hence it is a ring homomorphism by [F5]. Everything is built from products of manifolds and supplied collars, so no choice principle is used.