Unoriented and oriented bordism groups

Fix $n\ge 0$. By the cobordism equivalence relation (Smooth cobordism is an equivalence relation) the closed smooth $n$-manifolds are partitioned into cobordism classes (Unoriented smooth cobordism of closed manifolds). Let ${\mathrm{\Omega }}_n^O:=\{[M]:M\text{ a closed smooth }n\text{-manifold}\}$ be the set of these classes in the bounded model convention below, and let ${\mathrm{\Omega }}_n^{SO}:=\{[Q,o]:(Q,o)\text{ a closed oriented smooth }n\text{-manifold}\}$ be the set of oriented cobordism classes (Oriented smooth cobordism).

Set-size convention. For each $n$ use the set of all closed smooth $n$-manifold structures on subsets of $U_n={\mathbb{R}}^n\times \mathbb{N}$, and in the oriented theory include the orientation datum, before taking the quotient by cobordism. Every closed smooth $n$-manifold has such a model: compactness gives a finite chart cover $(V_j,{\phi }_j{)}_{j<k}$, and the map $x\mapsto ({\phi }_j(x),j)$, with $j$ the least index for which $x\in V_j$, is injective into $U_n$. Transport the topology, maximal atlas, and supplied orientation along this injection; its image is diffeomorphic to the original manifold. This uses only a finite chart cover, not a choice of a model for every manifold simultaneously. Two transported models are diffeomorphic, and a cylinder of Cylinders give reflexivity of cobordism with outgoing collar composed with a diffeomorphism's inverse shows they are cobordant (orientation-preservingly in the oriented case). Thus $[M]$ means the unique class of any such model. Products and disjoint unions are returned to this set of models in the same way; their class is independent of the transport.

The operation. Disjoint union of manifolds defines operations $[M]+[N]:=[M\bigsqcup N]\text{on }{\mathrm{\Omega }}_n^O,[Q,o]+[R,p]:=[Q\bigsqcup R,o\bigsqcup p]\text{on }{\mathrm{\Omega }}_n^{SO},$ where $o\bigsqcup p$ is the orientation of the disjoint union whose restriction to each summand is the given orientation, and finite disjoint unions carry the transported component atlases (Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds); for summands with boundary the same construction uses half-space charts. Finite union preserves compactness and combines finitely many countable bases, so this boundary extension requires no countable choice. The class of the empty $n$-manifold is the displayed zero $0$: it is null-cobordant and is a two-sided identity for the operation, since $M\bigsqcup \mathrm{\varnothing }$ is canonically diffeomorphic to $M$ (Null-cobordant closed manifolds).

Well-definedness. The operations are independent of the chosen representatives. If $W_i$ is a bordism from $M_i$ to $M_i^{\mathrm{\prime }}$ for $i=0,1$, then the disjoint union $W_0\bigsqcup W_1$ carries the canonical smooth structure of a compact $(n+1)$-manifold with boundary $\mathrm{\partial }(W_0\bigsqcup W_1)=(\mathrm{\partial }{W}_0)\bigsqcup (\mathrm{\partial }{W}_1)$, the collars ${\theta }_0^0\bigsqcup {\theta }_0^1$ and ${\theta }_1^0\bigsqcup {\theta }_1^1$ are smooth embeddings onto collar neighbourhoods of the boundary parts, and the decomposition of the boundary into the two parts is again open and closed; hence $W_0\bigsqcup W_1$ is a bordism from $M_0\bigsqcup M_1$ to $M_0^{\mathrm{\prime }}\bigsqcup M_1^{\mathrm{\prime }}$. In the oriented case, orienting the disjoint union by the given orientations makes the disjoint union of the oriented bordisms an oriented bordism between the disjoint unions, because the induced boundary orientation is computed componentwise and each summand carries its required sign. Thus $[M_0\bigsqcup M_1]=[M_0^{\mathrm{\prime }}\bigsqcup M_1^{\mathrm{\prime }}]$ whenever $[M_i]=[M_i^{\mathrm{\prime }}]$, and similarly in the oriented theory. The class of a finite disjoint union is computed from the canonical smooth structure on disjoint unions; no further choice is made.

Deferred axioms and the forgetful map. That these operations satisfy the group axioms (associativity, commutativity, identity and inverses) is not asserted here: it is proved in the following theorem, together with the finiteness of the inverse in the unoriented theory. The forgetful map ${\mathrm{\Omega }}_n^{SO}⟶{\mathrm{\Omega }}_n^O,[Q,o]⟼[Q],$ sends the oriented class of a closed oriented $n$-manifold to its underlying unoriented cobordism class; it is well defined because an oriented bordism between two oriented manifolds is in particular a bordism between their underlying manifolds, so oriented cobordant manifolds are cobordant.