Bordism group - Imported Knowlpedia Content

Definition

Fix $n\geq0$ and a bordism theory, such as unoriented or oriented smooth bordism. The $n$ -dimensional bordism group is the set of bordism-equivalence classes of closed $n$ -manifolds with the chosen structure, with operation

[M]+[N]=[M\sqcup N].

The identity is the class of the empty manifold. In unoriented bordism every class is its own inverse, since $M\sqcup M$ is the boundary of $M\times[0,1]$ . In oriented bordism the inverse of $[M]$ is $[-M]$ , the manifold with reversed orientation. The symmetry of disjoint union makes this an abelian group. Standard notations are $\Omega_n^O$ and $\Omega_n^{SO}$ .

Why the operation descends

Suppose $M_0$ is bordant to $M_1$ through $W$ , and $N_0$ is bordant to $N_1$ through $W'$ . Then $W\sqcup W'$ is a bordism from $M_0\sqcup N_0$ to $M_1\sqcup N_1$ . Hence addition is independent of representatives. Associativity and commutativity follow from canonical diffeomorphisms of disjoint unions, while the empty manifold acts as a unit. These geometric constructions establish the group laws directly Stong, Chapter I.

Grading and multiplication

Taking all dimensions together gives a graded abelian group

\Omega_*^O=\bigoplus_{n\geq0}\Omega_n^O

and similarly in the oriented or other structured theories. Cartesian product defines a graded multiplication $[M][N]=[M\times N]$ , producing a bordism ring when the structures are closed under products. This multiplication is additional structure; it is not the group operation, which remains disjoint union.

Dependence on tangential structure

Changing the allowed structure changes both the equivalence relation and the group. Forgetting orientation induces a homomorphism $\Omega_n^{SO}\to\Omega_n^O$ , but oriented null-bordism is stronger than unoriented null-bordism. Framed, spin, complex, and other bordism groups require the relevant structure to extend over every bordism. The notation $\Omega_n$ is therefore incomplete unless the underlying theory is understood.

Examples

The class of any boundary is zero by definition. In unoriented dimension zero, a closed manifold is a finite set of points, and a compact one-manifold has an even number of boundary points; therefore $\Omega_0^O\cong\mathbb Z/2$ . With orientations, signed point counts give $\Omega_0^{SO}\cong\mathbb Z$ . These examples also exhibit the different inverse conventions.

References

Robert E. Stong, Notes on Cobordism Theory, Princeton University Press, 1968. DOI record. Relevant: Chapter I, cobordism categories and bordism groups.
John Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. Publisher record. Relevant: Appendix B, cobordism.