nullcobordism

Nullcobordism is a concept in geometric topology describing when a closed manifold bounds a higher-dimensional compact manifold. Specifically, an n-dimensional closed manifold M is nullcobordant if there exists an (n+1)-dimensional manifold W with boundary ∂W = M. When additional structures are present—such as orientation, spin, or a complex structure—the requirement is that W and its boundary carry compatible structures, and nullcobordism is considered within the corresponding bordism group Ω_n^G.

Equivalently, a manifold represents the zero element in its bordism group. In oriented bordism, for example, M is nullcobordant iff its oriented bordism class [M] is zero in Ω_n^SO. Being nullcobordant means M can be realized as the boundary of some compact manifold, and this property is independent of the particular bounding manifold chosen.

Examples and obstructions are central to practice. The standard n-sphere S^n is nullcobordant because it is the boundary of the (n+1)-disk D^{n+1}. Conversely, in a given bordism theory, not every manifold is nullcobordant; many invariants detect nonbounding. In unoriented bordism, all Stiefel-Whitney numbers must vanish for nullcobordant manifolds. In oriented bordism, all Pontryagin numbers vanish for nullcobordant manifolds, and the signature (the L-genus) must be zero in dimensions divisible by four. Thus nullcobordism provides a practical criterion for when a manifold bounds a higher-dimensional one within a chosen structural framework.