Nullbordant

Nullbordant (null-bordant) is a term in bordism theory used to describe a closed manifold that bounds a higher-dimensional manifold. Specifically, a closed n-dimensional manifold M is nullbordant if there exists a compact (n+1)-dimensional manifold W with boundary ∂W = M, with the appropriate orientation in the oriented case. Equivalently, the bordism class of M is zero in the oriented bordism group Ω_n^SO.

Consequences and examples: If M is nullbordant, then it bounds and represents the zero element in its bordism group. A basic example is the n-sphere S^n, since S^n = ∂D^{n+1}. If M is nullbordant, the same holds for its orientation reversal −M. The concept extends to products: if M = ∂W, then M×P = ∂(W×P) for any compact manifold P, so nullbordancy is preserved under taking products with suitable manifolds.

Invariants and obstructions: If M is nullbordant, any bordism-invariant must vanish on M. For oriented bordism, invariants such as Pontryagin numbers and the signature vanish on nullbordant manifolds in the appropriate dimensions. Conversely, a nonzero invariant can obstruct nullbordancy and show that a manifold represents a nontrivial element of Ω_n^SO.

Relation to bordism theory: Nullbordant manifolds form the zero class in Ω_n^SO, and the collection of bordism classes across dimensions forms a graded Abelian group. The oriented bordism groups relate to the Thom spectrum MSO, providing a homotopical framework for bordism.