Witt bordism (${\Omega }_n^{\mathrm{Witt}}$)

Two closed oriented rational Witt spaces are Witt bordant if their signed disjoint union is the collared boundary of a compact oriented Witt space with collared boundary. Disjoint union gives the bordism group ${\Omega }_n^{\mathrm{Witt}}$.

Every odd-dimensional closed oriented Witt space bounds its closed cone: the new vertex has odd-dimensional link, so introduces no Witt obstruction. For a closed oriented two-dimensional pseudomanifold, normalize to a surface, fill each surface component by a handlebody, and attach a collar of the original space using the finite normalization map. Along the singular collar arcs the links are unions of circles; at their interior starting points the links are wedges of two-spheres. These links have zero middle intersection homology, so the construction is a three-dimensional Witt null-bordism.

For $k\ge 1$, the rational middle intersection pairing induces an isomorphism ${\Omega }_{4k}^{\mathrm{Witt}}\cong W(\mathbb{Q})$. The restriction $k>0$ matters: the degree-zero group is $\mathbb{Z}$. In positive dimension congruent to zero modulo four, a space is Witt null-bordant precisely when its middle intersection form is metabolic.

A nontrivial product of positive-dimensional closed oriented Witt spaces of total dimension four has metabolic rational middle intersection form. For dimensions $1+3$, the two Künneth summands in degree two pair only with one another, so either is a half-dimensional isotropic subspace. For dimensions $2+2$, the outside degree-two summands pair with one another, and the $I{H}^1\otimes I{H}^1$ summand has the tensor pairing of two symplectic forms. A Lagrangian in the first symplectic factor tensored with the entire second factor is a half-dimensional isotropic subspace of that summand. Adding one outside summand gives a Lagrangian in the complete middle form. The Witt bordism classification then makes the product null-bordant.