 Witt bordism classification

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.