Bordism groups here are geometric, not generalized homology constructions

The sets ${\mathrm{\Omega }}_n^O$ and ${\mathrm{\Omega }}_n^{SO}$ defined on this page are geometric: their elements are bordism classes of closed smooth manifolds, their operation is disjoint union, and their product (taken up later on this page) is the Cartesian product of manifolds (Unoriented and oriented bordism groups). Every construction on this page is a statement about manifolds, bordisms and their boundary data.

This page constructs no Thom spectrum, no ring spectrum and no generalized homology theory, and it asserts no excision, suspension or Mayer-Vietoris property for bordism. In particular the identification of these groups with stable homotopy groups of Thom spectra, for instance ${\mathrm{\Omega }}_n^O\cong {\pi }_n(MO),{\mathrm{\Omega }}_n^{SO}\cong {\pi }_n(MSO),$ together with the Pontryagin-Thom construction, the Thom transversality theorem and the bordism homology axioms, belongs to algebraic topology and to later pages of this library; the cited sources prove those statements, but nothing here depends on them. The remark fixes the seam so that consumers do not read a spectrum-level or homology-theoretic claim into the geometric definitions of this page.