using a cobordism and finding an embedding to some which gives a homotopy class of maps to the Thom space defined below. Showing the isomorphism of requires a little more work. Definition of Thom spectrum By definition, the Thom spectrum is a sequence of Thom spaces where we wrote for the universal vector bundle of rank n. The sequence forms a spectrum. A theorem of Thom says that is the unoriented cobordism ring; the proof of this theorem relies crucially on Thom’s transversality theorem. The lack of transversality prevents from computing cobordism rings of, say, topological manifolds from Thom spectra. See also Cobordism Cohomology operation Steenrod problem Hattori–Stong theorem Notes References A classic reference for differential topology, treating the link to Poincaré duality and the Euler class of Sphere bundles External links http://ncatlab.org/nlab/show/Thom+spectrum Akhil Mathew's blog posts: https://amathew.wordpress.com/tag/thom-space/ Algebraic topology Characteristic classes