Question

Hey everybody! I was wondering if anybody had available the calculation of the Oriented cobordism groups in dimensions higher than 10? Or if anybody knew if there is another kind of torsion beside 2-torsion in them? (e.g. I know that $\Omega^5$ is $\mathbb{Z}_2$, is there a group with n-torsion with n distinct from 2?). Thanx, Refferences are also appreciated....

Answer 1

There is no torsion other than 2-primary torsion in the oriented bordism ring. One has that after inverting 2, the oriented bordism ring is a polynomial algebra on generators in degrees which are multiples of 4: $$ \Omega^{SO}_*[1/2] = \mathbb{Z}[1/2, x_4, x_8, x_{12}, \ldots] $$ If I remember correctly, this (and the answers to many bordism-related questions) can be found in Stong's "Notes on cobordism theory".

Answer 2

I would also recommend Wall's Determination of the cobordism ring as a more primary source, it also contains the fact that all torsion is of order 2.