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....
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2 Answers
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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".
answered Jun 1, 2010 at 2:43
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$\begingroup$ Thanks, I looked up the reference and it was very helpful... I would still like to know, and sorry and if this is now redundant, if not having any but 2-primary torsion now implies all torsion being like $\mathbb{Z}_2$ or could you have $\mathbb{Z}_{2^n}$ for $n \neq 1$? $\endgroup$– JuanSOSCommented Jun 1, 2010 at 4:14
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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.
