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Do we know, or are there any References that list down complete oriented and unoriented Bordism Group $Ω_{n,O}(pt)$ and $Ω_{n,SO}(pt)$ of points $pt$ for dimensions $n=1,2,...,10$?

Here are some informations I know. And a useful Ref is here.

$$Ω_{1,O}(pt) =0,$$ $$Ω_{2,O}(pt) =\mathbb{Z}/2\mathbb{Z},$$ $$Ω_{3,O}(pt) = 0,$$ $$Ω_{4,O}(pt) =\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z},$$ $$Ω_{5,O}(pt) =\mathbb{Z}/2\mathbb{Z},$$ $$Ω_{6,O}(pt) =?,$$ $$Ω_{7,O}(pt) =?,$$ $$Ω_{8,O}(pt) =?,$$ $$Ω_{9,O}(pt) =?,$$ $$Ω_{10,O}(pt) =?,$$


$$Ω_{1,SO}(pt) =?,$$ $$Ω_{2,SO}(pt) =?,$$ $$Ω_{3,SO}(pt) = ?,$$ $$Ω_{4,SO}(pt) =?,$$ $$Ω_{5,SO}(pt) =\mathbb{Z}/2\mathbb{Z}?$$ $$Ω_{6,SO}(pt) =?,$$ $$Ω_{7,SO}(pt) =?,$$ $$Ω_{8,SO}(pt) =?,$$ $$Ω_{9,SO}(pt) =?,$$ $$Ω_{10,SO}(pt) =?,$$

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    $\begingroup$ The unoriented case is completely determined by Thom, the oriented case is by Wall in the reference you give (p.293). $\endgroup$
    – user43326
    Sep 16, 2017 at 18:48

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Unoriented cobordism: can be read off from the structure of the unoriented cobordism ring (calculated in Thom's thesis): $\Omega_6^O = (\mathbb Z/2)^3$, $\Omega_7^O = \mathbb Z/2$, $\Omega_8^O = (\mathbb Z/2)^5$, $\Omega_9^O = (\mathbb Z/2)^3$, and $\Omega_{10}^O = (\mathbb Z/2)^8$.

Oriented cobordism: in Milnor and Stasheff, Characteristic Classes, end of §17, p. 203. $\Omega_1^{SO} = 0$, $\Omega_2^{SO} = 0$, $\Omega_3^{SO} = 0$, $\Omega_4^{SO} = \mathbb Z$, $\Omega_5^{SO} = \mathbb Z/2$, $\Omega_6^{SO} = \Omega_7^{SO} = 0$, $\Omega_8^{SO} = \mathbb Z^2$, $\Omega_9^{SO} = (\mathbb Z/2)^2$, and $\Omega_{10}^{SO} = \mathbb Z/2$.

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  • $\begingroup$ Thanks, +1, are you saying that my above results from dimensions 1 to 5 coincide for unoriented cobordism and oriented cobordism? And my answers above were correct for 1 to 5? $\endgroup$
    – wonderich
    Sep 16, 2017 at 18:05
  • $\begingroup$ Yes, everything you stated in your question is correct. I didn't see that you were missing the lower-dimensional oriented cobordism groups; I'll go ahead and edit them into my answer. $\endgroup$ Sep 16, 2017 at 18:44
  • $\begingroup$ Thanks, +1, could you explain the unmatching relations between the $O$ and $SO$ for two sets of bordism groups? What is the intuition behind? Thanks! $\endgroup$
    – wonderich
    Sep 18, 2017 at 20:51
  • $\begingroup$ @wonderich I don't know what the unmatching relations are, and Google didn't help me at all, so you'll probably get more luck asking a separate question with a reference. $\endgroup$ Sep 18, 2017 at 21:13

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