Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)

Question

The classification of oriented compact smooth manifolds up to oriented cobordism is one of the landmarks of 20th century topology. The techniques used there form the part of the foundations of differential topology and stable homotopy theory.

It is a popular knowledge to find the oriented bordism groups $\Omega_d^{SO}$ in dimensions lower or equal to 8, http://www.map.mpim-bonn.mpg.de/Oriented_bordism, which we have $$\Omega_0^{SO}=\mathbb{Z}$$ $$\Omega_1^{SO}=0$$ $$\Omega_2^{SO}=0$$ $$\Omega_3^{SO}=0$$ $$\Omega_4^{SO}=\mathbb{Z}$$ $$\Omega_5^{SO}=\mathbb{Z}/2$$ $$\Omega_6^{SO}=0$$ $$\Omega_7^{SO}=0$$ $$\Omega_8^{SO}=\mathbb{Z} \oplus \mathbb{Z}$$ $$\Omega_9^{SO}=\mathbb{Z}/2 \oplus \mathbb{Z}/2$$ $$\Omega_{10}^{SO}=\mathbb{Z}/2$$ $$\Omega_{11}^{SO}=\mathbb{Z}/2$$

do we happen to know other dimensions in the literature? For any $d \leq 28$? Also are the manifold generators already known for those $d \leq 28$? Are manifold generators systematically constructible? References (precisely which pages) are surely welcome! Thank you in advance.

Answer 1

Most of the main results needed for this calculation can be found in Wall's paper "Determination of the oriented cobordism ring", but this note by Gwynne might be helpful to express this in more modern language. Here is the exective summary:

• All of the homotopy groups are a direct sum $\Bbb Z^r \oplus \Bbb (Z/2)^s$. Bordism classes of oriented manifolds are completely determined by their Pontrjagin and Stiefel-Whitney numbers.

• The mod-2 cohomology of $MSO$ is the same as the mod-2 cohomology of $BSO$, a polynomial ring on $w_2, w_3, \dots$ whose Poincare series is $$ \prod_{i \geq 2} \tfrac{1}{1-t^i}. $$

• Rationally, the ring is a polynomial algebra $\Bbb Q[x_4, x_8, x_{12}, \dots]$ on generators in degrees that are a power of $4$. This tells us the rank $r$ of each group. The Poincare series for the free part of $\Omega^{SO}_*$ is thus $$ p_{free}(t) = \prod_{j \geq 1} \tfrac{1}{1-t^{4i}}. $$

• $2$-locally, the bordism spectrum $MSO$ is a wedge of suspensions of Eilenberg--Mac Lane spectra $H\Bbb Z/2$ and $H\Bbb Z$. This allows us to write $$ H^*(MSO) \cong \bigoplus_\text{free summands}H^*(H\Bbb Z) \oplus \bigoplus_\text{torsion summands} H^*(H\Bbb Z/2). $$ Turning this into a Poincare series expression using the Poincare series for the cohomology of Eilenberg--Mac Lane spectra, we can solve for the Poincare series of the torsion part in $\Omega^{SO}_*$. $$ p_{tors}(t) = \left[(1-t) \prod_{k \geq 2, k \neq 2^i-1} \left(\tfrac{1}{1-t^k}\right)\right] - \left[\frac{1}{1+t}\prod_{k \geq 1}\left(\tfrac{1}{1-t^{4k}}\right)\right] $$

I asked Mathematica for a calculation of these groups out to degree 28. Assuming I didn't make a typo, here they are. $$ \begin{array}{c|l} n & \Omega^{SO}_n \\ \hline 0 & \Bbb Z\\ 1 & 0\\ 2 & 0\\ 3 & 0\\ 4 & \Bbb Z\\ 5 & \Bbb Z/2\\ 6 & 0\\ 7 & 0\\ 8 & \Bbb Z^2\\ 9 & (\Bbb Z/2)^2\\ 10 & \Bbb Z/2\\ 11 & \Bbb Z/2\\ 12 & \Bbb Z^3\\ 13 & (\Bbb Z/2)^4\\ 14 & (\Bbb Z/2)^2\\ 15 & (\Bbb Z/2)^3\\ 16 & \Bbb Z^5 \oplus \Bbb Z/2\\ 17 & (\Bbb Z/2)^8\\ 18 & (\Bbb Z/2)^5\\ 19 & (\Bbb Z/2)^7\\ 20 & \Bbb Z^7 \oplus (\Bbb Z/2)^{20}\\ 21 & (\Bbb Z/2)^{15}\\ 22 & (\Bbb Z/2)^{11}\\ 23 & (\Bbb Z/2)^{15}\\ 24 & \Bbb Z^{11} \oplus (\Bbb Z/2)^{10}\\ 25 & (\Bbb Z/2)^{28}\\ 26 & (\Bbb Z/2)^{22}\\ 27 & (\Bbb Z/2)^{31}\\ 28 & \Bbb Z^{15} \oplus (\Bbb Z/2)^{23}\\ \end{array} $$

(The OEIS doesn't seem like anybody interested in bordism theory has invested the effort into adding this type of information.)