Lower dimensional Pin cobordisms

Question

I'm studying Pin cobordism groups of a point for some low dimensions. I found a general result by Anderson, Brown, Peterson in Theorem 5.1 of their paper "Pin cobordism and related topics" http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42. Using the Pontryagin-Thom construction, they are able to get the following results:

1) The contribution to $\Omega_{*}^{Pin}$ of terms $\pi_{*}(\mathbb{RP}^{\infty} \wedge \textbf{K}(\mathbb{Z}_2,n))$ is a direct summand of $\mathbb{Z}_2$ in each dimension $\geq n$. 2) The contribution to $\Omega^{Pin}_*$ of terms $\pi_{*}(\mathbb{RP}^{\infty} \wedge \textbf{B}O \langle 8n \rangle)$ is as follows: $\mathbb{Z}_2$ in dim$8n+i, i \equiv 0, 1(8)$; 0 in dim$8n+i, i\equiv 3, 4, 5, 7(8)$; $\mathbb{Z}_{2^{4k+3}}$ in dim$8n+8k+2, k\geq0$; and $\mathbb{Z}_{2^{4k+4}}$ in dim$8n+8k+6, k\geq0$. 3) The contribution to $\Omega^{Pin}_{*}$ of terms $\pi_{*}(\mathbb{RP}^{\infty} \wedge \textbf{B}O \langle 8n+2 \rangle)$ is as follows: $\mathbb{Z}_2$ in dim$8n+2+i, i\equiv 1, 2, 5, 7(8)$; $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ in dim$8n+2+i, i\equiv6(8)$; 0 in dim$8n+2+i\equiv3(8)$; $\mathbb{Z}_{2^{4k+1}}$ in dim$8n+2+8k$, $k\geq0$; and $\mathbb{Z}_{2^{4k+2}}$ in dim$8n+2+8k+4$, $k\geq0$.

However, when I tried to apply the above theorem for some low dimensional cases, I get some apparent contradictions. For example, in dim=2, if we apply the above theorem, we get $\Omega^{Pin}_2 = \mathbb{Z}_8 \oplus \mathbb{Z}_2$, which contradicts the known result of $\mathbb{Z}_8$. (Somehow by considering contributions only from $\pi_{*}(\mathbb{RP}^{\infty} \wedge \textbf{B}O \langle 8n \rangle)$ and ignoring all other contributions, one gets the correct results up to dimension 7.) Also, for dim=22, when applying this theorem, one gets $\Omega^{Pin}_{22} = \mathbb{Z}_{2^{12}} \oplus \mathbb{Z}_{2^8} \oplus \mathbb{Z}_{2^4} \oplus \mathbb{Z}_{2^{10}} \oplus \mathbb{Z}_{2^6} \oplus \mathbb{Z}_{2^2}$, instead of $\Omega^{Pin}_{22} = \mathbb{Z}_{2^{12}} \oplus \mathbb{Z}_{2^8} \oplus \mathbb{Z}_{2^6} \oplus \mathbb{Z}_{2^4} \oplus \mathbb{Z}_{2^4} \oplus \mathbb{Z}_{2^2} \oplus \mathbb{Z}_{2^2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$, which is given as an example on page 467 of their paper.

Can anyone make sense of the above theorem?

Answer 1

The paper "Pin cobordism and related topics" http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42. only gives the homotopy groups of $RP^{\infty }\wedge X$ where $X$ is either $\Sigma ^nHZ/2$, $bo\langle 8n\rangle$ and $bo\langle 8n+2\rangle$, noting that they are summands of the homotopy groups of the spectrum $MPin$. However, it doesn't say how many copies of them actually show up. To find it out, you have to refer to the paper "Structure of Spin Cobordism Ring" by the same authors, Theorem 2.2 (and the definition of $J$'s on the top of the same page).

Thus $RP^{\infty }\wedge bo\langle 8n\rangle$ appears as many times as the number of the sequences $(j_1,\cdots ,j_k)$, $j_i> 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdot +j_k=2n$

and $RP^{\infty }\wedge bo\langle 8n+2\rangle$ appears as many times as the number of the sequences $(j_1,\cdots ,j_k)$, $j_i> 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdots ,+j_k=2n+1$

For the latter here is no such sequence when $n=0$, this is why there is no contribution of $bo\langle -2\rangle$.

As to the case of dimension 22 there will be one copy of $bo\langle 0\rangle$, one copy of $bo\langle 8\rangle$ and two copies of $bo\langle 16\rangle$ (one for the sequence $ (2,2)$ and another for $(4)$) as well as a copy of $bo\langle 10\rangle$ and two copies of $bo\langle 18\rangle$. Thus there is no contradiction.