4
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Could you give a source for "known results"? $\endgroup$ – user43326 Jun 24 '14 at 7:31
  • $\begingroup$ Yes. Kirby and Taylor give them up to dimension 4 math.berkeley.edu/~kirby/papers/… $\endgroup$ – Zitao Wang Jun 24 '14 at 15:54
  • $\begingroup$ It looks like Anderson, Brown and Peterson say that the contribution of $\pi _*(RP^{\infty }\vee HZ/2)$ is either $Z/2$ or $0$ (and not $Z/2$), this takes care of the problem in dimension 2. $\endgroup$ – user43326 Jun 24 '14 at 17:49
  • $\begingroup$ Well, the Z/2 here is from BO<8n+2>, which contributes $\mathbb{Z}_{2^{4k+1}}$ in dim 8n+2+8k. If you take n=k=0, you get Z/2. $\endgroup$ – Zitao Wang Jun 24 '14 at 19:50
  • $\begingroup$ I see. I guess $\pi _2(RP^{\infty }\wedge kO<2>)$ (or do they really mean $BO<2>$ ?) can be computed easily without Adams SS, so this is probably the first thing to check. $\endgroup$ – user43326 Jun 25 '14 at 5:32
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ This helps, thanks. Also $bo<8n-2>$ should be $bo<8n+2>$, and $j_i > 1$ instead of $j_i \geq 1$ right? Otherwise you get 2=2=1+1 so that $bo<8>$ appears twice. $\endgroup$ – Zitao Wang Jun 25 '14 at 17:15
  • $\begingroup$ You are right, I edited the answer. $\endgroup$ – user43326 Jun 25 '14 at 17:19
  • $\begingroup$ Everything works fine up to dimension 10. In dimension 22, seems that we are able to get all factors except $\mathbb{Z}_2 \oplus \mathbb{Z}_2$, it is a bit weird that we get this factor, since it comes from $BO<8n+2>$ in dim 8n+2+i, $i \equiv 6 (8)$, and dim=22 does not lie in this category. $K(\mathbb{Z}_2,n)$ can only contribute a single $\mathbb{Z}_2$, so it cannot be $K(\mathbb{Z}_2,n)$. $\endgroup$ – Zitao Wang Jun 25 '14 at 20:18
  • $\begingroup$ Each copie of $\Sigma ^nHZ/2$ can only contribute a single $Z/2$. However, there are many copies, and the number of copies is not said explicitely in Theorem 2.2 of "Structure of Spin Cobordism Ring". This is why you need to compare with the results for $\pi _*Mspin$ $\endgroup$ – user43326 Jun 25 '14 at 20:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.