Talk:Spin bordism

Throughout section $4.2$$<wikitex> Throughout section .2$, we should still have the condition that $J$ doesn't contain 4.2$, we should still have the condition that $J$$J$ doesn't contain $1$$1$. Right?

There also seems to be an issue with the basis for $\Omega^{Spin}_*\otimes\mathbb{Z}_2$$\Omega^{Spin}_*\otimes\mathbb{Z}_2$. The third dot point reads "$M_J\times\omega_8^k$$M_J\times\omega_8^k$ with $k \geq 0$$k \geq 0$, $n(J)$$n(J)$ odd". But $M_J$$M_J$ is only defined for $n(J)$$n(J)$ even.

I can think of two possibilities of what was meant:

1. "$n(J)$$n(J)$ odd" should be "$n(J)$$n(J)$ even". If this was what was intended, then this is covered by the first dot point with $i = 0$$i = 0$ so the third dot point is unnecessary.

2. $M_J$$M_J$ should be $N_J$$N_J$. If this is the case, then the fifth dot point is no longer needed (just take $k = 0$$k = 0$).

$. Right? There also seems to be an issue with the basis for $\Omega^{Spin}_*\otimes\mathbb{Z}_2$. The third dot point reads "$M_J\times\omega_8^k$ with $k \geq 0$, $n(J)$ odd". But $M_J$ is only defined for $n(J)$ even. I can think of two possibilities of what was meant: 1. "$n(J)$ odd" should be "$n(J)$ even". If this was what was intended, then this is covered by the first dot point with $i = 0$ so the third dot point is unnecessary. 2. $M_J$ should be $N_J$. If this is the case, then the fifth dot point is no longer needed (just take $k = 0$). 4.2, we should still have the condition that $J$$J$ doesn't contain $1$$1$. Right?

There also seems to be an issue with the basis for $\Omega^{Spin}_*\otimes\mathbb{Z}_2$$\Omega^{Spin}_*\otimes\mathbb{Z}_2$. The third dot point reads "$M_J\times\omega_8^k$$M_J\times\omega_8^k$ with $k \geq 0$$k \geq 0$, $n(J)$$n(J)$ odd". But $M_J$$M_J$ is only defined for $n(J)$$n(J)$ even.

I can think of two possibilities of what was meant:

1. "$n(J)$$n(J)$ odd" should be "$n(J)$$n(J)$ even". If this was what was intended, then this is covered by the first dot point with $i = 0$$i = 0$ so the third dot point is unnecessary.

2. $M_J$$M_J$ should be $N_J$$N_J$. If this is the case, then the fifth dot point is no longer needed (just take $k = 0$$k = 0$).