Here's an approach that works up to about dimension 7, outlined by Freed-Hopkins, §10, and explained in more detail by Campbell. It doesn't detect torsion away from 2, though.

There's a weak equivalence $\Sigma^{-1} \mathrm{MPin}^-\simeq \mathrm{MSpin}\wedge \mathrm{MTO}_1$, where $\mathrm{MTO}_1$ is a Madsen-Tillman spectrum, the Thom spectrum of the virtual vector bundle $(\underline{\mathbb R} - S)\to B\mathrm O_1$, where $\underline{\mathbb R}$ is the trivial line bundle and $S\to B\mathrm O_1$ is the tautological line bundle. Hence, to understand $\Omega_d^{\mathrm{Pin}^-}(B\mathbb Z/2)$, it suffices to understand the homotopy groups of $\mathrm{MSpin}\wedge\mathrm{MTO_1}\wedge B\mathbb Z/2$.

We'll use the Adams spectral sequence, but there's a key trick that makes it simpler. Let $\mathcal A(1)$ denote the subalgebra of the Steenrod algebra generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$. Then, Anderson, Brown, and Peterson proved that, as $\mathcal A$-modules,

$$ H^*(\mathrm{MSpin};\mathbb F_2)\cong \mathcal A\otimes_{\mathcal A(1)} (\mathbb F_2\oplus M),$$

where $M$ is a graded $\mathcal A(1)$-module which is $0$ in dimension less than 8.

Thus we can invoke a change-of-rings theorem for the $E_2$-page of the Adams spectral sequence for $\mathrm{MSpin}\wedge\mathrm{MTO_1}\wedge B\mathbb Z/2$: using the Adams grading, when $t -s < 8$,
\begin{align*}
E_2^{s,t} &= \mathrm{Ext}_{\mathcal A}^{s,t}(H^*(\mathrm{MSpin}\wedge\mathrm{MTO_1}\wedge B\mathbb Z/2; \mathbb F_2), \mathbb F_2)\\
&\cong \mathrm{Ext}_{\mathcal A}^{s,t}((A\otimes_{\mathcal A(1)} \mathbb F_2)\otimes H^*(\mathrm{MTO_1}\wedge B\mathbb Z/2; \mathbb F_2), \mathbb F_2)\\
&\cong \mathrm{Ext}_{\mathcal A(1)}^{s,t}(H^*(\mathrm{MTO_1}\wedge B\mathbb Z/2; \mathbb F_2), \mathbb F_2).
\end{align*}
Explicitly calculating this is tractable, because $\mathcal A(1)$ is small and we're only going up to dimension 7.

- The $\mathcal A(1)$-module structure on $\tilde H^*(B\mathbb Z/2; \mathbb F_2)$ is standard, and Campbell describes it in Example 3.3 of his
paper.
- Campbell also calculates the $\mathcal A(1)$-module structure on $H^*(\mathrm{MTO}_1; \mathbb F_2)$, and describes the answer in Example 6.6 and Figure 6.4.

This post imported from StackExchange MathOverflow at 2017-09-14 13:29 (UTC), posted by SE-user Arun Debray