Cross posted from [here](https://math.stackexchange.com/questions/3391037/d3-in-the-atiyah-hirzebruch-spectral-sequence-for-twisted-ko) after no responses and a bounty being placed on the question.


Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-Hirzebruch spectral sequence:


$E\_2^{p,q}:=H^p(X;h^q(\ast))\Rightarrow h^{p+q}(X)$.


In the case of complex topological $K$-theory, i.e $KU^n(X)$, the differentials admit nice descriptions in terms of higher cohomology operations (i.e. $d^3=Sq^3$, the Steenrod square of degree $3$). For twisted $KU$ we have that $d^3(-)=Sq^3(-)+ \lambda\smile (-)$ where $H\in H^3(X;\mathbb{Z})$ is the class of the twist.


This may be a naive question but is there a similar description for real topological (twisted) $K$-theory, $KO$? I am particularly interested in $d^3$.



$d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$
Here are the first, more straightforward, differentials in the AHSS
$$H^p(X;KO^q(\ast)) \Rightarrow KO^{p+q}(X)$$
for real K-theory. Note $KO^q(\ast)$ is
$$
\begin{cases}
\Bbb Z &\text{if }q = 8k,\\
\Bbb Z/2 &\text{if }q = 8k-1,\\
\Bbb Z/2 &\text{if }q = 8k-2,\\
\Bbb Z &\text{if }q = 8k-4,\\
0 &\text{otherwise.}
\end{cases}
$$


* For $q=8k$ the map
$$d\_2: H^\ast(X;\Bbb Z) \to H^{\ast+2}(X;\Bbb Z/2)$$
is $Sq^2 \circ r\_2$, where $r\_2$ is reduction from integer cohomology to mod-2 cohomology.
* For $q=8k-1$ the map
$$d\_2: H^\ast(X;\Bbb Z/2) \to H^{\ast+2}(X;\Bbb Z/2)$$
is $Sq^2$. (These two differentials are what Denis Nardin alluded to in the comments.)
* For $q=8k-2$ the map
$$d\_3: H^\ast(X;\Bbb Z/2) \to H^{\ast+3}(X;\Bbb Z)$$
is $\beta\_2 \circ Sq^2$, where $\beta\_2$ is the Bockstein from mod-2 cohomology to integer cohomology.
* For $q=8k-4$ the map
$$d\_5: H^\ast(X;\Bbb Z) \to H^{\ast+5}(X;\Bbb Z)$$
is $\beta\_2 \circ Sq^4 \circ r\_2 + \beta\_3 \circ P^1 \circ r\_3$, where $P^1$ is a Steenrod operation on mod-3 cohomology. (Or perhaps this is $-d\_5$, depending on choice of generators.)


The only remaining $d\_3$ differential, from $E\_3^{p,8k}$ to $E\_3^{p+3,8k-2}$, is given by a map
$$
\Phi\_{1,1} \circ r\_2: \ker(Sq^2 \circ r\_2) \to \mathrm{coker}(Sq^2).
$$
This operation $\Phi\_{1,1}$ is a so-called secondary cohomology operation, associated to the relation $Sq^2 Sq^2 + Sq^3 Sq^1 = 0$ between cohomology operations. There is no straightforward description of $\Phi\_{1,1}$ in terms of primary operations. In the same way that $Sq^1$ in $H^\*(X)$ detects the presence of degree-2 attaching maps $S^n \to S^n$ among the cells of $X$, and $Sq^2$ detects the presence of the Hopf map $\eta$ in an attaching map $S^{n+1} \to S^n$, the operation $\Phi\_{1,1}$ detects the presence of $\eta \circ \eta$ in an attaching map $S^{n+2} \to S^n$.


I am not certain of the twisted K-theory versions of some of these, but this is a deficiency in my knowledge of real twisted K-theory.


