# $d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$

Cross posted from here 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 $$d^3$$.

• One problem I see is that since KO is not even, now there's also a non-trivial $d_2$ detecting multiplication by $\eta$ (so $Sq^2$) in some degrees at least. Oct 23 '19 at 12:55

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.

• Is $\Phi_{1,1}\circ r_2$ going to be equal to $0$ when evaluated on a class when $p=0$? Dec 15 '19 at 13:48
• @samhughes Yes. You can prove this by naturality, comparing with the AHSS for a point Dec 16 '19 at 4:03
• Perfect, thank you! Dec 16 '19 at 13:39

Expressions for $$d^2$$ and $$d^3$$ of the twisted Atiyah-Hirzebruch spectral sequence can be found in Grady, Daniel, and Hisham Sati. "Twisted differential KO-theory." arXiv preprint arXiv:1905.09085 (2019) Proposition 18, where the twist is given by elements $$(\sigma_1,\sigma_2)\in H^1(X,\mathbb Z/2)\times H^2(X,\mathbb Z/2)\cong H^*(X,\tau_{\le 1}bgl_1 KO)$$:

• The $$E_2$$-page is given by twisted cohomology $$H^p(X;(KO^q)_{\sigma_1})$$, with $$\mathbb Z/2$$ acting by the sign representation on all groups
• The only nonvanishing $$d_2$$ are \begin{align*} d_2^{p,-8t}:H^p(X;\mathbb Z_{\sigma_1})\to H^{p+2}(X;\mathbb Z/2)&, x\mapsto \operatorname Sq^2(r(x)) + \sigma_1\cup Sq^1(r(x)) + \sigma_2\cup r(x)\\ d_2^{p,-8t-1}:H^p(X;\mathbb Z/2)\to H^{p+2}(X;\mathbb Z/2)&, x\mapsto \operatorname Sq^2(x) + \sigma_1\cup Sq^1(x) + \sigma_2\cup x \end{align*}
• The only canonically defined $$d_3$$ is $$d_3^{p,-8t-2}:H^p(X;\mathbb Z/2)\to H^{p+2}(X;\mathbb Z_{\sigma_1}), x\mapsto \operatorname \beta(Sq^2(x)) + \beta(\sigma_2\cup r(x))\\$$

Here $$r: H^p(X;\mathbb Z_{\sigma_1}) \to H^p(X;\mathbb Z/2)$$ and $$\beta: H^p(X;\mathbb Z/2) \to H^{p+1}(X;\mathbb Z_{\sigma_1})$$ are reduction mod 2 and the corresponding Bockstein.

The above twists are those defined by geometry, i.e. by the inclusion $$\operatorname {sLine}^\otimes\to \operatorname {Pic}(\operatorname{sVect}^\otimes)$$ of superlines as tensor-invertible super vector spaces, i.e. the twists arising from super gerbes'' on $$X$$. More generally, twists for $$KO$$-theory are classified by $$H^*(X;bgl_1 KO)\cong H^1(X,\mathbb Z/2)\times H^2(X,\mathbb Z/2)\oplus H^*(X,\tau_{> 1}bgl_1 KO)$$. By degree considerations, you can see that the ones in the third group don't change the above differentials (for $$d_3$$, use that the multiplication $$KO_2\otimes KO_{8t+2}\to KO_{8t+4}$$ vanishes).

The only remaining canonically defined differential is $$d_5^{p,-8t-4}:H^p(X;\mathbb Z_{\sigma_1})\to H^{p+4}(X;\mathbb Z_{\sigma_1})$$ I don't know anything about it besides the untwisted part, which you can find in Tyler Lawson's answer.