# Third differential in Atiyah Hirzebruch spectral sequence

Does any one know why $d_3: H^* (X, K^0(point))\rightarrow H^{*+3}(X,K^0(point))$ is actually extended $Sq^3$ to $\mathbb{Z}$ coefficient.

• An interesting thing to note, for people finding this question in the future, is that the same sequence for $KU/2$, that is $H^*(X, KU/2^0(pt. ))\to H^{*+3}(X, KU/2^0(pt.)$ is infact $Sq^2Sq^1+Sq^3$. The nontriviality of the Bockstein on $Sq^2Sq^1$ obstructs the lifting of its effect to the integral sequence.
– pax
Mar 6, 2017 at 19:58

This follows from the following considerations:

1. This differential in the Atiyah-Hirzebruch spectral sequence must be a stable cohomology operation for general nonsense reasons (the first nonvanishing differential always is, no matter what the generalized cohomology theory is).

2. There are exactly two stable cohomology operations $H^*(X) \to H^{*+3}(X)$ with integer coefficients. One of them is zero, and the other is $\beta \circ Sq^2 \circ r$, where $r$ is reduction mod 2 and $\beta$ is the Bockstein from mod-2 cohomology to integer cohomology. This comes from a calculation of the cohomology of Eilenberg-Mac Lane spaces, which describe all possible cohomology operations; for n sufficiently large we have $H^{n+3}(K(\mathbb{Z},n)) = \mathbb{Z}/2$.

3. The $d_3$ differential is not the zero cohomology operation. For this, it suffices to find one space for which this differential is nontrivial (and you can find this by actually calculating the complex K-groups). I believe that you can find this for $\mathbb{RP}^2 \times \mathbb{RP}^4$; perhaps someone more industrious can flesh this out?

• But computing the K group of $\mathbb{R}P^k$ needs to apply AHSS. Actually at the appendix of the paper by Atiyah, titled Analytic cycle on complex manifold, He computed $d_{2p−1}$ in terms of Steenrod p-power by attaching a cell to some$\Sigma^k\matbb{C}P^n$ and using some relation of differentials and chern character which I don't quite understand. But I expect for d3 there should be some straight forward example Apr 24, 2011 at 5:11
• Why do the differentials for the AHSS have to be a stable cohomology operation? Do you have a reference? Aug 15, 2017 at 3:51
• @54321user I believe that I learned that from Adams. To be a stable cohomology operation it merely needs to (a) be a natural transformation (which is true because the Atiyah-Hirzebruch spectral sequence is natural), and (b) be compatible with the suspension isomorphism (which one can show straightforwardly when the (based) space is a point and then progressively bootstrapped up to the result for a general space). Aug 18, 2017 at 23:07
• @TylerLawson I’m trying to parse this... I can understand how this shows the d2 differential must be a cohomology operation, but even if the d2 differential vanishes at X, it seems to me that because it doesn’t vanish for all spaces, it results that the E3 page is not a cohomology theory and so the d3 differential is not a natural transformation of cohomology theories and so I don’t see how to conclude that it’s a cohomology operation May 25, 2019 at 21:01
• @TimCampion Right. The reason it works here is that the d2 differential does vanish for all spaces, because the K-groups of a point are concentrated on even degrees. May 26, 2019 at 13:21

A pretty direct argument was given by Frank Adams in the proof of 16.6 (page 336) in part III of his 1974 Chicago lectures (MR0402720). Thinking of the Atiyah--Hirzebruch spectral sequence for $K^*(X)$ as arising from the Postnikov tower $\{P^n ku\}$ of ku (Adams calls this spectrum $bu$), row 0 and row 2 come from the layers $HZ$ and $\Sigma^2 HZ$, sitting in a cofiber sequence $\Sigma^2 HZ \to P^2 ku \to HZ \to \Sigma^3 HZ$. The $d_3$-differential is induced by the third map, i.e., the first $k$-invariant of $ku$. To see that the $k$-invariant is the integral lift $\beta Sq^2$ of $Sq^3$, i.e., not zero, Adams looks at the third space in the $ku$-spectrum, namely $SU$, and notices that $\beta Sq^2 \ne 0$ in $H^6(K(Z, 3); Z)$ but $H^6(SU; Z) = 0$. This implies that the $k$-invariant is nonzero.

Okay, I can't pass up the chance to try and be more industrious than Tyler (this is really a comment on Tyler's answer).

I'll try to explain why there's a non-zero $d_3$ differential in the AHSS for $\mathbb{R}P^2 \times \mathbb{R}P^4$.

The K-theory of $\mathbb{R}P^{2k}$ is $\mathbb{Z} \oplus \mathbb{Z}/2^k$ in dimension 0 and is trivial in dimension 1. Applying the Kunneth Theorem tells you that $K^0 (\mathbb{R}P^2 \times \mathbb{R}P^4) = \mathbb{Z}\oplus \mathbb{Z}/2\oplus \mathbb{Z}/4\oplus \mathbb{Z}/2$. Comparing with the cohomology of $\mathbb{R}P^2 \times \mathbb{R}P^4$, one sees that one factor of $\mathbb{Z}/2$ appearing on the line $y-x = 0$ in the E_2 page of the AHSS has to be killed by a differential.

After the 3rd page, all differentials coming in (or out) of the line $y-x = 0$ in the AHSS start (or end) at trivial groups. So there must be a non-zero differential on the 3rd page. It's not clear to me which one it is, but I haven't thought about the multiplicative structure.

• But computing the K group of $\mathbb{R}P^k$ needs to apply AHSS. Actually at the appendix of the paper by Atiyah, titled Analytic cycle on complex manifold, He computed $d_{2p-1}$ in terms of Steenrod p-power by attaching a cell to some $\Sigma^k \matbb{C}P^n$ and using some relation of differentials and chern character which I don't quite understand. But I expect for $d_3$ there should be some straight forward example. Apr 24, 2011 at 5:06
• Sam, are you saying that my argument might be circular, because one would need to understand the $d_3$ differential already in order to compute the K-theory of $RP^2$ and $RP^4$? In both cases, there's no room for a non-zero differential (on any page). One does have to solve the extension problem in order to see that $K^0 (RP^4)$ is $Z\oplus Z/4$ and not $Z\oplus Z/2 \oplus Z/2$. It seems unlikely to me that this could require knowing something about $d_3$, since it's zero in this case. Apr 24, 2011 at 6:07
• According to this question math.stackexchange.com/questions/26438/… math.stackexchange, there's a computation of $K^*(RP^n)$ on pp. 100-110 of Atiyah's K-theory book. I'm not sure what's involved; I don't think I've ever gone through the computation. Apr 24, 2011 at 6:09
• For what I wrote above, one really does need to know that the torsion group is $Z/4$ and not $Z/2\oplus Z/2$. I was actually specifically interested in figuring out how to solve this extension problem the day this question appeared; I'll look into it further. Apr 24, 2011 at 6:11
• In this case, you can solve the extension problem using Chern classes; namely, you show twice a line bundle is not stably trivial by showing that it has nontrivial $w_2$. Apr 24, 2011 at 8:39