KU representability and the skeletal–Postnikov d-three comparison

For a based finite CW complex $Y$ the finite-rank complement theorem and stable classification identify ${\tilde{K}}^0(Y)$ with the based homotopy set $[Y,\mathbb{Z}\times BU]$: every virtual class has the form $[E]-[{\epsilon }^N]$; adding trivial summands does not change the stabilized classifying map; and a homotopy of classifying maps gives the corresponding stable bundle isomorphism. The common-summand relation therefore identifies both directions naturally.

The adjoint structure maps of $KU$ are equivalences, so $KU$ represents a cohomology theory; the degree-zero bundle-classification identification, the suspension adjunction and the natural Bott maps identify $K{U}^n(X,A)$ with the two-periodic $K^n(X,A)$ in every degree, because both connecting maps are induced by the same quotient map followed by suspension.

If the chosen $KU$ coefficient row is odd, its source and target groups are zero and the comparison assertion is vacuous. Thus fix $p\ge 1$ on a nonzero row and Bott-translate it to an even index $q\le -2$. Put $t=p+q$. A bidegree-$(p,q)$ class has total cohomological degree $t$, so its representing space is $k{u}_t$, not $k{u}_{-q}$. If $t<0$, this means the explicitly defined iterated loop space ${\mathrm{\Omega }}^{-t}k{u}_0$ in [A5]. The spectrum-space indexing gives ${\pi }_p(k{u}_t)={\pi }_{p-t}(ku)={\pi }_{-q}(ku)\cong \mathbb{Z},$ ${\pi }_{p+1}(k{u}_t)={\pi }_{1-q}(ku)=0,{\pi }_{p+2}(k{u}_t)={\pi }_{2-q}(ku)\cong \mathbb{Z}\mathrm{.}$ The relative quotient $X^{p+2}\mathrm{/}{X}^{p-1}$ is $(p-1)$-connected, so a based representing map has trivial data in lower Postnikov degrees; for $p=1$ its base component is specified. Thus the first possible obstruction from this coefficient class is the invariant joining the displayed ${\pi }_p$ and ${\pi }_{p+2}$.

We identify that obstruction with the skeletal $d_3$ directly. Represent a cellular $E_1^{p,q}$ class by a based map $a:X^p\mathrm{/}{X}^{p-1}\to k{u}_t$. Its restriction to each $p$-cell is the element of ${\pi }_p(k{u}_t)$ in step 1.3. The skeletal exact-couple formula in [A4] says that $a$ survives to $E_3$ exactly when a representative extends, after the allowed $E_1$ boundary adjustment, over $X^{p+2}\mathrm{/}{X}^{p-1}$. The first extension obstruction, on $(p+1)$-cells, is the cellular coboundary $d_{1}a$. After it vanishes, the next obstruction lies in ${\pi }_{p+1}(k{u}_t)=0$, so extension over $(p+2)$-cells is possible. Fix such an extension $\tilde{a}$. On each $(p+3)$-cell, the attaching sphere followed by $\tilde{a}$ gives an element of ${\pi }_{p+2}(k{u}_t)$. These elements form the next cellular obstruction cochain. They are also the components of the generalized-cohomology connecting map for the cofiber sequence $X^{p+2}\mathrm{/}{X}^{p-1}\to X^{p+3}\mathrm{/}{X}^{p-1}\to \bigvee S^{p+3}$: on one cell, that connecting map is restriction to its attaching sphere, followed by suspension. This connecting map is precisely the pair boundary represented by $j(x)$ in [A4]. Therefore its class in the $E_3$ target is $d_3[a]$. Changing $\tilde{a}$ changes the cochain by a coboundary or the preceding exact-couple indeterminacy; the intervening odd coefficient groups make $d_2=0$.

Bott-translate any coefficient row to $q\le -2$. Since $E^q(\ast )={\pi }_{-q}(E)$ for a representing spectrum, $ku\to KU$ is an isomorphism on the coefficient groups in the source row $q$ and target row $q-2$. It is also an isomorphism on the intervening odd rows, both of which are zero. Hence its map of skeletal exact couples induces isomorphisms on the relevant $E_2$ and $E_3$ source and target groups and commutes with $d_3$.

To identify the operation rather than just the obstruction group, write $-q=2k$ with $k\ge 1$. Iterating the Bott map of [A3] gives ${\beta }^k:{\mathrm{\Sigma }}^{2k}ku\to ku$, an isomorphism on ${\pi }_j$ for $j\ge 2k$. At representing-space index $t=p-2k$, its source space is $k{u}_p$ and its target is $k{u}_t$; it is an isomorphism on ${\pi }_i$ for $i\ge p$, while $k{u}_p$ has no homotopy below degree $p$ in its based component. Thus $k{u}_p\to k{u}_t$ is the relevant $p$-connective cover. The quotient $X^{p+3}\mathrm{/}{X}^{p-1}$ is $(p-1)$-connected, so the lower Postnikov stages of $k{u}_t$ carry only the null map on it; the lifting obstruction of step 2.1 is transported from the first marked Postnikov fibration of $k{u}_p$. For $p\ge 2$, The first connective complex K-theory Postnikov invariant is integral Sq-three identifies its universal class with ${\beta }_{\mathbb{Z}}S{q}^2{\rho }_2$ on $H^p(-;\mathbb{Z})$. Its evaluation on the class represented by $a$ is consequently the obstruction cochain class of step 2.1. This derives the Bott-row translation from the actual Bott map and does not infer it merely from equality of homotopy groups.

The representing-space and exact-couple comparison in step 2.1 applies componentwise for every $p\ge 1$; no class in $H^p$ is evaluated on a space indexed by $-q$. At $p=1$, the base component has ${\pi }_1=\mathbb{Z}$, ${\pi }_2=0$ and ${\pi }_3=\mathbb{Z}$. The relevant universal class lies in $H^4(K(\mathbb{Z},1);\mathbb{Z})=H^4(S^1;\mathbb{Z})=0$, agreeing with the instability vanishing of $S{q}^2$ on degree-one classes. Thus this edge case has zero obstruction and zero $d_3$ without invoking the $m\ge 2$ space-invariant clause of The first connective complex K-theory Postnikov invariant is integral Sq-three. For a finite CW pair $(X,A)$, apply the same argument to its finite based cofiber: $X\mathrm{/}A$ if $A\mathrm{\ne }\mathrm{\varnothing }$, or $X_{+}$ if $A=\mathrm{\varnothing }$. Represented cohomology identifies these with the relative and absolute groups respectively, preserving the skeletal filtration and connecting maps.

Combining steps 1.3 and 2.1–3.2 identifies the skeletal $d_3$ of the $K$-AHSS with the relevant Postnikov obstruction in the total-degree space $k{u}_{p+q}$, equivalently the coefficient-row translate of the first stable $ku$ invariant. Step 1.2 identifies the local $K$-groups with those represented by $KU$, and step 3.2 covers all $p\ge 1$ and finite CW pairs.

Steps 1.2 and 4.1 give the asserted representability, the $E_3$ comparison in nonpositive coefficient rows and the identification of the skeletal $d_3$ with the Postnikov obstruction.