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 in Maunder's comparison is $k{u}_t$, not $k{u}_{-q}$. The spectrum-space indexing gives ${\pi }_p(k{u}_t)={\pi }_{p-t}(ku)={\pi }_{-q}(ku)\cong \mathbb{Z},$ ${\pi }_{p+1}(k{u}_t)={\pi }_{1-q}(ku)=0,{\pi }_{p+2}(k{u}_t)={\pi }_{2-q}(ku)\cong \mathbb{Z}\mathrm{.}$ Apply [A5] to the component containing the representing map. It gives an isomorphism from $E_2$ onward between the skeletal AHSS of [A4] and the Postnikov spectral sequence for $k{u}_t$, commuting with $d_3$.

In the Postnikov spectral sequence of step 1.3, the vanishing of ${\pi }_{p+1}(k{u}_t)$ makes the first possible differential out of this bidegree the operation represented by the relevant class $k_{p+3}$ linking ${\pi }_p(k{u}_t)$ to ${\pi }_{p+2}(k{u}_t)$. This need not be the first Postnikov invariant of the whole space $k{u}_t$: by [A3] it is the Bott translate, in this coefficient row, of the first stable $ku$ invariant. By the definition and lifting theorem cited in [A5], evaluating it on a cellular cocycle is the primary obstruction to lifting the corresponding map through that Postnikov stage: on each oriented $(p+3)$-cell it is obtained from the attaching map, and changing the partial lift changes the obstruction cochain by a coboundary. Maunder's differential-compatible comparison transports exactly this obstruction class to the skeletal $d_3$.

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 $d_3$.

Maunder's comparison and the relevant Postnikov obstruction in step 2.1 apply componentwise for every $p\ge 1$, including $p=1$; no class in $H^p$ is evaluated on a space indexed by $-q$. For a finite CW pair $(X,A)$, apply the reduced comparison to the finite quotient $X\mathrm{/}A$; represented cohomology identifies this with the relative group and preserves the skeletal filtration and connecting maps.

Combining steps 1.3, 2.1 and 2.2 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 3.1 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.