Let $X$ be a finite CW-complex such that its $K$-theory $K^*(X)$ is, as a $\mathbb{Z}$-algebra, generated by $a_1, \cdots, a_n$ which are represented by reduced line bundles $L_1-1, \cdots, L_n-1$ satisfying $L_i^{\otimes 2}\oplus 1\cong L_i^{\oplus 2}$ (implying that $a_i^2=0$) for $1\leq i\leq n$. Note that $a_i$ may be a torsion element. We say $a\in K^{-1}(X)$ is represented by a reduced line bundle if, through the identification $K^{-1}(X)=\widetilde{K}^0(SX)$, $a$ is represented by a reduced line bundle on $SX$. Examples of such finite CW-complexes include $S^2$ and $\mathbb{RP}^2$, any product of them, and $S^2\times_{\mathbb{Z}_2}S^2$ where $\mathbb{Z}_2$ acts on $S^2$ by antipodal map.

  1. Is it true that the Atiyah-Hirzebruch spectral sequence for $K^*(X)$ collapses on the $E_2$-page, and there are no extension problems? It seems that $\text{Sq}^1$ and $\text{Sq}^2$ vanish on $X$, and $d_3$, being the third integral Steenrod square, seems to vanish. I am not sure how to go about the higher differentials and the extension problems.

  2. Is it true that the `Chern character map' \begin{align*} \text{ch}: K^*(X)&\to H^*(X, \mathbb{Z})\\ a_i&\mapsto \text{ch}(a_i)=c_1(L_i) \end{align*} is a ring isomorphism?

Note that the condition $L_i^{\otimes 2}\oplus 1\cong L_i^{\oplus 2}$ implies that $c_1(L_i)^2=0$, and it still makes sense to define the `Chern character' of any generator $\text{ch}(a_i)=\text{ch}(L_i)-1=e^{c_1(L_i)}-1=c_1(L_i)$ and product of generators \begin{align*} \text{ch}(a_ia_j)&=\text{ch}(L_i\otimes L_j\oplus 1-(L_i\oplus L_j))\\ &=\left(1+c_1(L_i)+c_1(L_j)+\frac{(c_1(L_i)+c_1(L_j))^2}{2}+\cdots\right)+1-1-c_1(L_i)-1-c_1(L_j)\\ &=c_1(L_i)c_1(L_j)\\ &=\text{ch}(a_i)\text{ch}(a_j) \end{align*} even for those $a_i$ which are torsion $K$-theory classes.


1 Answer 1


This is not true. One of the main reasons is that, because you've only specified properties about $K^*(X)$, that leaves a lot of room for $H^*(X;\Bbb Z)$ to have information which doesn't ultimately contribute to $K$-theory.

Finding an example is a little more work. The mod-2 Moore space $M(\Bbb Z/2, n)$ is the $(n-1)$-fold reduced suspension of $\Bbb RP^2$; it has the property that its reduced homology is $\Bbb Z/2$ in dimension $n$ and $0$ elsewhere. These have the property that for sufficiently large $n$, there is a map $$ v: M(\Bbb Z/2, n+8) \to M(\Bbb Z/2, n) $$ which induces an isomorphism $$ v: K^* M(\Bbb Z/2, n) \to K^* M(\Bbb Z/2, n+8) $$ on $K$-theory (real or complex $K$-theory, in fact). These maps were constructed by Adams in his study of the $J$-homomorphism ("On the groups J(X) IV").

As a result, we can take the mapping cone $$ M(\Bbb Z/2, n+8) \to M(\Bbb Z/2, n) \to Cv $$ and get a finite complex $Cv$ whose integral cohomology is $\Bbb Z/2$ in dimensions $(n+1)$ and $(n+10)$ and zero elsewhere; this means that the Atiyah-Hirzebruch spectral sequence starts nontrivially. However, the fact that $v$ was an isomorphism on $K$-theory means that $Cv$ has the same $K$-theory as a point. The space $Cv$ then satisfies your criteria for not very interesting reasons.

The fact that $Cv$ has no $K$-theory means that the Atiyah-Hirzebruch spectral sequence has to erase everything except the $0$-line (in fact, there is a $d_9$ differential). This also means that the Chern character doesn't have a hope of being an isomorphism because the source ring has no torsion and the target does.


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