| Résume | Résumé : This is joint work with Ilan Hirshberg.
In 1993, Blackadar asked for which $n$, $k$, $d$, and $r$ there is a unital homomorphism $C (S^{2 n}, M_r) \to C (S^{2 k}, M_{d r})$ which is injective on $K_0$. There are "obvious" examples constructed as follows. For $m \in {\mathbb{N}}$, $K^1 (S^{2 m - 1})$ is generated by the class of a unitary in $C (S^{2 m - 1}, M_m)$, and from this one gets homomorphisms $C (S^{1}) \to C (S^{2 m - 1}, M_m)$ which are injective on $K_1$. One can then take (odd) suspensions, increase the matrix size in the codomain, and tensor everything with $M_r$. Ordered $K_0$ suggests that many more might exist. In particular, Blackadar asked whether there is a unital homomorphism $C (S^{2 n}, M_{2 n}) \to C (S^{4 n}, M_{4 n})$ which is injective on $K_0$.
The answer is no. We prove that the only choices of $n$, $k$, $d$, and $r$ for which such a homomorphism exists are those which occur in the examples above. The proof uses indecomposability in the rational cohomology ring of the space of homomorphisms from $C (S^{2 n})$ to $M_d$.
This material is based upon work partially supported by the US National Science Foundation under Grant DMS-2400332. |
