What is known about spaces $X$ with the property that $K^\*(\text{point})\to K^\*(X)$ is an isomorphism?


The same question for $K$-homology $K\_\*(X)\to K\_\*(\text{point})$; I don't even know whether these conditions are equivalent.


Note that replacing $K$-theory with integral homology one gets very interesting (I think) class of spaces, studied in "[Acyclic spaces](http://www.ma.huji.ac.il/~farjoun/acyclic%20spaces.pdf)" by E. Dror.



Which spaces have trivial K-theory?
I'll just give an answer for finite complexes $X$. The condition $\widetilde{K}^\*(X)=0$ only depends on the suspension spectrum of $X$ so this is naturally regarded as a question in stable homotopy theory. The condition also implies that $\widetilde{H}^\*(X;\mathbb{Q})=0$ and thus that $n.1\_X=0$ as a stable map for some $n>0$. From this it follows that $X$ splits stably as a wedge of finitely many $p$-torsion finite spectra for different primes $p$, so we are really looking at a question in $p$-local stable homotopy theory. The condition $\widetilde{K}^\*(X)=0$ then says that $X$ has chromatic type at least two. For the general theory of chromatic type of finite spectra you can read Ravenel's book "Nilpotence in stable homotopy theory". For type two, it is possible to be a little more explicit than for higher types. For example, Adams constructed a certain self-map of the mod $p$ Moore spectrum which induces an isomorphism in $K$-theory, so the cofibre of that map has type two.


