Reduced real connective K theory of a point

Question

I’ve just started reading about reduced real connective k-theory, denoted $\widetilde{ko}_*$. I’m familiar with real k-theory and complex k-theory, they’re reduced counterparts, and the definition of $ko_*$. However, I can’t seem to get straight the easiest case of $\widetilde{ko}_*(pt)$. For example, what is $\widetilde{ko}_i(pt)$ for low values of i? I imagine that we only have to do things up to $i=8$ by Bott periodicity.

Answer 1

Any reduced homology theory evaluates to $0$ on a point, in particular for reduced connective K-theory. More interesting is to ask what $\widetilde{ko}_*(S^0)$ is? By the definition of connective covers of spectra, if $* \geq 0$ then $\widetilde{ko}_*(S^0)=\pi_*(ko)=\pi_*(KO)=\widetilde{KO}_*(S^0)$; $0$ otherwise.

It is important to note this argument only works in the case of $\widetilde{ko}_*(S^0)$ and not for a general space.