results are seen in the computations of [@LLM:quat] for a real variety $X$ with the property that its complexification is the quaternionic projective space (see Example \[exR\]).

In [@Teh:HT] Teh proves a conditional Harnack-Thom type theorem for the homotopy groups of reduced algebraic cycles on $X$ which holds under the assumption that these homotopy groups are all finitely generated and they are zero in high degrees. In the case of divisors he shows that $$\pi _k\mcal{R} _{d-1}(X)=0$$ when $k\geq 3$ for any smooth projective real variety $X$ of dimension $d$.

The main theorem of this paper provides this vanishing in general and should be viewed as a massive generalization of both the classical vanishing of singular homology groups of a manifold in degree larger than the manifold and of the vanishing results discussed above.

\[mth\] Let $X$ be a quasi-projective real variety. Then $$\pi_{k}\mcal{R}_{q}(X) = 0$$ for $k\geq \dim X -q +1$.

In the case of divisors our result improves the previously known vanishing range. The case of real projective space described above shows that the theorem’s vanishing range is optimal.

The homotopy groups of reduced algebraic cycles $R _q(X)$ define a homology theory for real quasi-projective varieties $X$ introduced 