What are the homotopy groups of the oriented Grassmannian $Gr^{+}(p,q)$ (p-planes in $R^{p+q}$) $\pi_{r}(Gr^{+}(p,q))$, $r \le pq$?
Do you know any references on the web about it?
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7$\begingroup$ This is about as hard as computing the homotopy groups of spheres; for example, $Gr^{+}(2, 2)$ is $S^2 \times S^2$. $\endgroup$– Qiaochu YuanCommented Apr 7, 2016 at 15:50
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2$\begingroup$ What do you mean by $r=<p.q$? $\endgroup$– Tom GoodwillieCommented Apr 7, 2016 at 17:02
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$\begingroup$ Perhaps a better place to start, Renan, is what do you need to know about these groups, or what information would help you? $\endgroup$– Ryan BudneyCommented Apr 11, 2016 at 2:53
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$\begingroup$ I mean r less or equal than the dimension of the grassmannia Gr (p,q) $\endgroup$– Renan AssimosCommented Apr 11, 2016 at 16:29
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$\begingroup$ Prof Budney, I just would like to know, if there exists such a description, the r-homotopy groups of the Grassmannian for r less or equal than the dim of the grassmannian itself. Sorry if I don't manage to be more specific, but it is exactly because I don't understand what can appears there.... $\endgroup$– Renan AssimosCommented Apr 11, 2016 at 16:37
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