2

Consider real Grassmanians $$\mathrm{Gr}(n,m)=O(n+m)/O(n)\times O(m),$$ what are the homotopy groups $\pi_k[\mathrm{Gr}(n,m)]$ for generic $k,m,n$?

(I did some search and only found the stable case following Bott periodicity and $k=2$ case discussed in a related question. If the generic formula is not available, I would at least like to know the cases of $n=4$, $m=6$ and $k=0,1,2,3,4,5$. Using exact sequence, it seems that I can only determine $\pi_5[\mathrm{Gr}(4,6)]=\mathbb{Z}_2\times\mathbb{Z}_2$ (is that correct?).)

Bernard
  • 179,256
Everett You
  • 315
  • 3
  • 9
  • $Gr(n,m)$ has the homotopy type of a finite CW complex with finite $\pi_1$. The only simply-connected finite CW complexes whose homotopy is completely known are tori. There's a small gap there, but I suspect it's also true that the only connected finite CW complexes with finite $\pi_1$ whose homotopy is completely known are still just the tori. So you will not get a complete answer. There should be a suitable range where the homotopy is known though. – tcamps Apr 21 '21 at 06:31
  • @tcamps: Tori aren't simply connected. The only finite CW complexes for which all homotopy groups are known are those which are contractible or have contractible universal cover. – Michael Albanese Jun 17 '21 at 22:16

0 Answers0