[ALGTOP-L] pi_* of homogenous spaces

marek at mat.uni.torun.pl marek at mat.uni.torun.pl
Tue May 18 11:58:12 EDT 2010

On May 14 Professor Brayton Gray wrote:
Here is a question that came to me from a physicist:

  Does there exist a homogenous space G/H with G a finite dimensional
  compact Lie group and H a closed subgroup such that pi_3 has an
  element x of order 2 which when co0mposed with eta is a nonzero
  element of pi_4?

I have been informed by Juno Mukai on the following:
G=SO(5), H=SO(3). Then, G/H=V_{5,2}=M^4 cup e^7,
M^4=E^2RP^2 (Z_2 Moore space).
And pi_3( ) cong pi_3(M^4) cong Z_2, pi_4(M^4) cong Z_2.
x=i: S^3 --> V_{5,2} (inclusion) generates pi_3(V_{5,2}
and (i circ eta_3) generates pi_4(V_{5,2}).

Marek Golasinski

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