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Homotopy groups of flag manifolds

Flag manifolds

To define the flag manifold M = M(n₁ ,..., n_k), let n = n₁ +...+ n_k. Then M is defined to be the quotient of SO(n) by SO(n₁) x...x SO(n_k).

Strictly speaking, these are oriented flag manifolds, and the unoriented flag manifolds are defined by replacing SO with O.

This concept includes

• The Lie groups SO(n) = M(1 ,..., 1) = M(1ⁿ)
• The spheres S^n-1 = M(1, n-1)
• The Grassmannian manifolds G_k(Rⁿ) = M(k, n-k)
• The Stiefel varieties V_k(Rⁿ) = M(1^k, n-k)

Note that the order of the numbers n_i in M(n₁ ,..., n_k) is irrelevant (up to homeomorphism).

Why should I care?

My motivation comes from theoretical physics. Suppose that the universe is n dimensional, that at every point there is a vector, this vector field is continuous, and that these vectors want to be of unit length, in the sense that this minimises some sort of energy. Suppose these vectors started off randomly at the big bang, and then settled down, locally minimising the energy. Then there will be isolated points where the vector is zero, simply because pi_n-1S^n-1=Z. These points come in two kinds, which we may as well call positive and negative. Likes repel, unlikes attract, and when unlikes get together they annihilate with a release of energy. This should sound familiar.

Now suppose that we postulate, instead of 1 vector field trying to be S^n-1, we have n vector fields trying to be unit length and normal to each other, in other words trying to be SO(n). Now if pi_n-k-1SO(n) is non-trivial we get whole k-dimensional manifolds in n-dimensional space which have singular frames.

These two cases are M(1, n-1) and M(1ⁿ), and the concept generalises naturally to any flag manifold M. If pi_n-k-1M is non-trivial we get singular k-dimensional manifolds. If k=1, this should sound like string theory.

If the free part of pi_n-k-1M is Z then we get two kinds of k-dimensional manifold, as before, and it makes sense to call these positive and negative. If the free part of pi_n-k-1M is Z^d then there are d properties which can separately be positive or negative on each manifold.

If pi_n-k-1M has no free part, then the only singular manifolds annihilate themselves, which can just about work for k=0, but seems very unstable beyond that. If pi_n-k-1M has both a free part and a torsion part, however, the situation changes. Now the k-manifolds are stable, because of the free part, but they also carry a modular charge from the torsion part. This should sound like charged particles which also carry a (mod 2) spin.

Homotopy groups

This table gives all the homotopy groups pi_i of the flag manifolds M up to dimension n = 10. At least, it would if I could calculate them, but in fact it only gives the free parts. The homotopy groups in question are all of the form Z^dA, where A is a finite abelian group. This table specifies only d.

For each M, there are only finitely many i for which pi_i(M) is infinite, so the table is complete, in the sense that for each M, any homotopy group not given is finite.

The table

n=2       i=1
        11: 1
n=3       i=1 2 3
       111: 0 0 1
        12: 0 1 1
n=4       i=1 2 3
      1111: 0 0 2
       112: 0 1 2
        13: 0 0 1
        22: 0 2 2
n=5       i=1 2 3 4 5 6 7
     11111: 0 0 1 0 0 0 1
      1112: 0 1 1 0 0 0 1
       113: 0 0 0 0 0 0 1
       122: 0 2 1 0 0 0 1
        14: 0 0 0 1 0 0 1
        23: 0 1 0 0 0 0 1
n=6       i=1 2 3 4 5 6 7
    111111: 0 0 1 0 1 0 1
     11112: 0 1 1 0 1 0 1
      1113: 0 0 0 0 1 0 1
      1122: 0 2 1 0 1 0 1
       114: 0 0 0 1 1 0 1
       123: 0 1 0 0 1 0 1
        15: 0 0 0 0 1
       222: 0 3 1 0 1 0 1
        24: 0 1 0 1 1 0 1
        33: 0 0 0 1 1 0 1
n=7       i=1 2 3 4 5 6 7 8 91011
   1111111: 0 0 1 0 0 0 1 0 0 0 1
    111112: 0 1 1 0 0 0 1 0 0 0 1
     11113: 0 0 0 0 0 0 1 0 0 0 1
     11122: 0 2 1 0 0 0 1 0 0 0 1
      1114: 0 0 0 1 0 0 1 0 0 0 1
      1123: 0 1 0 0 0 0 1 0 0 0 1
       115: 0 0 0 0 0 0 0 0 0 0 1
      1222: 0 3 1 0 0 0 1 0 0 0 1
       124: 0 1 0 1 0 0 1 0 0 0 1
       133: 0 0 0 1 0 0 1 0 0 0 1
        16: 0 0 0 0 0 1 0 0 0 0 1
       223: 0 2 0 0 0 0 1 0 0 0 1
        25: 0 1 0 0 0 0 0 0 0 0 1
        34: 0 0 0 2 0 0 1 0 0 0 1
n=8       i=1 2 3 4 5 6 7 8 91011
  11111111: 0 0 1 0 0 0 2 0 0 0 1
   1111112: 0 1 1 0 0 0 2 0 0 0 1
    111113: 0 0 0 0 0 0 2 0 0 0 1
    111122: 0 2 1 0 0 0 2 0 0 0 1
     11114: 0 0 0 1 0 0 2 0 0 0 1
     11123: 0 1 0 0 0 0 2 0 0 0 1
      1115: 0 0 0 0 0 0 1 0 0 0 1
     11222: 0 3 1 0 0 0 2 0 0 0 1
      1124: 0 1 0 1 0 0 2 0 0 0 1
      1133: 0 0 0 1 0 0 2 0 0 0 1
       116: 0 0 0 0 0 1 1 0 0 0 1
      1223: 0 2 0 0 0 0 2 0 0 0 1
       125: 0 1 0 0 0 0 1 0 0 0 1
       134: 0 0 0 2 0 0 2 0 0 0 1
        17: 0 0 0 0 0 0 1
      2222: 0 4 1 0 0 0 2 0 0 0 1
       224: 0 2 0 1 0 0 2 0 0 0 1
       233: 0 1 0 1 0 0 2 0 0 0 1
        26: 0 1 0 0 0 1 1 0 0 0 1
        35: 0 0 0 1 0 0 1 0 0 0 1
        44: 0 0 0 3 0 0 2 0 0 0 1
n=9       i=1 2 3 4 5 6 7 8 9101112131415
 111111111: 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
  11111112: 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1
   1111113: 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1
   1111122: 0 2 1 0 0 0 1 0 0 0 1 0 0 0 1
    111114: 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1
    111123: 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1
     11115: 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
    111222: 0 3 1 0 0 0 1 0 0 0 1 0 0 0 1
     11124: 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1
     11133: 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1
      1116: 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1
     11223: 0 2 0 0 0 0 1 0 0 0 1 0 0 0 1
      1125: 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1
      1134: 0 0 0 2 0 0 1 0 0 0 1 0 0 0 1
       117: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
     12222: 0 4 1 0 0 0 1 0 0 0 1 0 0 0 1
      1224: 0 2 0 1 0 0 1 0 0 0 1 0 0 0 1
      1233: 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1
       126: 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1
       135: 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1
       144: 0 0 0 3 0 0 1 0 0 0 1 0 0 0 1
        18: 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
      2223: 0 3 0 0 0 0 1 0 0 0 1 0 0 0 1
       225: 0 2 0 0 0 0 0 0 0 0 1 0 0 0 1
       234: 0 1 0 2 0 0 1 0 0 0 1 0 0 0 1
        27: 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
       333: 0 0 0 2 0 0 1 0 0 0 1 0 0 0 1
        36: 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1
        45: 0 0 0 2 0 0 0 0 0 0 1 0 0 0 1
n=10      i=1 2 3 4 5 6 7 8 9101112131415
1111111111: 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1
 111111112: 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1
  11111113: 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1
  11111122: 0 2 1 0 0 0 1 0 1 0 1 0 0 0 1
   1111114: 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1
   1111123: 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1
    111115: 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1
   1111222: 0 3 1 0 0 0 1 0 1 0 1 0 0 0 1
    111124: 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1
    111133: 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1
     11116: 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1
    111223: 0 2 0 0 0 0 1 0 1 0 1 0 0 0 1
     11125: 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1
     11134: 0 0 0 2 0 0 1 0 1 0 1 0 0 0 1
      1117: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1
    112222: 0 4 1 0 0 0 1 0 1 0 1 0 0 0 1
     11224: 0 2 0 1 0 0 1 0 1 0 1 0 0 0 1
     11233: 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1
      1126: 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1
      1135: 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1
      1144: 0 0 0 3 0 0 1 0 1 0 1 0 0 0 1
       118: 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1
     12223: 0 3 0 0 0 0 1 0 1 0 1 0 0 0 1
      1225: 0 2 0 0 0 0 0 0 1 0 1 0 0 0 1
      1234: 0 1 0 2 0 0 1 0 1 0 1 0 0 0 1
       127: 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
      1333: 0 0 0 2 0 0 1 0 1 0 1 0 0 0 1
       136: 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1
       145: 0 0 0 2 0 0 0 0 1 0 1 0 0 0 1
        19: 0 0 0 0 0 0 0 0 1
     22222: 0 5 1 0 0 0 1 0 1 0 1 0 0 0 1
      2224: 0 3 0 1 0 0 1 0 1 0 1 0 0 0 1
      2233: 0 2 0 1 0 0 1 0 1 0 1 0 0 0 1
       226: 0 2 0 0 0 1 0 0 1 0 1 0 0 0 1
       235: 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1
       244: 0 1 0 3 0 0 1 0 1 0 1 0 0 0 1
        28: 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1
       334: 0 0 0 3 0 0 1 0 1 0 1 0 0 0 1
        37: 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1
        46: 0 0 0 2 0 1 0 0 1 0 1 0 0 0 1
        55: 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1
          i=1 2 3 4 5 6 7 8 9101112131415

Calculation

The main tool was the long exact sequence which results from the following fibration.

Suppose we partition the numbers n₁ ,..., n_k into m subsets (n₁ ,..., n_k(1)) ,..., (n_k(m-1)+1 ,..., n_k(m)).

We define s_i = n_k(i-1)+1 +...+ n_k(i) (where k(0)=0).

Then there is a fibration F -> E -> B, where

• F = M(n₁ ,..., n_k(1)) x...x M(n_k(m-1)+1 ,..., n_k(m))
• E = M(n₁ ,..., n_k)
• B = M(s₁ ,..., s_m)

To start the process off, three sources of information were used, all available from [1].

• The homotopy groups of the spheres S^n-1 = M(1, n-1) are well-known.
• Bott periodicity specifies pi_i SO(n) = pi_i M(1ⁿ), when i < n-2.
• When i = n-k, we know pi_i V_k(Rⁿ) = pi_i M(1^k, n-k).

Accuracy

The tables were calculated by hand and then typed in by hand. I have done my best to eliminate errors, but please let me know of any which you find.

References

[1] Fibre Bundles (third edition), Dale Husemoller, Graduate Texts in Mathematics 20, Springer-Verlag.

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