A minimal set of generators for the cohomology is:
  • w4(r8) of degree 4
  • w1(r5) of degree 1
  • w1(r6) of degree 1
  • w1(r7) of degree 1
The Steenrod operations are as follows:
  • Sq1(w4(r8)) = 0
  • Sq2(w4(r8)) = w4(r8)w1(r7)2
  • Sq3(w4(r8)) = 0
Here is a minimal system of equations:
  • (1)          w1(r5)2 + w1(r5)w1(r6) + w1(r6)2 + w1(r5)w1(r7) + w1(r6)w1(r7) = 0
  • Sq1(1)          w1(r5)2w1(r6) + w1(r5)w1(r6)2 + w1(r5)2w1(r7) + w1(r6)2w1(r7) + w1(r5)w1(r7)2 + w1(r6)w1(r7)2 = 0
Stiefel-Whitney classes:
  • w1(r1) = w1(r6) + w1(r7)
  • w1(r2) = w1(r5) + w1(r7)
  • w1(r3) = w1(r5) + w1(r6)
  • w1(r4) = w1(r5) + w1(r6) + w1(r7)
  • w1(r5) = w1(r5)
  • w1(r6) = w1(r6)
  • w1(r7) = w1(r7)
  • w2(r8) = w1(r7)2
  • w4(r8) = w4(r8)
Chern classes:
  • c1(r1) = w1(r6)2 + w1(r7)2
  • c1(r2) = w1(r5)2 + w1(r7)2
  • c1(r3) = w1(r5)2 + w1(r6)2
  • c1(r4) = w1(r5)2 + w1(r6)2 + w1(r7)2
  • c1(r5) = w1(r5)2
  • c1(r6) = w1(r6)2
  • c1(r7) = w1(r7)2
  • c1(r8) = w1(r7)2
  • c2(r8) = w4(r8)
  • c1(r9) = w1(r7)2
  • c2(r9) = w4(r8)
The algebra of Milnor constants is generated by:
  • c2(r8)
  • c1(r1) + c1(r4)
  • c1(r2) + c1(r4)
  • c1(r1) + c1(r2) + c1(r4)
  • Note: the only algebraic cycles in the cohomology are Chern classes.
The results above have been produced during the second run in April, 2008.