A minimal set of generators for the cohomology is:
  • w1(r6) of degree 1
  • w1(r7) of degree 1
  • w1(r13) of degree 1
  • w4(r14) of degree 4
The Steenrod operations are as follows:
  • Sq1(w4(r14)) = 0
  • Sq2(w4(r14)) = w1(r7)2w4(r14)
  • Sq3(w4(r14)) = 0
Here is a minimal system of equations:
  • (1)          w1(r6)2 + w1(r7)2 + w1(r6)w1(r13) + w1(r7)w1(r13) = 0
  • (2)          w1(r6)3 + w1(r6)2w1(r7) + w1(r6)w1(r7)2 + w1(r7)3 + w1(r6)2w1(r13) + w1(r13)3 = 0
Stiefel-Whitney classes:
  • w1(r1) = w1(r6) + w1(r7)
  • w1(r2) = w1(r6) + w1(r7) + w1(r13)
  • w1(r3) = w1(r13)
  • w1(r4) = w1(r7) + w1(r13)
  • w1(r5) = w1(r6) + w1(r13)
  • w1(r6) = w1(r6)
  • w1(r7) = w1(r7)
  • w1(r8) = w1(r13)
  • w2(r8) = 0
  • w1(r9) = w1(r13)
  • w2(r9) = w1(r6)2 + w1(r6)w1(r13)
  • w1(r10) = w1(r13)
  • w2(r10) = w1(r7)2 + w1(r13)2
  • w1(r11) = w1(r13)
  • w2(r11) = w1(r7)2 + w1(r13)2
  • w1(r12) = w1(r13)
  • w2(r12) = w1(r6)2 + w1(r7)2 + w1(r6)w1(r13) + w1(r13)2
  • w1(r13) = w1(r13)
  • w2(r13) = w1(r6)2 + w1(r7)2 + w1(r6)w1(r13) + w1(r13)2
  • w2(r14) = w1(r7)2
  • w4(r14) = w4(r14)
  • w2(r15) = w1(r7)2
  • w4(r15) = w4(r14)
  • w2(r18) = w1(r7)2
  • w4(r18) = w4(r14)
  • w2(r19) = w1(r7)2
  • w4(r19) = w4(r14)
Chern classes:
  • c1(r1) = w1(r6)2 + w1(r7)2
  • c1(r2) = w1(r6)2 + w1(r7)2 + w1(r13)2
  • c1(r3) = w1(r13)2
  • c1(r4) = w1(r7)2 + w1(r13)2
  • c1(r5) = w1(r6)2 + w1(r13)2
  • c1(r6) = w1(r6)2
  • c1(r7) = w1(r7)2
  • c1(r8) = w1(r13)2
  • c2(r8) = 0
  • c1(r9) = w1(r13)2
  • c2(r9) = w1(r6)4 + w1(r6)2w1(r13)2
  • c1(r10) = w1(r13)2
  • c2(r10) = w1(r6)4 + w1(r6)2w1(r13)2
  • c1(r11) = w1(r13)2
  • c2(r11) = w1(r6)4 + w1(r6)2w1(r13)2
  • c1(r12) = w1(r13)2
  • c2(r12) = 0
  • c1(r13) = w1(r13)2
  • c2(r13) = 0
  • c1(r14) = w1(r7)2
  • c2(r14) = w4(r14)
  • c1(r15) = w1(r7)2
  • c2(r15) = w4(r14)
  • c1(r16) = w1(r7)2
  • c2(r16) = w4(r14)
  • c1(r17) = w1(r7)2
  • c2(r17) = w4(r14)
  • c1(r18) = w1(r7)2
  • c2(r18) = w4(r14)
  • c1(r19) = w1(r7)2
  • c2(r19) = w4(r14)
  • c1(r20) = w1(r7)2
  • c2(r20) = w4(r14)
  • c1(r21) = w1(r7)2
  • c2(r21) = w4(r14)
The algebra of Milnor constants is generated by:
  • c2(r14)
  • c1(r1) + c1(r2) + c1(r4)
  • c1(r1) + c1(r2)
  • 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.