A minimal set of generators for the cohomology is:
  • w1(r6) of degree 1
  • w1(r7) of degree 1
  • w1(r5) of degree 1
  • w4(r34) of degree 4
The Steenrod operations are as follows:
  • Sq1(w4(r34)) = 0
  • Sq2(w4(r34)) = w1(r6)w1(r7)w4(r34) + w1(r6)w1(r5)w4(r34) + w1(r5)2w4(r34)
  • Sq3(w4(r34)) = 0
Here is a minimal system of equations:
  • (1)          w1(r6)2 + w1(r7)2 = 0
  • (2)          w1(r6)3 + w1(r6)2w1(r7) + w1(r6)2w1(r5) + w1(r6)w1(r5)2 = 0
Stiefel-Whitney classes:
  • w1(r1) = w1(r6) + w1(r7)
  • w1(r2) = w1(r7) + w1(r5)
  • w1(r3) = w1(r6) + w1(r5)
  • w1(r4) = w1(r6) + w1(r7) + w1(r5)
  • w1(r5) = w1(r5)
  • w1(r6) = w1(r6)
  • w1(r7) = w1(r7)
  • w2(r8) = 0
  • w2(r10) = w1(r6)2 + w1(r5)2
  • w2(r12) = w1(r5)2
  • w2(r14) = w1(r6)2
  • w2(r16) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • w2(r17) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • w2(r18) = w1(r6)w1(r7) + w1(r6)w1(r5)
  • w2(r19) = w1(r6)w1(r7) + w1(r6)w1(r5)
  • w2(r20) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5)
  • w2(r21) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5)
  • w2(r22) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • w2(r23) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • w2(r32) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • w4(r32) = w4(r34)
  • w2(r33) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • w4(r33) = w4(r34)
  • w2(r34) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • w4(r34) = w4(r34)
  • w2(r35) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • w4(r35) = w4(r34)
Chern classes:
  • c1(r1) = 0
  • c1(r2) = w1(r6)2 + w1(r5)2
  • c1(r3) = w1(r6)2 + w1(r5)2
  • c1(r4) = w1(r5)2
  • c1(r5) = w1(r5)2
  • c1(r6) = w1(r6)2
  • c1(r7) = w1(r6)2
  • c1(r8) = 0
  • c1(r9) = 0
  • c1(r10) = w1(r6)2 + w1(r5)2
  • c1(r11) = w1(r6)2 + w1(r5)2
  • c1(r12) = w1(r5)2
  • c1(r13) = w1(r5)2
  • c1(r14) = w1(r6)2
  • c1(r15) = w1(r6)2
  • c1(r16) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c1(r17) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c1(r18) = w1(r6)w1(r7) + w1(r6)w1(r5)
  • c1(r19) = w1(r6)w1(r7) + w1(r6)w1(r5)
  • c1(r20) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5)
  • c1(r21) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5)
  • c1(r22) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c1(r23) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c1(r24) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c1(r25) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c1(r26) = w1(r6)w1(r7) + w1(r6)w1(r5)
  • c1(r27) = w1(r6)w1(r7) + w1(r6)w1(r5)
  • c1(r28) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5)
  • c1(r29) = w1(r6)2 + w1(r6)w1(r7) + w1(r6)w1(r5)
  • c1(r30) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c1(r31) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c1(r32) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c2(r32) = w4(r34)
  • c1(r33) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c2(r33) = w4(r34)
  • c1(r34) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c2(r34) = w4(r34)
  • c1(r35) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c2(r35) = w4(r34)
  • c1(r36) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c2(r36) = w4(r34)
  • c1(r37) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c2(r37) = w4(r34)
  • c1(r38) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c2(r38) = w4(r34)
  • c1(r39) = w1(r6)w1(r7) + w1(r6)w1(r5) + w1(r5)2
  • c2(r39) = w4(r34)
The algebra of Milnor constants is generated by:
  • c2(r32)
  • c1(r2) + c1(r16)
  • c1(r4)
  • 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.