A minimal set of generators for the cohomology is:
  • w2(r26) of degree 2
  • w1(r13) of degree 1
  • w1(r14) of degree 1
  • w4(r36) of degree 4
  • w1(r15) of degree 1
  • w1(r11) of degree 1
The Steenrod operations are as follows:
  • Sq1(w2(r26)) = 0
  • Sq1(w4(r36)) = 0
  • Sq2(w4(r36)) = w1(r14)2w4(r36)
  • Sq3(w4(r36)) = 0
Here is a minimal system of equations:
  • (1)          w1(r14)2 + w1(r11)2 = 0
  • (2)          w1(r13)2 + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15) + w1(r15)2 = 0
  • Sq1(2)          w1(r13)2w1(r14) + w1(r13)w1(r14)2 + w1(r13)2w1(r15) + w1(r14)2w1(r15) + w1(r13)w1(r15)2 + w1(r14)w1(r15)2 = 0
Stiefel-Whitney classes:
  • w1(r1) = w1(r14) + w1(r15)
  • w1(r2) = w1(r13) + w1(r15)
  • w1(r3) = w1(r13) + w1(r14)
  • w1(r4) = w1(r15) + w1(r11)
  • w1(r5) = w1(r14) + w1(r11)
  • w1(r6) = w1(r13) + w1(r11)
  • w1(r7) = w1(r13) + w1(r14) + w1(r15) + w1(r11)
  • w1(r8) = w1(r13) + w1(r14) + w1(r11)
  • w1(r9) = w1(r13) + w1(r15) + w1(r11)
  • w1(r10) = w1(r14) + w1(r15) + w1(r11)
  • w1(r11) = w1(r11)
  • w1(r12) = w1(r13) + w1(r14) + w1(r15)
  • w1(r13) = w1(r13)
  • w1(r14) = w1(r14)
  • w1(r15) = w1(r15)
  • w2(r16) = w2(r26) + w1(r13)2 + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • w2(r17) = w2(r26) + w1(r14)2
  • w2(r18) = w2(r26) + w1(r13)2
  • w2(r19) = w2(r26) + w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • w2(r24) = w2(r26) + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • w2(r25) = w2(r26) + w1(r13)2 + w1(r14)2
  • w2(r26) = w2(r26)
  • w2(r27) = w2(r26) + w1(r13)2 + w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • w2(r32) = w1(r14)2
  • w4(r32) = w2(r26)2 + w2(r26)w1(r14)2 + w4(r36)
  • w2(r33) = w1(r14)2
  • w4(r33) = w2(r26)2 + w2(r26)w1(r14)2 + w4(r36)
  • w2(r36) = w1(r14)2
  • w4(r36) = w4(r36)
  • w2(r37) = w1(r14)2
  • w4(r37) = w4(r36)
Chern classes:
  • c1(r1) = w1(r13)2 + w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r2) = w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r3) = w1(r13)2 + w1(r14)2
  • c1(r4) = w1(r13)2 + w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r5) = 0
  • c1(r6) = w1(r13)2 + w1(r14)2
  • c1(r7) = w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r8) = w1(r13)2
  • c1(r9) = w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r10) = w1(r13)2 + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r11) = w1(r14)2
  • c1(r12) = w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r13) = w1(r13)2
  • c1(r14) = w1(r14)2
  • c1(r15) = w1(r13)2 + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r16) = w2(r26) + w1(r13)2 + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r17) = w2(r26) + w1(r14)2
  • c1(r18) = w2(r26) + w1(r13)2
  • c1(r19) = w2(r26) + w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r20) = w2(r26) + w1(r14)2
  • c1(r21) = w2(r26) + w1(r13)2 + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r22) = w2(r26) + w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r23) = w2(r26) + w1(r13)2
  • c1(r24) = w2(r26) + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r25) = w2(r26) + w1(r13)2 + w1(r14)2
  • c1(r26) = w2(r26)
  • c1(r27) = w2(r26) + w1(r13)2 + w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r28) = w2(r26) + w1(r13)2 + w1(r14)2
  • c1(r29) = w2(r26) + w1(r13)w1(r14) + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r30) = w2(r26) + w1(r13)2 + w1(r13)w1(r14) + w1(r14)2 + w1(r13)w1(r15) + w1(r14)w1(r15)
  • c1(r31) = w2(r26)
  • c1(r32) = w1(r14)2
  • c2(r32) = w2(r26)2 + w2(r26)w1(r14)2 + w4(r36)
  • c1(r33) = w1(r14)2
  • c2(r33) = w2(r26)2 + w2(r26)w1(r14)2 + w4(r36)
  • c1(r34) = w1(r14)2
  • c2(r34) = w2(r26)2 + w2(r26)w1(r14)2 + w4(r36)
  • c1(r35) = w1(r14)2
  • c2(r35) = w2(r26)2 + w2(r26)w1(r14)2 + w4(r36)
  • c1(r36) = w1(r14)2
  • c2(r36) = w4(r36)
  • c1(r37) = w1(r14)2
  • c2(r37) = w4(r36)
  • c1(r38) = w1(r14)2
  • c2(r38) = w4(r36)
  • c1(r39) = w1(r14)2
  • c2(r39) = w4(r36)
The algebra of Milnor constants is generated by:
  • c1(r2) + c1(r8) + c1(r16)
  • c1(r1)c1(r16) + c1(r2)c1(r16) + c1(r8)c1(r16) + c1(r16)2 + c2(r32)
  • w1(r13)w1(r14)3 + w1(r13)2w1(r14)w1(r11) + w1(r13)w1(r14)2w1(r11) + w1(r13)2w1(r11)2 which is not a combination of Chern classes; it is nilpotent of order 2
  • c1(r8)
  • c1(r2) + c1(r8)
  • c1(r1) + c1(r2) + c1(r8)
  • w1(r13)2w1(r14)w1(r15) + w1(r13)2w1(r15)w1(r11) which is not a combination of Chern classes; it is nilpotent of order 2
  • w1(r13)w1(r14)2w1(r15) + w1(r13)w1(r14)w1(r15)w1(r11) which is not a combination of Chern classes; it is nilpotent of order 2
The results above have been produced during the second run in April, 2008.