Table of the homotopy groups of the suspensions of the projective plane

From My thesis, University of Rochester, 1995.

In the following table,

  • an integer n indicates a cyclic group Z/nZ of order n,
  • "infty" indicates the infinite cyclic group Z,
  • the symbol "+" indicates the direct sum of the (abelian) groups,
  • nk indicates the direct sum of k-copies of Z/nZ.

  • Pn(2)=\Sigman-2RP2, the (n-2)-fold suspension of the real projective plane RP2

    pin+k(Pn(2)) for k<9

    k\n

    n=3

    n=4

    n=5

    n=6

    n=7

    n=8

    n=9

    n=10

    n=11

    n>11

    k=-1

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    k=0

    4

    2

    2

    2

    2

    2

    2

    2

    2

    2

    k=1

    4

    4

    4

    4

    4

    4

    4

    4

    4

    4

    k=2

    23

    2+4

    2+4

    22

    22

    22

    22

    22

    22

    22

    k=3

    25

    22

    22

    22

    2

    2

    2

    2

    2

    2

    k=4

    22+42+8

    23

    22

    8

    2

    0

    0

    0

    0

    0

    k=5

    27+4

    23

    2+4

    22

    2

    2

    2

    2

    2

    2

    k=6

    210

    22+4

    22+4

    22

    22

    2+4

    2+4

    22

    22

    22

    k=7

    25+43+8

    23+4

    23

    23

    23+4

    24

    24

    24

    23

    23

    k=8

    214+42

    25+4

    22+42

    24+4

    2+42+8

    23+42

    23+42

    2+42+8

    22+42

    2+42