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,

n^k indicates the direct sum of k-copies of Z/nZ.

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

pi_n+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

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

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

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

pi_n+k(Pⁿ(2)) for k<9