Self-Homotopy of the Double Suspension of the Real 6-Projective Space

Abstract

Abstract. Let P n be the real n-dimensional projective space. We determine the group structure of the self-homotopy set of the double suspension of P n where n is 3, 4, 5 and 6 using the ideas and methods of the second author (The suspension order of the real even dimensional projective space, J. Math. Kyoto Univ. 43(4) (2003), 755-769).

Groups of Self-Equivalences of Suspended Real Projective Spaces

“…Finally, we consider the structure of self-equivalence group of k P 6 . From Proposition 2.5 and Theorem 3.7 of [4], Theorem 3.4 of [3], Theorem 4.5 of [2] and Section 2, we have…”

“…Then we have xι •η 2 = xη 2 from the equation (xι •η 2 ) = (xη 2 ) and the first equality follows. Since the kernel of : 2 . Hence, xι • ξ 2 ≡ xξ 2 mod 2ξ 2 = 0 and the second equality follows.…”

“…The group [ k P 4 , k P 4 ] is given by Theorem 6.8 of [8], Proposition 3.2 of [2], Theorem 2.4 of [3] and Section 2 as follows: If we take (a, b, c) = (ι + ζ 3 , 3ι, −ι), an easy analogous computation shows that…”

“…The group [ 2 P n , 2 P n ] for 3 ≤ n ≤ 6, the stable group [ n P n , n P n ] for n = 4, 6, the group [ k P 6 , k P 6 ] for k = 4, 5 and [ P n , P n ] for n = 3, 4 were given in [2,3,4,8], respectively. The others are calculated in Section 2.…”

“…2 and (Z 2 ) 4 for k = 2, 3 and k ≥ 4, respectively. (iv) E( k P 6 ) is isomorphic to (Z 2 ) 6 , D 4 × (Z 2 ) 3 , (Z 2 ) 5 and (Z 2 ) 4 for k = 2, 3, 4…”