Homotopy spheres III (Ex)

Assume Adam's result that the $J$$<wikitex>; Assume Adam's result that the $J$-homomorphism, $J \colon \pi_{8k+\epsilon}(SO) \to \pi_{8k+\epsilon}^S$, is injective for $\epsilon = 0, 1$ and for all $k$, \cite{Adams1966|Theorems 1.1 and 1.3}. {{beginthm|Exercise|\cite{Kervaire&Milnor1963|Theorem 3.1}}} Show that every homotopy sphere $\Sigma \in \Theta_{8k+\epsilon}$ is stably parallelisable. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]J$-homomorphism, $J \colon \pi_{8k+\epsilon}(SO) \to \pi_{8k+\epsilon}^S$$J \colon \pi_{8k+\epsilon}(SO) \to \pi_{8k+\epsilon}^S$, is injective for $\epsilon = 0, 1$$\epsilon = 0, 1$ and for all $k$$k$, [Adams1966, Theorems 1.1 and 1.3].

Exercise 0.1 [Kervaire&Milnor1963, Theorem 3.1]. Show that every homotopy sphere $\Sigma \in \Theta_{8k+\epsilon}$$\Sigma \in \Theta_{8k+\epsilon}$ is stably parallelisable.

References

• [Adams1966] J. F. Adams, On the groups $J(X)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5L6gfb$$J(X)$. IV, Topology 5 (1966), 21–71. MR0198470 (33 #6628) Zbl 0145.19902
• [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505