Homotopy spheres I (Ex)
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In the following, you should use the [[Wikipedia:H-cobordism|$h$-cobordism Theorem]], \cite{Smale1962a|Theorem 3.1}. | In the following, you should use the [[Wikipedia:H-cobordism|$h$-cobordism Theorem]], \cite{Smale1962a|Theorem 3.1}. | ||
{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
| − | # Show that every homotopy $n$-sphere $\Sigma$ is diffeomorphic to | + | # Show that every homotopy $n$-sphere $\Sigma$ is diffeomorphic to a manifold $$ \Sigma_f : = D^n \cup_f D^n$$ where $f \colon S^{n-1} \cong S^{n-1}$ is a diffeomorphism. |
# Using 1. show that the group of homotopy $n$-spheres, $\Theta_n$, can be identified with the set of oriented diffeomorphism classes of smooth structures on $S^n$, at least for $n \geq 6$. | # Using 1. show that the group of homotopy $n$-spheres, $\Theta_n$, can be identified with the set of oriented diffeomorphism classes of smooth structures on $S^n$, at least for $n \geq 6$. | ||
# What happens for $n = 5$? | # What happens for $n = 5$? | ||
Latest revision as of 23:23, 26 August 2013
In the following, you should use the 
-cobordism Theorem, [Smale1962a, Theorem 3.1].
Exercise 0.1.
- Show that every homotopy
-sphere 
is diffeomorphic to a manifold where
is a diffeomorphism.
- Using 1. show that the group of homotopy -spheres,
, can be identified with the set of oriented diffeomorphism classes of smooth structures on 
, at least for 
.
- What happens for
?
- If
is a homotopy -sphere,
, show that 
is diffeomorphic to : this is [Kervaire&Milnor1963, Lemma 2.4].
[edit] References
- [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
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
