Exotic spheres
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== Introduction == | == Introduction == | ||
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+ | Let $\Theta_{n} := \{ \Sigma^n \simeq S^n \}$ denote the set of oriented diffeomorphism classes of closed, smooth n-manifolds homotopy equivalent to $S^n$. | ||
+ | |||
I should like to start this exotic spheres page by a [http://www.maths.ed.ac.uk/~aar/exotic.htm link] to my exotic spheres home page. | I should like to start this exotic spheres page by a [http://www.maths.ed.ac.uk/~aar/exotic.htm link] to my exotic spheres home page. | ||
This already has a large collection of original source material, but some of it would be inappropriate to put on the Manifold Atlas | This already has a large collection of original source material, but some of it would be inappropriate to put on the Manifold Atlas | ||
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A little later Shimada {{cite|Shimada1957}} used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. | A little later Shimada {{cite|Shimada1957}} used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. | ||
− | By Adams' solution of the Hopf-invariant 1 problem, {{cite|Adams1960}}, dimensions n = 7 and 15 are the only dimensions where an n-sphere can be fibre over an m-sphere for m<n. | + | By Adams' solution of the [[Wikipedia:Hopf_invariant| Hopf-invariant]] 1 problem, {{cite|Adams1960}}, dimensions n = 7 and 15 are the only dimensions where an n-sphere can be fibre over an m-sphere for m<n. |
</wikitex> | </wikitex> | ||
=== Plumbing === | === Plumbing === | ||
+ | <wikitex>; | ||
+ | As special case of the following construction goes back at least to {{cite|Milnor1959}} | ||
+ | Let $i = 1 \dots n$, let $(p_i, q_i)$ be pairs of positive integers such that $p_i + q_i + 2 = n$ and let $\alpha_i \in \pi_{p_i}(SO(q_i+1))$ by the clutching functions of $D^{q_i+1}$-bundles $D(\alpha_i)$. Let $G$ be a graph with verticies $\{v_1, \dots, v_n\}$ such that $E_{ij}$, the edge set between $v_i$ and $v_j$, is non-empty only if $p_i = q_j$. We form the manifold $W = W(G,\{\alpha_i\})$ from the disjoint union of the $D(\alpha_i)$ by identifying $D^{p_i+1} \times D^{q_i+1}$ and $D^{q_j+1} \times D^{p_j+1}$. | ||
+ | If $G$ is simply connected the $\Sigma(G, \{\alpha_i \}) : = \partial W$ is often a homotopy sphere. | ||
+ | We establish some notation for bundles and graphs: | ||
+ | * let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$-sphere, | ||
+ | * let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k))$, $k > 4s$, denote a generator, | ||
+ | * let $E_8$ denote the $E_8$-graph, | ||
+ | * let $T$ denote the graph with two vertices and one edge connecting them. | ||
− | + | The we have the following exotic spheres. | |
+ | * $\Sigma(E_8, \{\tau_{2k}, \dots \tau_{2k}\})$, the Milnor sphere, generates $bP_{4k}$ $k>1$. | ||
+ | * $\Sigma(T, \{\tau_{2k+1}, \tau_{2k+1}\})$, the Kervaire sphere, generates $bP_{4k+2}$ | ||
+ | * $\Sigma(T, \gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Z_2$. Here $\eta_7 : S^8 \to S^7$ is essential. | ||
</wikitex> | </wikitex> | ||
+ | === Twisting === | ||
+ | <wikitex>; | ||
+ | By {{cite|Cerf1970}} and {{cite|Smale1962a}} there is an isomorphism $\Theta_{n+1} \cong \Gamma_{n+1}$ for $n \geq 5$ where $\Gamma_{n+1} = \pi_0(\Diff_+(S^n))$ is the group of isotopy classes of orientation preserving diffeomorphisms of $S^n$. The map is given by | ||
+ | $$ \Gamma_{n+1} \to \Theta_{n+1}, [f] \longmapsto D^{n+1} \cup_f (-D^{n+1}).$$ | ||
+ | Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$ which are not isotopic to the identity. | ||
+ | </wikitex>; | ||
+ | </wikitex> | ||
== Invariants == | == Invariants == | ||
<wikitex>; | <wikitex>; | ||
− | Signature, Kervaire invaiant, $\alpha$-invariant | + | Signature, Kervaire invaiant, $\alpha$-invariant, Eels-Kuiper invariant, $s$-invariant. |
</wikitex> | </wikitex> | ||
Revision as of 15:08, 21 November 2009
Contents |
1 Introduction
Let denote the set of oriented diffeomorphism classes of closed, smooth n-manifolds homotopy equivalent to .
I should like to start this exotic spheres page by a link to my exotic spheres home page.
This already has a large collection of original source material, but some of it would be inappropriate to put on the Manifold Atlas
directly.
Andrew Ranicki
2 Construction and examples
Sphere bundles
The first known examples of exotic spheres were discovered by Milnor in [Milnor1956]. They are the total spaces of certain 3-sphere bundles over the 4-sphere as we now explain: the group parametrises linear -sphere bundles over ...
A little later Shimada [Shimada1957] used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres.
By Adams' solution of the Hopf-invariant 1 problem, [Adams1960], dimensions n = 7 and 15 are the only dimensions where an n-sphere can be fibre over an m-sphere for m<n.
2.1 Plumbing
As special case of the following construction goes back at least to [Milnor1959] Let , let be pairs of positive integers such that and let by the clutching functions of -bundles . Let be a graph with verticies such that , the edge set between and , is non-empty only if . We form the manifold from the disjoint union of the by identifying and .
If is simply connected the is often a homotopy sphere.
We establish some notation for bundles and graphs:
- let denote the tangent bundle of the -sphere,
- let , , denote a generator,
- let denote the -graph,
- let denote the graph with two vertices and one edge connecting them.
The we have the following exotic spheres.
- , the Milnor sphere, generates .
- , the Kervaire sphere, generates
- , generates
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. Here is essential.
2.2 Twisting
By [Cerf1970] and [Smale1962a] there is an isomorphism for where is the group of isotopy classes of orientation preserving diffeomorphisms of . The map is given by
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of which are not isotopic to the identity.;
</wikitex>
3 Invariants
Signature, Kervaire invaiant, -invariant, Eels-Kuiper invariant, -invariant.
4 Classification
[Kervaire&Milnor1963], [Levine1983]
5 Further discussion
... is welcome
6 References
- [Adams1960] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. MR0141119 (25 #4530) Zbl 0096.17404
- [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
- [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
- [Levine1983] J. P. Levine, Lectures on groups of homotopy spheres, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., 1126 (1983), 62–95. MR802786 (87i:57031) Zbl 0576.57028
- [Milnor1956] J. Milnor, On manifolds homeomorphic to the -sphere, Ann. of Math. (2) 64 (1956), 399–405. MR0082103 (18,498d) Zbl 0072.18402
- [Milnor1959] J. Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962–972. MR0110107 (22 #990) Zbl 0111.35501
- [Shimada1957] N. Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957), 59–69. MR0096223 (20 #2715) Zbl 0145.20303
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
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