Exotic spheres

1 Introduction

Let $\Theta_{n} := \{ \Sigma^n \simeq S^n \}$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; 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 link to [http://www.maths.ed.ac.uk/~aar/exotic.htm http://www.maths.ed.ac.uk/~aar/exotic.htm] to my exotic spheres home page. This already has a large collection of original source material. <br> Andrew Ranicki </wikitex> == Construction and examples == <wikitex>; === Sphere bundles === The first known examples of exotic spheres were discovered by Milnor in {{cite|Milnor1956}}. They are the total spaces of certain 3-sphere bundles over the 4-sphere as we now explain: the group $\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $-sphere bundles over $S^4$ ... 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 [[Wikipedia:Hopf_invariant| Hopf-invariant]] 1 problem, {{cite|Adams1960}}, dimensions n = 1, 3, 7 and 15 are the only dimensions where an n-sphere can be fibre over an m-sphere for m<n. </wikitex> === 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))$ be 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} = \Zz_2$. Here $\eta_7 : S^8 \to S^7$ is essential. </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}).$$ Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$ which are not isotopic to the identity. </wikitex>; == Invariants == <wikitex>; Signature, Kervaire invaiant, $\alpha$-invariant, Eels-Kuiper invariant, $s$-invariant. </wikitex> == Classification == <wikitex>; {{cite|Kervaire&Milnor1963}}, {{cite|Levine1983}} </wikitex> == Further discussion == <wikitex>; ... is welcome </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\Theta_{n} := \{ \Sigma^n \simeq S^n \}$ denote the set of oriented diffeomorphism classes of closed, smooth n-manifolds homotopy equivalent to $S^n$$S^n$.

I should like to start this exotic spheres page by a link to http://www.maths.ed.ac.uk/~aar/exotic.htm to my exotic spheres home page. This already has a large collection of original source material.
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 $\pi_3(SO(4)) \cong \Zz \oplus \Zz$$\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$$3$-sphere bundles over $S^4$$S^4$ ...

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 = 1, 3, 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 $i = 1 \dots n$$i = 1 \dots n$, let $(p_i, q_i)$$(p_i, q_i)$ be pairs of positive integers such that $p_i + q_i + 2 = n$$p_i + q_i + 2 = n$ and let $\alpha_i \in \pi_{p_i}(SO(q_i+1))$$\alpha_i \in \pi_{p_i}(SO(q_i+1))$ be the clutching functions of $D^{q_i+1}$$D^{q_i+1}$-bundles $D(\alpha_i)$$D(\alpha_i)$. Let $G$$G$ be a graph with verticies $\{v_1, \dots, v_n\}$$\{v_1, \dots, v_n\}$ such that $E_{ij}$$E_{ij}$, the edge set between $v_i$$v_i$ and $v_j$$v_j$, is non-empty only if $p_i = q_j$$p_i = q_j$. We form the manifold $W = W(G,\{\alpha_i\})$$W = W(G,\{\alpha_i\})$ from the disjoint union of the $D(\alpha_i)$$D(\alpha_i)$ by identifying $D^{p_i+1} \times D^{q_i+1}$$D^{p_i+1} \times D^{q_i+1}$ and $D^{q_j+1} \times D^{p_j+1}$$D^{q_j+1} \times D^{p_j+1}$.

If $G$$G$ is simply connected the $\Sigma(G, \{\alpha_i \}) : = \partial W$$\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))$$\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$$n$-sphere,
• let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k))$$\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k))$, $k > 4s$$k > 4s$, denote a generator,
• let $E_8$$E_8$ denote the $E_8$$E_8$-graph,
• let $T$$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}\})$$\Sigma(E_8, \{\tau_{2k}, \dots \tau_{2k}\})$, the Milnor sphere, generates $bP_{4k}$$bP_{4k}$k>1$.

• $\Sigma(T, \{\tau_{2k+1}, \tau_{2k+1}\})$$\Sigma(T, \{\tau_{2k+1}, \tau_{2k+1}\})$, the Kervaire sphere, generates $bP_{4k+2}$$bP_{4k+2}$

• $\Sigma(T, \gamma_{7}^9, \eta_7\tau_8)$$\Sigma(T, \gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$$\Theta_{16} = \Zz_2$. Here $\eta_7 : S^8 \to S^7$$\eta_7 : S^8 \to S^7$ is essential.

2.2 Twisting

By [Cerf1970] and [Smale1962a] there is an isomorphism $\Theta_{n+1} \cong \Gamma_{n+1}$$\Theta_{n+1} \cong \Gamma_{n+1}$ for $n \geq 5$$n \geq 5$ where $\Gamma_{n+1} = \pi_0(\Diff_+(S^n))$$\Gamma_{n+1} = \pi_0(\Diff_+(S^n))$ is the group of isotopy classes of orientation preserving diffeomorphisms of $S^n$$S^n$. The map is given by

$\displaystyle \Gamma_{n+1} \to \Theta_{n+1}, ~~~~~[f] \longmapsto D^{n+1} \cup_f (-D^{n+1}).$ $$\displaystyle \Gamma_{n+1} \to \Theta_{n+1}, ~~~~~[f] \longmapsto D^{n+1} \cup_f (-D^{n+1}).$$

Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$$S^n$ which are not isotopic to the identity.;

3 Invariants

Signature, Kervaire invaiant, $\alpha$$\alpha$-invariant, Eels-Kuiper invariant, $s$$s$-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 $7$$7$-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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