Exotic spheres

(Difference between revisions)

Jump to: navigation, search

Revision as of 01:20, 5 February 2010

This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

By a homotopy sphere ${{Stub}} == Introduction == <wikitex>; By a homotopy sphere $\Sigma^n$ we mean a closed smooth oriented n-manifold homotopy equivalent to $S^n$. The manifold $\Sigma^n$ is called an exotic sphere if it is not diffeomorphic to $S^n$. By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension $n \geq 5$ is homeomorphic to $S^n$: this statement holds in dimension 2 by the classification of [[Surfaces|surfaces]] and was famously proven in dimension 4 in {{cite|Freedman1982}} and in dimension 3 by Perelman. We define $$\Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$$ to be the set of oriented diffeomorphism classes of homotopy spheres. [[Wikipedia:Connected_sum|Connected sum]] makes $\Theta_n$ into an abelian group with inverse given by reversing orientation. An important subgroup of $\Theta_n$ is $bP_{n+1}$ which consists of those homotopy spheres which bound parallelisable manifolds. </wikitex> == Construction and examples == <wikitex>; The first exotic spheres discovered were certain 3-sphere bundles over the 4-sphere, {{cite|Milnor1956}}. Following this discovery there was a rapid development of techniques which construct exotic spheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting. </wikitex> === Plumbing === <wikitex>; As special case of the following construction goes back at least to {{cite|Milnor1959}}. Let $i \in \{1, \dots, n\}$, 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 over $S^{p_i + 1}$ $$ D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$$ Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that 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 $D^{q_j+1} \times D^{p_j+1}$ for each edge in $G$. If $G$ is simply connected then $$\Sigma(G, \{\alpha_i \}) : = \partial W$$ is often a homotopy sphere. We establish some notation for graphs, bundles and define * let $T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$, * let $E_8$ denote the $E_8$-graph, * let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$-sphere, * let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$, $k > 4s$, denote a generator, * let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$, denote a generator: * let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism, **$S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$, * let $\eta_n : S^{n+1} \to S^n$ be essential. Then we have the following exotic spheres. * $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$, the Milnor sphere, generates $bP_{4k}$, $k>1$. * $\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$, the Kervaire sphere, generates $bP_{4k+2}$. * $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$. **For general $k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic. * $\Sigma^8(\gamma_3^5, \eta_3\tau_4)$, generates $\Theta_8 = \Zz_2$. * $\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$. </wikitex> === Brieskorn varieties === <wikitex>; Let $z = (z_0, \dots , z_n)$ be a point in $\Cc^{n+1}$ and let $a = (a_0, \dots, a_n)$ be a string of n+1 positive integers. Given the complex variety $V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$ and the $\epsilon$-sphere $S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$ for small $\epsilon$, we define the closed smooth oriented (2n-1)-manifolds $$ W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$$ The manifolds $W^{2n-1}(a)$ are often called Brieskorn varieties. By construction, every $W^{2n-1}(a)$ lies in $S^{2n+1}$ and so bounds a parallelisable manifold. In {{cite|Brieskorn1966}} and {{cite|Brieskorn1966a}} (see also {{cite|Hirzebruch&Mayer1968}}), it is shown in particular that all homotopy spheres in $bP_{4k-1}$ and $bP_{4k-2}$ can be realised as $W(a)$ for some $a$. Let , \dots, 2$ be a string of 2k-1 2's in a row with $k \geq 2$, then there are diffeomorphisms $$ W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$$ $$ W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$$ </wikitex> === Sphere bundles === <wikitex>; The first known examples of exotic spheres were discovered by Milnor in {{cite|Milnor1956}}. They are the total spaces of certain 3-[[Wikipedia:Sphere_bundle#Sphere_bundles|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 $S^4$ where a pair $(m, n)$ gives rise to a bundle with Euler number $n$ and first Pontrjagin class (n+2m)$: here we orient $S^4$ and so identify $H^4(S^4; \Zz) = \Zz$. If we set $n = 1$ then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold $\Sigma^7_{m, 1}$, the total space of the bundle $(m, 1)$, is a homotopy sphere. Milnor first used a $\Zz_7$-invariant, called the $\lambda$-invariant, to show, e.g. that $\Sigma^7_{1, 2}$ is not diffeomorphic to $S^7$. A little later Kervaire and Milnor {{cite|Kervaire&Milnor1963}} proved that $\Theta_7 \cong \Zz_{28}$ and Eells and Kuiper {{cite|Eells&Kuiper1962}} defined a refinement of the $\lambda$-invariant, now called the Eells-Kuiper $\mu$-invariant, which in particular gives $$ \Sigma^7_{m, 1} = -(m(m-1)/56)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$$ 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. In this case $\pi_7(SO(8)) \cong \Zz \oplus \Zz$ and the bundle $(m, n)$ has Euler number $n$ and second Pontrjagin class (n+2m)$. Moreover $\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$ where the $\Zz_{8,128}$-summand is $bP_{16}$ as explained below. Results of {{cite|Wall1962a}} and {{cite|Eells&Kuiper1962}} combine to show that $$ \Sigma^{15}_{m, 1} = -(m(m-1)/16,256)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$$ *By Adams' solution of the [[Wikipedia:Hopf_invariant| Hopf-invariant]] 1 problem, {{cite|Adams1958}} and {{cite|Adams1960}}, the dimensions n = 3, 7 and 15 are the only dimensions in which a topological n-sphere can be fibre over an m-sphere for 0 < m < n. </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 $$ \Gamma_{n+1} \to \Theta_{n+1}, ~~~~~[f] \longmapsto \Sigma_{f} := D^{n+1} \cup_f (-D^{n+1}).$$ Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$ which are not isotopic to the identity. We give such a construction which probably goes back to Milnor: so far the earliest reference found is the problem list of the 1963 Seattle topology conference {{cite|Lashof1965}}. Represent $\alpha \in \pi_p(SO(q))$ and $\beta \in \pi_q(SO(p))$ by smooth compactly supported functions $\alpha : \Rr^p \to SO(q)$ and $\beta : \Rr^q \to SO(p)$ and define the following self-diffeomorphisms of $\Rr^p \times \Rr^q$ $$ F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),$$ $$ F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),$$ $$s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.$$ If follows that $s(\alpha, \beta)$ is compactly supported and so extends uniquely to a diffeomrphism of $S^{p+q}$. In this way we obtain a bilinear pairing $$ \sigma : \pi_q(SO(q)) \otimes \pi_q(SO(p)) \longrightarrow \Theta_{p+q+1}, ~~~~(\alpha, \beta) \longmapsto \Sigma_{s(\alpha, \beta)}$$ such that $$ \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).$$ In particular for $k=1, 2$ we see that $\sigma(\gamma_{4k-1}', \gamma_{4k-1}') \cong -\Sigma_M$ generates $bP_{4k}$. </wikitex> == Invariants == <wikitex>; Signature, Kervaire invariant, $\alpha$-invariant, Eels-Kuiper invariant, $s$-invariant. </wikitex> == Classification == <wikitex>; For $n =1, 2$ and $, $\Theta_n = \{ S^n \}$. For $n = 4$, $\Theta_4$ is unknown. We therefore concentrate on higher dimensions. For $n \geq 5$, the group of exotic n-spheres $\Theta_n$ fits into the following long exact sequence, first discovered in {{cite|Kervaire&Milnor1963}} (more details can also be found in {{cite|Levine1983}} and {{cite|Lück2001}}): $$ \dots \stackrel{\eta_{n+1}}{\longrightarrow} \pi_{n+1}(G/O) \stackrel{\sigma_{n+1}}{\longrightarrow} L_{n+1}(e) \stackrel{\omega_{n+1}}{\longrightarrow} \Theta_n \stackrel{\eta_n}{\longrightarrow} \pi_n(G/O) \stackrel{\sigma_n}{\longrightarrow} L_n(e) \to \dots~.$$ Here $L_n(e)$ is the n-th [[Wikipedia:L-theory|L-group]] of the the trivial group: $L_n(e) = \Zz, 0, \Zz/2, 0$ as i = 0, 1, 2 or 3 modulo 4 and the sequence ends at $L_5(e) = 0$. Also $O$ is the stable orthogonal group and $G$ is the stable group of homtopy self-equivalences of the sphere. There is a fibration $O \to G \to G/O$ and the groups $\pi_n(G/O)$ fit into the homtopy long exact sequence $$ \dots \to \pi_n(O) \to \pi_n(G) \to \pi_n(G/O) \to \pi_{n-1}(O) \to \pi_{n-1}(G) \to \dots $$ of this fibration. The homomorphism $J_n: \pi_n(O) \to \pi_n^S \cong \pi_n(G)$ is the [[Wikipedia:J-homomorphism|stable J-homomorphism]]. In particular, by {{cite|Serre1951}} the groups $\pi_i(G)$ are finite and by {{cite|Bott1959}}, {{cite|Adams1966}} and {{cite|Quillen1971}} the domain, image and kernel of $J_n$ have been completely determined. An important result in {{cite|Kervaire&Milnor1963}} is that the homomorphism $\sigma_{4k}$ is nonzero. The above sequence then gives {{beginthm|Theorem|{{cite|Kervaire&Milnor1963}}}} For $n \geq 5$, the group $\Theta_n$ is finite. Moreover there is an exact sequence $$ 0 \longrightarrow bP_{n+1} \longrightarrow \Theta_{n} \longrightarrow Coker(J_n) \stackrel{K_n}{\longrightarrow} C_n \longrightarrow 0 $$ where $bP_{n+1} := {Im}(\omega_{n+1})$, the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if $n$ is even. Moreover $C_n = 0$ unless $n = 4k+2$ when it is <script type="math/tex">\Sigma^n$ we mean a closed smooth oriented n-manifold homotopy equivalent to $S^n$ . The manifold $\Sigma^n$ is called an exotic sphere if it is not diffeomorphic to $S^n$ . By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension $n \geq 5$ is homeomorphic to $S^n$ : this statement holds in dimension 2 by the classification of surfaces and was famously proven in dimension 4 in [Freedman1982] and in dimension 3 by Perelman. We define

$\displaystyle \Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$ $\displaystyle \Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$

to be the set of oriented diffeomorphism classes of homotopy spheres. Connected sum makes $\Theta_n$ into an abelian group with inverse given by reversing orientation. An important subgroup of $\Theta_n$ is $bP_{n+1}$ which consists of those homotopy spheres which bound parallelisable manifolds.

2 Construction and examples

The first exotic spheres discovered were certain 3-sphere bundles over the 4-sphere, [Milnor1956]. Following this discovery there was a rapid development of techniques which construct exotic spheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting.

2.1 Plumbing

As special case of the following construction goes back at least to [Milnor1959].

Let $i \in \{1, \dots, n\}$ , 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 over $S^{p_i + 1}$

$\displaystyle D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$ $\displaystyle D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$

Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that 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 $D^{q_j+1} \times D^{p_j+1}$ for each edge in $G$ . If $G$ is simply connected then

$\displaystyle \Sigma(G, \{\alpha_i \}) : = \partial W$ $\displaystyle \Sigma(G, \{\alpha_i \}) : = \partial W$

is often a homotopy sphere. We establish some notation for graphs, bundles and define

let $T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$ ,
let $E_8$ denote the $E_8$ -graph,
let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$ -sphere,
let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$ , $k > 4s$ , denote a generator,
let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$ , denote a generator:
let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism,
- $S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$ ,
let $\eta_n : S^{n+1} \to S^n$ be essential.

Then we have the following exotic spheres.

$\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$ , the Milnor sphere, generates $bP_{4k}$ , $k>1$ .
$\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$ , the Kervaire sphere, generates $bP_{4k+2}$ .
$\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$ .
- For general $k$ , $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic.
$\Sigma^8(\gamma_3^5, \eta_3\tau_4)$ , generates $\Theta_8 = \Zz_2$ .
$\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$ , generates $\Theta_{16} = \Zz_2$ .

2.2 Brieskorn varieties

Let $z = (z_0, \dots , z_n)$ be a point in $\Cc^{n+1}$ and let $a = (a_0, \dots, a_n)$ be a string of n+1 positive integers. Given the complex variety $V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$ and the $\epsilon$ -sphere $S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$ for small $\epsilon$ , we define the closed smooth oriented (2n-1)-manifolds

$\displaystyle W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$ $\displaystyle W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$

The manifolds $W^{2n-1}(a)$ are often called Brieskorn varieties. By construction, every $W^{2n-1}(a)$ lies in $S^{2n+1}$ and so bounds a parallelisable manifold. In [Brieskorn1966] and [Brieskorn1966a] (see also [Hirzebruch&Mayer1968]), it is shown in particular that all homotopy spheres in $bP_{4k-1}$ and $bP_{4k-2}$ can be realised as $W(a)$ for some $a$ . Let $2, \dots, 2$ be a string of 2k-1 2's in a row with $k \geq 2$ , then there are diffeomorphisms

$\displaystyle W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$ $\displaystyle W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$

$\displaystyle W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$ $\displaystyle W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$

2.3 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 $\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$ -sphere bundles over $S^4$ where a pair $(m, n)$ gives rise to a bundle with Euler number $n$ and first Pontrjagin class $2(n+2m)$ : here we orient $S^4$ and so identify $H^4(S^4; \Zz) = \Zz$ . If we set $n = 1$ then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold $\Sigma^7_{m, 1}$ , the total space of the bundle $(m, 1)$ , is a homotopy sphere. Milnor first used a $\Zz_7$ -invariant, called the $\lambda$ -invariant, to show, e.g. that $\Sigma^7_{1, 2}$ is not diffeomorphic to $S^7$ . A little later Kervaire and Milnor [Kervaire&Milnor1963] proved that $\Theta_7 \cong \Zz_{28}$ and Eells and Kuiper [Eells&Kuiper1962] defined a refinement of the $\lambda$ -invariant, now called the Eells-Kuiper $\mu$ -invariant, which in particular gives

$\displaystyle \Sigma^7_{m, 1} = -(m(m-1)/56)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$ $\displaystyle \Sigma^7_{m, 1} = -(m(m-1)/56)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$

Shimada [Shimada1957] used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. In this case $\pi_7(SO(8)) \cong \Zz \oplus \Zz$ and the bundle $(m, n)$ has Euler number $n$ and second Pontrjagin class $6(n+2m)$ . Moreover $\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$ where the $\Zz_{8,128}$ -summand is $bP_{16}$ as explained below. Results of [Wall1962a] and [Eells&Kuiper1962] combine to show that

$\displaystyle \Sigma^{15}_{m, 1} = -(m(m-1)/16,256)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$ $\displaystyle \Sigma^{15}_{m, 1} = -(m(m-1)/16,256)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$

By Adams' solution of the Hopf-invariant 1 problem, [Adams1958] and [Adams1960], the dimensions n = 3, 7 and 15 are the only dimensions in which a topological n-sphere can be fibre over an m-sphere for 0 < m < n.

2.4 Twisting

By [Cerf1970] and [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

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

Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$ which are not isotopic to the identity. We give such a construction which probably goes back to Milnor: so far the earliest reference found is the problem list of the 1963 Seattle topology conference [Lashof1965].

Represent $\alpha \in \pi_p(SO(q))$ and $\beta \in \pi_q(SO(p))$ by smooth compactly supported functions $\alpha : \Rr^p \to SO(q)$ and $\beta : \Rr^q \to SO(p)$ and define the following self-diffeomorphisms of $\Rr^p \times \Rr^q$

$\displaystyle F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),$ $\displaystyle F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),$

$\displaystyle F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),$ $\displaystyle F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),$

$\displaystyle s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.$ $\displaystyle s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.$

If follows that $s(\alpha, \beta)$ is compactly supported and so extends uniquely to a diffeomrphism of $S^{p+q}$ . In this way we obtain a bilinear pairing

$\displaystyle \sigma : \pi_q(SO(q)) \otimes \pi_q(SO(p)) \longrightarrow \Theta_{p+q+1}, ~~~~(\alpha, \beta) \longmapsto \Sigma_{s(\alpha, \beta)}$ $\displaystyle \sigma : \pi_q(SO(q)) \otimes \pi_q(SO(p)) \longrightarrow \Theta_{p+q+1}, ~~~~(\alpha, \beta) \longmapsto \Sigma_{s(\alpha, \beta)}$

such that

$\displaystyle \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).$ $\displaystyle \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).$

In particular for $k=1, 2$ we see that $\sigma(\gamma_{4k-1}', \gamma_{4k-1}') \cong -\Sigma_M$ generates $bP_{4k}$ .

3 Invariants

Signature, Kervaire invariant, $\alpha$ -invariant, Eels-Kuiper invariant, $s$ -invariant.

4 Classification

For $n =1, 2$ and $3$ , $\Theta_n = \{ S^n \}$ . For $n = 4$ , $\Theta_4$ is unknown. We therefore concentrate on higher dimensions.

For $n \geq 5$ , the group of exotic n-spheres $\Theta_n$ fits into the following long exact sequence, first discovered in [Kervaire&Milnor1963] (more details can also be found in [Levine1983] and [Lück2001]):

$\displaystyle \dots \stackrel{\eta_{n+1}}{\longrightarrow} \pi_{n+1}(G/O) \stackrel{\sigma_{n+1}}{\longrightarrow} L_{n+1}(e) \stackrel{\omega_{n+1}}{\longrightarrow} \Theta_n \stackrel{\eta_n}{\longrightarrow} \pi_n(G/O) \stackrel{\sigma_n}{\longrightarrow} L_n(e) \to \dots~.$ $\displaystyle \dots \stackrel{\eta_{n+1}}{\longrightarrow} \pi_{n+1}(G/O) \stackrel{\sigma_{n+1}}{\longrightarrow} L_{n+1}(e) \stackrel{\omega_{n+1}}{\longrightarrow} \Theta_n \stackrel{\eta_n}{\longrightarrow} \pi_n(G/O) \stackrel{\sigma_n}{\longrightarrow} L_n(e) \to \dots~.$

Here $L_n(e)$ is the n-th L-group of the the trivial group: $L_n(e) = \Zz, 0, \Zz/2, 0$ as i = 0, 1, 2 or 3 modulo 4 and the sequence ends at $L_5(e) = 0$ . Also $O$ is the stable orthogonal group and $G$ is the stable group of homtopy self-equivalences of the sphere. There is a fibration $O \to G \to G/O$ and the groups $\pi_n(G/O)$ fit into the homtopy long exact sequence

$\displaystyle \dots \to \pi_n(O) \to \pi_n(G) \to \pi_n(G/O) \to \pi_{n-1}(O) \to \pi_{n-1}(G) \to \dots$ $\displaystyle \dots \to \pi_n(O) \to \pi_n(G) \to \pi_n(G/O) \to \pi_{n-1}(O) \to \pi_{n-1}(G) \to \dots$

of this fibration. The homomorphism $J_n: \pi_n(O) \to \pi_n^S \cong \pi_n(G)$ is the stable J-homomorphism. In particular, by [Serre1951] the groups $\pi_i(G)$ are finite and by [Bott1959], [Adams1966] and [Quillen1971] the domain, image and kernel of $J_n$ have been completely determined. An important result in [Kervaire&Milnor1963] is that the homomorphism $\sigma_{4k}$ is nonzero. The above sequence then gives

For $n \geq 5$ , the group $\Theta_n$ is finite. Moreover there is an exact sequence

$\displaystyle 0 \longrightarrow bP_{n+1} \longrightarrow \Theta_{n} \longrightarrow Coker(J_n) \stackrel{K_n}{\longrightarrow} C_n \longrightarrow 0$ $\displaystyle 0 \longrightarrow bP_{n+1} \longrightarrow \Theta_{n} \longrightarrow Coker(J_n) \stackrel{K_n}{\longrightarrow} C_n \longrightarrow 0$

where $bP_{n+1} := {Im}(\omega_{n+1})$ , the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if $n$ is even. Moreover $C_n = 0$ unless $n = 4k+2$ when it is $0$ or $\Zz/2$ .

The groups $Coker(J_n)$ are known for $n$ up to approximately 62. In general their determination is a very hard problem. Modulo this problem we see two remaining problems in the determination of $\Theta_n$ : an extension problem and the comptutation of the order of the groups $bP_{n+1}$ and $C_n$ . We discuss these in turn.

If $n \neq 2^{j} - 3$ the Kervaire-Milnor extension splits:

$\displaystyle \Theta_n \cong bP_{n+1} \oplus Ker(K_n).$ $\displaystyle \Theta_n \cong bP_{n+1} \oplus Ker(K_n).$

The map $K_{4k+2} : Coker(J_{4k+2}) \to \Zz/2$ is the Kervaire invariant and by definition $C_{4k+2} = Im(K_{4k+2})$ . By the long exact sequence above we have

The group $bP_{4k+2}$ is either $\Zz/2$ or $0$ . Moreover the following are equivalent:

$bP_{4k+2} = 0$ ,
the Kervaire sphere $\Sigma^{4k+2}_K$ is diffeomorphic to the standard sphere,
there is a framed manifold with Kervaire invariant 1: $C_{4k+2} \cong \Zz/2$ .

Conversely the following are equivalent:

$bP_{4k+2} = \Zz/2$
the Kervaire sphere $\Sigma^{4k+2}_K$ is not diffeomorphic to the standard sphere,
there is no framed manifold with Kervaire invariant 1: $C_{4k+2} \cong 0$ .

4.1 The orders of bP_4k and bP_4k+2

The group $bP_{4k}$ is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of $Im(J_{4k-1}) \subset \pi_{4k-1}(G)$ . Adams determined the latter group up to a factor of two which was settled by Quillen with a positive solution to the Adams conjecture.

Let $a_k = (3-(-1)^k)/2$ , let $B_k$ be the k-th Bernoulli number (topologist indexing) and for $x \in \Qq$ let $Num(x)$ denote the numerator of $x$ expressed in lowest form. Then for $k \geq 2$ , the order of $bP_{4k}$ is

$\displaystyle t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).$ $\displaystyle t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).$

Note that $Num(B_k/4k)$ is odd so the 2-primary order of $bP_{4k}$ is $a_k \cdot 2^{2k-2}$ while the odd part is $(2^{2k-1}-1) \cdot Num(B_k/4k)$ . Modulo the Adams conjecture the proof appeared in [Kervaire&Milnor1963, Section 7]. Detailed treatments can also be found in [Levine1983, Section 3] and [Lück2001, Chapter 6].

The next theorem describes the situation for $bP_{4k+2}$ which is now almost completely understood as well. References for the theorem are given in the remark which follows it.

The order of $bP_{4k+2}$ is given as follows:

$bP_2 = bP_{6} = bP_{14} = bP_{30} = bP_{62} = 0$ ,
$bP_{126} = 0$ or $\Zz/2$ ,
$bP_{4k+2} = \Zz/2$ else.

The following is a chronological list of determinations of $bP_{4k+2}$ :

$bP_{10} = \Zz/2$ , [Kervaire1960a].
$bP_{6} = bP_{14} = 0$ [Kervaire&Milnor1963].
$bP_{8k+2} = \Zz/2$ , [Anderson&Brown&Peterson1966a].
$bP_{30} = 0$ , [Mahowald&Tangora1967].
$bP_{4k+2} = \Zz/2$ unless $4k+2 = 2^j - 2$ [Browder1969].
$bP_{62} = 0$ , [Barratt&Jones&Mahowald1984].
$bP_{2^j - 2} = \Zz/2$ for $j \geq 8$ , [Hill&Hopkins&Ravenel2009].

5 Further discussion

$\displaystyle \def\curv{1.5pc}% Adjust the curvature of the curved arrows here \xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here \pi_n(\Top/O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(G) \\ & \pi_n(G/O) \ar[dr] \ar[ur] && \pi_{n-1}(\Top) \ar[dr] \ar[ur] \\ \pi_n(G) \ar[ur] \ar@/d\curv/[rr] && \pi_n(G/\Top) \ar[ur] \ar@/d\curv/[rr] && \pi_{n-1}(\Top/O) }$

6 External references

Wikipedia article on exotic spheres
http://www.maths.ed.ac.uk/~aar/exotic.htm Andrew Ranicki's exotic sphere home page, with many of the original papers.

7 References

[Adams1958] J. F. Adams, On the nonexistence of elements of Hopf invariant one, Bull. Amer. Math. Soc. 64 (1958), 279–282. MR0097059 (20 #3539) Zbl 0178.26106
[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
[Adams1966] J. F. Adams, On the groups $J(X)$ . IV, Topology 5 (1966), 21–71. MR0198470 (33 #6628) Zbl 0145.19902
[Anderson&Brown&Peterson1966a] D. W. Anderson, E. H. Brown and F. P. Peterson, $\SU$ -cobordism, $KO$ -characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54–67. MR0189043 (32 #6470) Zbl 0137.42802
[Barratt&Jones&Mahowald1984] M. G. Barratt, J. D. S. Jones and M. E. Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension $62$ , J. London Math. Soc. (2) 30 (1984), no.3, 533–550. MR810962 (87g:55025) Zbl 0606.55010
[Bott1959] R. Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR0110104 (22 #987) Zbl 0129.15601
[Brieskorn1966] E. Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2 (1966), 1–14. MR0206972 (34 #6788) Zbl 0145.17804
[Brieskorn1966a] E. V. Brieskorn, Examples of singular normal complex spaces which are topological manifolds, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 1395–1397. MR0198497 (33 #6652) Zbl 0144.45001
[Browder1969] W. Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. (2) 90 (1969), 157–186. MR0251736 (40 #4963) Zbl 0198.28501
[Brumfiel1968] G. Brumfiel, On the homotopy groups of $BPL$ and $PL/O$ , Ann. of Math. (2) 88 (1968), 291–311. MR0234458 (38 #2775) Zbl 0179.28601
[Brumfiel1969] G. Brumfiel, On the homotopy groups of $BPL$ and $PL/O$ . II, Topology 8 (1969), 305–311. MR0248830 (40 #2080) Zbl 0179.28601
[Brumfiel1970] G. Brumfiel, The homotopy groups of $BPL$ and $PL/O$ . III, Michigan Math. J. 17 (1970), 217–224. MR0271938 (42 #6819) Zbl 0201.55901
[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
[Eells&Kuiper1962] J. Eells and N. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93–110. MR0156356 (27 #6280) Zbl 0119.18704
[Freedman1982] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no.3, 357–453. MR679066 (84b:57006) Zbl 0528.57011
[Hill&Hopkins&Ravenel2009] M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, (2009). Available at the arXiv:0908.3724.
[Hirzebruch&Mayer1968] F. Hirzebruch and K. H. Mayer, $O(n)$ -Mannigfaltigkeiten, exotische Sphären und Singularitäten, Springer-Verlag, Berlin, 1968. MR0229251 (37 #4825) Zbl 0172.25304
[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
[Kervaire1960a] M. A. Kervaire, A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34 (1960), 257–270. MR0139172 (25 #2608) Zbl 0145.20304
[Lashof1965] R. Lashof, Problems in differential and algebraic topology. Seattle Conference, 1963, Ann. of Math. (2) 81 (1965), 565–591. MR0182961 (32 #443) Zbl 0137.17601
[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
[Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020
[Mahowald&Tangora1967] M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349–369. MR0214072 (35 #4924) Zbl 0213.24901
[Milnor1956] J. Milnor, On manifolds homeomorphic to the $7$ -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
[Quillen1971] D. Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR0279804 (43 #5525) Zbl 0219.55013
[Serre1951] J. Serre, Homologie singulière des espaces fibrès. Applications, Ann. of Math. (2) 54 (1951), 425–505. MR0045386 (13,574g) Zbl 0045.26003
[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
[Wall1962a] C. T. C. Wall, Classification of $(n-1)$ -connected $2n$ -manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR0145540 (26 #3071) Zbl 0218.57022

$ or $\Zz/2$. {{endthm}} The groups $Coker(J_n)$ are known for $n$ up to approximately 62. In general their determination is a very hard problem. Modulo this problem we see two remaining problems in the determination of $\Theta_n$: an extension problem and the comptutation of the order of the groups $bP_{n+1}$ and $C_n$. We discuss these in turn. {{beginthm|Theorem|{{cite|Brumfiel1968}}, {{cite|Brumfiel1969}}, {{cite|Brumfiel1970}}}} If $n \neq 2^{j} - 3$ the Kervaire-Milnor extension splits: $$\Theta_n \cong bP_{n+1} \oplus Ker(K_n).$$ {{endthm}} The map $K_{4k+2} : Coker(J_{4k+2}) \to \Zz/2$ is the Kervaire invariant and by definition $C_{4k+2} = Im(K_{4k+2})$. By the long exact sequence above we have {{beginthm|Theorem|{{cite|Kervaire&Milnor1963|Section 8}}}} The group $bP_{4k+2}$ is either $\Zz/2$ or $\Sigma^n$ we mean a closed smooth oriented n-manifold homotopy equivalent to $S^n$ $S^n$ . The manifold $\Sigma^n$ $\Sigma^n$ is called an exotic sphere if it is not diffeomorphic to $S^n$ $S^n$ . By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension $n \geq 5$ $n \geq 5$ is homeomorphic to $S^n$ $S^n$ : this statement holds in dimension 2 by the classification of surfaces and was famously proven in dimension 4 in [Freedman1982] and in dimension 3 by Perelman. We define

$\displaystyle \Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$ $\displaystyle \Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$

to be the set of oriented diffeomorphism classes of homotopy spheres. Connected sum makes $\Theta_n$ into an abelian group with inverse given by reversing orientation. An important subgroup of $\Theta_n$ is $bP_{n+1}$ which consists of those homotopy spheres which bound parallelisable manifolds.

2 Construction and examples

The first exotic spheres discovered were certain 3-sphere bundles over the 4-sphere, [Milnor1956]. Following this discovery there was a rapid development of techniques which construct exotic spheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting.

2.1 Plumbing

As special case of the following construction goes back at least to [Milnor1959].

Let $i \in \{1, \dots, n\}$ , 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 over $S^{p_i + 1}$

$\displaystyle D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$ $\displaystyle D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$

Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that 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 $D^{q_j+1} \times D^{p_j+1}$ for each edge in $G$ . If $G$ is simply connected then

$\displaystyle \Sigma(G, \{\alpha_i \}) : = \partial W$ $\displaystyle \Sigma(G, \{\alpha_i \}) : = \partial W$

is often a homotopy sphere. We establish some notation for graphs, bundles and define

let $T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$ ,
let $E_8$ denote the $E_8$ -graph,
let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$ -sphere,
let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$ , $k > 4s$ , denote a generator,
let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$ , denote a generator:
let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism,
- $S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$ ,
let $\eta_n : S^{n+1} \to S^n$ be essential.

Then we have the following exotic spheres.

$\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$ , the Milnor sphere, generates $bP_{4k}$ , $k>1$ .
$\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$ , the Kervaire sphere, generates $bP_{4k+2}$ .
$\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$ .
- For general $k$ , $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic.
$\Sigma^8(\gamma_3^5, \eta_3\tau_4)$ , generates $\Theta_8 = \Zz_2$ .
$\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$ , generates $\Theta_{16} = \Zz_2$ .

2.2 Brieskorn varieties

Let $z = (z_0, \dots , z_n)$ be a point in $\Cc^{n+1}$ and let $a = (a_0, \dots, a_n)$ be a string of n+1 positive integers. Given the complex variety $V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$ and the $\epsilon$ -sphere $S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$ for small $\epsilon$ , we define the closed smooth oriented (2n-1)-manifolds

$\displaystyle W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$ $\displaystyle W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$

The manifolds $W^{2n-1}(a)$ are often called Brieskorn varieties. By construction, every $W^{2n-1}(a)$ lies in $S^{2n+1}$ and so bounds a parallelisable manifold. In [Brieskorn1966] and [Brieskorn1966a] (see also [Hirzebruch&Mayer1968]), it is shown in particular that all homotopy spheres in $bP_{4k-1}$ and $bP_{4k-2}$ can be realised as $W(a)$ for some $a$ . Let $2, \dots, 2$ be a string of 2k-1 2's in a row with $k \geq 2$ , then there are diffeomorphisms

$\displaystyle W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$ $\displaystyle W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$

$\displaystyle W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$ $\displaystyle W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$

2.3 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 $\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$ -sphere bundles over $S^4$ where a pair $(m, n)$ gives rise to a bundle with Euler number $n$ and first Pontrjagin class $2(n+2m)$ : here we orient $S^4$ and so identify $H^4(S^4; \Zz) = \Zz$ . If we set $n = 1$ then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold $\Sigma^7_{m, 1}$ , the total space of the bundle $(m, 1)$ , is a homotopy sphere. Milnor first used a $\Zz_7$ -invariant, called the $\lambda$ -invariant, to show, e.g. that $\Sigma^7_{1, 2}$ is not diffeomorphic to $S^7$ . A little later Kervaire and Milnor [Kervaire&Milnor1963] proved that $\Theta_7 \cong \Zz_{28}$ and Eells and Kuiper [Eells&Kuiper1962] defined a refinement of the $\lambda$ -invariant, now called the Eells-Kuiper $\mu$ -invariant, which in particular gives

$\displaystyle \Sigma^7_{m, 1} = -(m(m-1)/56)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$ $\displaystyle \Sigma^7_{m, 1} = -(m(m-1)/56)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$

Shimada [Shimada1957] used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. In this case $\pi_7(SO(8)) \cong \Zz \oplus \Zz$ and the bundle $(m, n)$ has Euler number $n$ and second Pontrjagin class $6(n+2m)$ . Moreover $\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$ where the $\Zz_{8,128}$ -summand is $bP_{16}$ as explained below. Results of [Wall1962a] and [Eells&Kuiper1962] combine to show that

$\displaystyle \Sigma^{15}_{m, 1} = -(m(m-1)/16,256)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$ $\displaystyle \Sigma^{15}_{m, 1} = -(m(m-1)/16,256)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$

By Adams' solution of the Hopf-invariant 1 problem, [Adams1958] and [Adams1960], the dimensions n = 3, 7 and 15 are the only dimensions in which a topological n-sphere can be fibre over an m-sphere for 0 < m < n.

2.4 Twisting