# Exotic spheres

## 1 Introduction

By a homotopy sphere $\Sigma^n$${{Stub}} == Introduction == ; 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. == Construction and examples == ; 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. === Plumbing === ; 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. === 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, following {{cite|Milnor1968}} we define the closed smooth oriented (n-2)-connected (2n-1)-manifold 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} 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}. === Sphere bundles === ; 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)/2)\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)/2)\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. === Twisting === ; 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 diffeomorphism 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}. == Invariants == ; Finding invariants of exotic sphere \Sigma which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold W with \partial W \cong \Sigma. In this case finding an intrinsic definition and or computation of the relevant invariant can also be subtle. We begin by listing some invariants which are equal for all exotic spheres. {{beginthm|Proposition}} Let \Sigma be a closed smooth manifold homeomorphic to the n-sphere. Then # there is an isomorphism of tangent bundles T\Sigma \cong TS^n, # the signature of \Sigma vanishes, # the Kervaire invariant of (\Sigma, F) is zero for every framing of \Sigma. (To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to S^n.) {{endthm}} {{beginrem|Remark}} The analogue of the first statement for the stable tangent bundle was proven in \cite{Kervaire&Milnor1963|Theorem 3.1}. A proof of the unstable statement is given in \cite{Ray&Pedersen1980|Lemma 1.1}. The next two statements are obvious since both the signature and Kervaire invariant are defined to be zero if n = 2k+1 and via a symmetric or quadratic form on H_k(\Sigma; \Zz) = 0 if n = 2k. {{endrem}} === Bordism classes === ; As every homotopy sphere is stably parallelisable, homotopy spheres admit [[B-Bordism|B-structures]] for any B. If B is such that [S^n, F] \mapsto 0 \in \Omega_n^B for any stable framing F of S^n, then we obtain a well-defined homomorphism \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F]. * If B = BO\langle k \rangle for [n/2] + 1 < k < n+2 then \Omega_n^B is isomorphic to almost framed bordism and the homomorphism \eta^B is the same thing as the \eta: \Theta_n \to \pi_n(G/O) in Theorem \ref{thm-ses}. * Perhaps surprisingly \eta_n^{\Spin} \neq 0 for all n = 8k+1, 8k+2, as explained in the next subsection. * In general determining \eta^B is a hard an interesting problem. * B-coboundaries for elements of Ker(\eta^B_n) are often used to define invariants of B-null bordant homotopy spheres. === The α-invariant === ; In dimensions n > 1, every exotic sphere \Sigma has a unique Spin structure and from above we have the homomorphism \eta_n^{\Spin} : \Theta_n \to \Omega_n^{\Spin}. Recall the \alpha-invariant homomorphism \alpha : \Omega_*^{\Spin} \to KO^{-*} and that there are isomorphisms KO^{-8k-1} \cong KO^{-8k-2} \cong \Zz/2 for all k \geq 1. {{beginthm|Theorem|\cite{Anderson&Brown&Peterson1967}}} We have \eta_n^{\Spin}(\Sigma) = 0 if and only if \alpha \circ \eta_n^{\Spin}(\Sigma) = 0 and \eta_n^{\Spin} \neq 0 if and only if n = 8k+1 or k+2. {{endthm}} {{beginrem|Remark}} Exotic spheres \Sigma with \alpha(\Sigma) \neq 0 are often called Hitchin spheres, after \cite{Hitchin1974}: see the discussion of curvature [[#Curvature on exotic spheres|below]]. {{endrem}} === The Eells-Kuiper invariant === ; === The s-invariant === ; == Classification == ; 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 n = 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}}}}\label{thm-ses} 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 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

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

to be the set of oriented diffeomorphism classes of homotopy spheres. Connected sum makes $\Theta_n$$\Theta_n$ into an abelian group with inverse given by reversing orientation. An important subgroup of $\Theta_n$$\Theta_n$ is $bP_{n+1}$$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\}$$i \in \{1, \dots, n\}$, let $(p_i, q_i)$$(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}$$D^{q_i+1}$-bundles over $S^{p_i + 1}$$S^{p_i + 1}$

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

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

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

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

• let $T$$T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$$\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$,
• let $E_8$$E_8$ denote the $E_8$$E_8$-graph,
• 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)) \cong \Zz$$\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$, $k > 4s$$k > 4s$, denote a generator,
• let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$$\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$, denote a generator:
• let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$$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}$$S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$$k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$$S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$$k > 2$,
• let $\eta_n : S^{n+1} \to S^n$$\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$$\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$, the Milnor sphere, generates $bP_{4k}$$bP_{4k}$, $k>1$$k>1$.
• $\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$$\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$, the Kervaire sphere, generates $bP_{4k+2}$$bP_{4k+2}$.
• $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$$\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$$k = 1, 2$.
• For general $k$$k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$$\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic.
• $\Sigma^8(\gamma_3^5, \eta_3\tau_4)$$\Sigma^8(\gamma_3^5, \eta_3\tau_4)$, generates $\Theta_8 = \Zz_2$$\Theta_8 = \Zz_2$.
• $\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$$\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$$\Theta_{16} = \Zz_2$.

### 2.2 Brieskorn varieties

Let $z = (z_0, \dots , z_n)$$z = (z_0, \dots , z_n)$ be a point in $\Cc^{n+1}$$\Cc^{n+1}$ and let $a = (a_0, \dots, a_n)$$a = (a_0, \dots, a_n)$ be a string of n+1 positive integers. Given the complex variety $V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$$V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$ and the $\epsilon$$\epsilon$-sphere $S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$$S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$ for small $\epsilon$$\epsilon$, following [Milnor1968] we define the closed smooth oriented $(n-2)$$(n-2)$-connected $(2n-1)$$(2n-1)$-manifold

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

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

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

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

$\displaystyle \Sigma^7_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$

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$$\pi_7(SO(8)) \cong \Zz \oplus \Zz$ and the bundle $(m, n)$$(m, n)$ has Euler number $n$$n$ and second Pontrjagin class $6(n+2m)$$6(n+2m)$. Moreover $\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$$\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$ where the $\Zz_{8,128}$$\Zz_{8,128}$-summand is $bP_{16}$$bP_{16}$ as explained below. Results of [Wall1962a] and [Eells&Kuiper1962] combine to show that

$\displaystyle \Sigma^{15}_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$
• 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}$$\Theta_{n+1} \cong \Gamma_{n+1}$ for $n \geq 5$$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

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

Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$$S^n$ which are not isotopic to the identity. 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))$$\alpha \in \pi_p(SO(q))$ and $\beta \in \pi_q(SO(p))$$\beta \in \pi_q(SO(p))$ by smooth compactly supported functions $\alpha : \Rr^p \to SO(q)$$\alpha : \Rr^p \to SO(q)$ and $\beta : \Rr^q \to SO(p)$$\beta : \Rr^q \to SO(p)$ and define the following self-diffeomorphisms of $\Rr^p \times \Rr^q$$\Rr^p \times \Rr^q$

$\displaystyle F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),$
$\displaystyle F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),$
$\displaystyle s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.$

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

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

such that

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

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

## 3 Invariants

Finding invariants of exotic sphere $\Sigma$$\Sigma$ which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold $W$$W$ with $\partial W \cong \Sigma$$\partial W \cong \Sigma$. In this case finding an intrinsic definition and or computation of the relevant invariant can also be subtle.

We begin by listing some invariants which are equal for all exotic spheres.

Proposition 3.1. Let $\Sigma$$\Sigma$ be a closed smooth manifold homeomorphic to the n-sphere. Then

1. there is an isomorphism of tangent bundles $T\Sigma \cong TS^n$$T\Sigma \cong TS^n$,
2. the signature of $\Sigma$$\Sigma$ vanishes,
3. the Kervaire invariant of $(\Sigma, F)$$(\Sigma, F)$ is zero for every framing of $\Sigma$$\Sigma$.

(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to $S^n$$S^n$.)

Remark 3.2. The analogue of the first statement for the stable tangent bundle was proven in [Kervaire&Milnor1963, Theorem 3.1]. A proof of the unstable statement is given in [Ray&Pedersen1980, Lemma 1.1]. The next two statements are obvious since both the signature and Kervaire invariant are defined to be zero if $n = 2k+1$$n = 2k+1$ and via a symmetric or quadratic form on $H_k(\Sigma; \Zz) = 0$$H_k(\Sigma; \Zz) = 0$ if $n = 2k$$n = 2k$.

### 3.1 Bordism classes

As every homotopy sphere is stably parallelisable, homotopy spheres admit $B$$B$-structures for any $B$$B$. If $B$$B$ is such that $[S^n, F] \mapsto 0 \in \Omega_n^B$$[S^n, F] \mapsto 0 \in \Omega_n^B$ for any stable framing $F$$F$ of $S^n$$S^n$, then we obtain a well-defined homomorphism

$\displaystyle \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F].$
• If $B = BO\langle k \rangle$$B = BO\langle k \rangle$ for $[n/2] + 1 < k < n+2$$[n/2] + 1 < k < n+2$ then $\Omega_n^B$$\Omega_n^B$ is isomorphic to almost framed bordism and the homomorphism $\eta^B$$\eta^B$ is the same thing as the $\eta: \Theta_n \to \pi_n(G/O)$$\eta: \Theta_n \to \pi_n(G/O)$ in Theorem 4.1.
• Perhaps surprisingly $\eta_n^{\Spin} \neq 0$$\eta_n^{\Spin} \neq 0$ for all $n = 8k+1, 8k+2$$n = 8k+1, 8k+2$, as explained in the next subsection.
• In general determining $\eta^B$$\eta^B$ is a hard an interesting problem.
• $B$$B$-coboundaries for elements of $Ker(\eta^B_n)$$Ker(\eta^B_n)$ are often used to define invariants of $B$$B$-null bordant homotopy spheres.

### 3.2 The α-invariant

In dimensions $n > 1$$n > 1$, every exotic sphere $\Sigma$$\Sigma$ has a unique Spin structure and from above we have the homomorphism $\eta_n^{\Spin} : \Theta_n \to \Omega_n^{\Spin}$$\eta_n^{\Spin} : \Theta_n \to \Omega_n^{\Spin}$. Recall the $\alpha$$\alpha$-invariant homomorphism $\alpha : \Omega_*^{\Spin} \to KO^{-*}$$\alpha : \Omega_*^{\Spin} \to KO^{-*}$ and that there are isomorphisms $KO^{-8k-1} \cong KO^{-8k-2} \cong \Zz/2$$KO^{-8k-1} \cong KO^{-8k-2} \cong \Zz/2$ for all $k \geq 1$$k \geq 1$.

Theorem 3.3 [Anderson&Brown&Peterson1967]. We have $\eta_n^{\Spin}(\Sigma) = 0$$\eta_n^{\Spin}(\Sigma) = 0$ if and only if $\alpha \circ \eta_n^{\Spin}(\Sigma) = 0$$\alpha \circ \eta_n^{\Spin}(\Sigma) = 0$ and $\eta_n^{\Spin} \neq 0$$\eta_n^{\Spin} \neq 0$ if and only if $n = 8k+1$$n = 8k+1$ or $8k+2$$8k+2$.

Remark 3.4. Exotic spheres $\Sigma$$\Sigma$ with $\alpha(\Sigma) \neq 0$$\alpha(\Sigma) \neq 0$ are often called Hitchin spheres, after [Hitchin1974]: see the discussion of curvature below.

## 4 Classification

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

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

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

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

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

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

Theorem 4.1 [Kervaire&Milnor1963]. For $n \geq 5$$n \geq 5$, the group $\Theta_n$$\Theta_n$ is finite. Moreover there is an exact sequence

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

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

The groups $Coker(J_n)$$Coker(J_n)$ are known for $n$$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$$\Theta_n$: an extension problem and the comptutation of the order of the groups $bP_{n+1}$$bP_{n+1}$ and $C_n$$C_n$. We discuss these in turn.

Theorem 4.2 [Brumfiel1968], [Brumfiel1969], [Brumfiel1970]. If $n \neq 2^{j} - 3$$n \neq 2^{j} - 3$ the Kervaire-Milnor extension splits:

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

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

Theorem 4.3 [Kervaire&Milnor1963, Section 8]. The group $bP_{4k+2}$$bP_{4k+2}$ is either $\Zz/2$$\Zz/2$ or $0$$0$. Moreover the following are equivalent:

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

Conversely the following are equivalent:

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

### 4.1 The orders of bP4k and bP4k+2

The group $bP_{4k}$$bP_{4k}$ is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of $Im(J_{4k-1}) \subset \pi_{4k-1}(G)$$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.

Theorem 4.4. Let $a_k = (3-(-1)^k)/2$$a_k = (3-(-1)^k)/2$, let $B_k$$B_k$ be the k-th Bernoulli number (topologist indexing) and for $x \in \Qq$$x \in \Qq$ let $Num(x)$$Num(x)$ denote the numerator of $x$$x$ expressed in lowest form. Then for $k \geq 2$$k \geq 2$, the order of $bP_{4k}$$bP_{4k}$ is

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

Remark 4.5. Note that $Num(B_k/4k)$$Num(B_k/4k)$ is odd so the 2-primary order of $bP_{4k}$$bP_{4k}$ is $a_k \cdot 2^{2k-2}$$a_k \cdot 2^{2k-2}$ while the odd part is $(2^{2k-1}-1) \cdot Num(B_k/4k)$$(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}$$bP_{4k+2}$ which is now almost completely understood as well. References for the theorem are given in the remark which follows it.

Theorem 4.6. The group $bP_{4k+2}$$bP_{4k+2}$ is given as follows:

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

Remark 4.7. The following is a chronological list of determinations of $bP_{4k+2}$$bP_{4k+2}$:

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

## 5 Further discussion

### 5.1 Curvature on exotic spheres

Gromoll-Meyer proved that a certain exotic 7-sphere can be realized as a biquotient of the compact Lie group Sp(2) and has a Riemannian metric of nonnegative sectional curvature. It is not known whether there exist exotic spheres with Riemannian metrics of positive sectional curvature. For a recent review of which exotic spheres admit metrics of various sorts of positive curvature see [Joachim&Wraith2008].

### 5.2 The Kervaire-Milnor braid

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

## 6 PL manifolds admitting no smooth structure

Let $W^{2n}$$W^{2n}$ be a plumbing manifold as described above. By a simple version of the Alexander trick, there is a homemorphism $f \colon \partial W \cong S^{2n-1}$$f \colon \partial W \cong S^{2n-1}$ and so we can form the closed topological manifold

$\displaystyle \bar W : = W \cup_f D^{2n}.$

If $\partial W$$\partial W$ is exotic then it turns out that $\bar W$$\bar W$ is a topological manifold which admits no smooth structure!

[Kervaire1960a] shows that $\bar W^{10}$$\bar W^{10}$ is non-smoothable and the arugments there work for all odd $n$$n$ so long as the Kervaire sphere is exotic.

When $n$$n$ is even the proof is more complicated: one first need's Novikov's theorem that the rational Pontrjagin classes of a topological manifold are homeomorphism invariants [Novikov1965b]. Prior to Novikvo's result, some weaker statements were known. For example, when $n=4$$n=4$ and $W$$W$ is the total space of a $D^4$$D^4$-bundle over $S^4$$S^4$ as above and if $\partial W = \Sigma_{m, 1}$$\partial W = \Sigma_{m, 1}$ then by [Tamura1961] $\bar W$$\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$$m(m-1)/2 \equiv 0$ mod $4$$4$.</wikitex>[1]; Applying Novikov's theorem we know that $\bar W$$\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$$m(m-1)/2 \equiv 0$ mod $56$$56$.

## 8 Footnotes

1. Note that Tamura uses a different identification $\pi_3(SO(4)) \cong \Zz \oplus \Zz$$\pi_3(SO(4)) \cong \Zz \oplus \Zz$ from the one used above.