Exotic spheres

Revision as of 16:51, 5 January 2010

1 Introduction

A homotopy sphere $<!-- 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 --> {{Stub}} == Introduction == <wikitex>; A homotopy sphere $\Sigma^n$ is a closed, smooth 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. </wikitex> == Construction and examples == <wikitex>; Exotic spheres may be constructed in a variety of ways. </wikitex> === Brieskorn varieties === <wikitex>; </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 $\Sigma_{m, 1}$, 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 $\Sigma_{1, 1}$ is not diffeomorphic to $S^7$. A little later Kervaire and Milnor {{cite|Kervaire&Milnor1963}} show that $\Theta_7 \cong \Zz/28$ and Eells and Kuiper {{cite|Eells&Kuiper1962}} showed that $$ \Sigma_{m, 1} = \frac{m(m-1)}{56} \in \Zz/28 \cong \Theta_7.$$ 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. 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 where an n-sphere can be fibre over an m-sphere for 0 < m < n. </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. * $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\})$, the Milnor sphere, generates $bP_{4k}$, $k>1$. * $\Sigma^{4k+1}(T; \{\tau_{2k+1}, \tau_{2k+1}\})$, 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. * $\Sigma^8(T; \{\gamma_3^5, \eta_3\tau_4\})$, generates $\Theta_8 = \Zz_2$. * $\Sigma^{16}(T; \{\gamma_{7}^9, \eta_7\tau_8\})$, generates $\Theta_{16} = \Zz_2$. </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 $\sigma(\gamma_{4k-1}', \gamma_{4k-1}')$ generates $bP_{4k}$ for $k = 1, 2$. </wikitex> == Invariants == <wikitex>; Signature, Kervaire invariant, $\alpha$-invariant, Eels-Kuiper invariant, $s$-invariant. </wikitex> == Classification == <wikitex>; For $n \geq 5$, group of exotic spheres $\Theta_n$ fits into the following long exact sequence which stops at $L_5(e) = 0$: $$ \dots \to \pi_{n+1}(G/O) \to L_{n+1}(e) \to \Theta_n \to \pi_n(G/O) \to L_n(e) \to \dots .$$ The existence of the above sequence was proven in {{cite|Kervaire&Milnor1963}}. More details can also be found in {{cite|Levine1983}}. </wikitex> == Further discussion == <wikitex>; $$ \def\curv{1.5pc}% Adjust the curvature of the curved arrows here \xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here \pi_n(\Top/O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(G) \ & \pi_n(G/O) \ar[dr] \ar[ur] && \pi_{n-1}(\Top) \ar[dr] \ar[ur] \ \pi_n(G) \ar[ur] \ar@/d\curv/[rr] && \pi_n(G/\Top) \ar[ur] \ar@/d\curv/[rr] && \pi_{n-1}(\Top/O) } $$ </wikitex> == External references == * [http://en.wikipedia.org/wiki/Exotic_sphere Wikipedia article on exotic spheres] * [http://www.maths.ed.ac.uk/~aar/exotic.htm http://www.maths.ed.ac.uk/~aar/exotic.htm] Andrew Ranicki's exotic sphere home page, with many of the original papers. == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]\Sigma^n$ is a closed, smooth 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

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

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.

2 Construction and examples

Exotic spheres may be constructed in a variety of ways.

2.1 Brieskorn varieties

2.2 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 $\Sigma^7_{1, 1}$ is not diffeomorphic to $S^7$. A little later Kervaire and Milnor [Kervaire&Milnor1963] show that $\Theta_7 \cong \Zz_{28}$ and Eells and Kuiper [Eells&Kuiper1962] showed that

$$\displaystyle \Sigma^7_{m, 1} = \frac{m(m-1)}{56} \in \Zz/28 \cong \Theta_7.$$

• 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 and results of [Wall1962] and [Eells&Kuiper1962] combine to show that

$$\displaystyle \Sigma^{15}_{m, 1} = \frac{m(m-1)}{8,128} \in \Zz_{8,128} \cong bP_{16} \subset \Theta_{15}.$$

• 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.

2.3 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 $S^{p_i + 1}$

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

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

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

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.

• $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\})$, the Milnor sphere, generates $bP_{4k}$, $k>1$.
• $\Sigma^{4k+1}(T; \{\tau_{2k+1}, \tau_{2k+1}\})$, 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.
• $\Sigma^8(T; \{\gamma_3^5, \eta_3\tau_4\})$, generates $\Theta_8 = \Zz_2$.
• $\Sigma^{16}(T; \{\gamma_{7}^9, \eta_7\tau_8\})$, generates $\Theta_{16} = \Zz_2$.

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

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

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 $\Rr^p \times \Rr^q$

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

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

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

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

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

such that

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

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

3 Invariants

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

4 Classification

For $n \geq 5$, group of exotic spheres $\Theta_n$ fits into the following long exact sequence which stops at $L_5(e) = 0$:

$$\displaystyle \dots \to \pi_{n+1}(G/O) \to L_{n+1}(e) \to \Theta_n \to \pi_n(G/O) \to L_n(e) \to \dots .$$

The existence of the above sequence was proven in [Kervaire&Milnor1963]. More details can also be found in [Levine1983].

5 Further discussion

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
• [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
• [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
• [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
• [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
• [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
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803