Exotic spheres
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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 | 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 | ||
$$\Theta_{n}:=\{[\Sigma^n]|\Sigma^n\simeq S^n \}$$ | $$\Theta_{n}:=\{[\Sigma^n]|\Sigma^n\simeq S^n \}$$ | ||
− | to be the set of oriented | + | to be the set of oriented $h$-cobordism 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> | </wikitex> | ||
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* $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$. | * $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ 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. | **For general $k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic. | ||
− | * $\Sigma^8(\gamma_3^5, \ | + | * $\Sigma^8(\gamma_3^5, \eta_3S\gamma'_3)$, generates $\Theta_8 = \Zz_2$. |
− | * $\Sigma^{16}(\gamma_{7}^9, \ | + | * $\Sigma^{16}(\gamma_{7}^9, \eta_7S\gamma'_7)$, generates $\Theta_{16} = \Zz_2$. |
</wikitex> | </wikitex> | ||
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===The order of bP<sub>4k+2</sub> === | ===The order of bP<sub>4k+2</sub> === | ||
<wikitex>; | <wikitex>; | ||
− | The situation for $bP_{4k+2}$ | + | The situation for $bP_{4k+2}$ is now almost completely understood as well. References for the theorem are given in the remark which follows it. |
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
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* Andrew Ranicki's exotic sphere home page, with many of the original papers: [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: [http://www.maths.ed.ac.uk/~aar/exotic.htm http://www.maths.ed.ac.uk/~aar/exotic.htm] | ||
** Including some [http://www.maths.ed.ac.uk/~aar/papers/km-it.pdf original correspondence between Kervaire and Milnor] | ** Including some [http://www.maths.ed.ac.uk/~aar/papers/km-it.pdf original correspondence between Kervaire and Milnor] | ||
− | *[http://www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres] | + | *[http://www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres]: slides from a presentation by [http://www.nilesjohnson.net/ Nile Johsnon] at the [http://www.ima.umn.edu/2011-2012/SW1.30-2.1.12/ Second Abel conference] in honor of [[Wikipedia:John Milnor|John Milnor]] |
[[Category:Manifolds]] | [[Category:Manifolds]] | ||
[[Category:Highly-connected manifolds]] | [[Category:Highly-connected manifolds]] | ||
[[Category:Surgery]] | [[Category:Surgery]] |
Latest revision as of 01:04, 23 November 2022
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
By a homotopy sphere we mean a closed smooth oriented n-manifold homotopy equivalent to
. The manifold
is called an exotic sphere if it is not diffeomorphic to
. By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension
is homeomorphic to
: 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 \}](/images/math/c/b/7/cb71ea20d95b5de0597d392b267be3e6.png)
to be the set of oriented -cobordism classes of homotopy spheres. Connected sum makes
into an abelian group with inverse given by reversing orientation. An important subgroup of
is
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 , let
be pairs of positive integers such that
and let
be the clutching functions of
-bundles over
![\displaystyle D^{q_i+1}\to D(\alpha_i) \to S^{p_i+1}.](/images/math/4/7/0/470c81dd4044e9fc54c82a845fdb9320.png)
Let be a graph with vertices
such that the edge set between
and
, is non-empty only if
. We form the manifold
from the disjoint union of the
by identifying
and
for each edge in
. If
is simply connected then
![\displaystyle \Sigma(G, \{\alpha_i \}):= \partial W](/images/math/b/5/b/b5b30b7ab7757d59d96727727c327ddb.png)
is often a homotopy sphere. We establish some notation for graphs, bundles and define
- let
denote the graph with two vertices and one edge connecting them and define
,
- let
denote the
-graph,
- let
denote the tangent bundle of the
-sphere,
- let
,
, denote a generator,
- let
, denote a generator:
- let
be the suspension homomorphism,
for
and
for
,
- let
be essential.
Then we have the following exotic spheres.
-
, the Milnor sphere, generates
,
.
-
, the Kervaire sphere, generates
.
-
is the inverse of the Milnor sphere for
.
- For general
,
is exotic.
- For general
-
, generates
.
-
, generates
.
2.2 Brieskorn varieties
Let be a point in
and let
be a string of n+1 positive integers. Given the complex variety
and the
-sphere
for small
, following [Milnor1968]
we define the closed smooth oriented
-connected
-manifold
![\displaystyle W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.](/images/math/c/0/e/c0e3bb8916c52378e049004dd164baf8.png)
The manifolds are often called Brieskorn varieties. By construction, every
lies in
and so bounds a parallelisable manifold. In [Brieskorn1966, Korollar 2] (see also [Brieskorn1966a] and [Hirzebruch&Mayer1968]), it is shown that all homotopy spheres in
and
can be realised as
for some
. Let
be a string of 2k-1 2's in a row with
, then there are diffeomorphisms
![\displaystyle W^{4k-1}(3, 6r+1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},](/images/math/e/5/e/e5e80750c335535911ec4bab24ad44ed.png)
![\displaystyle W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.](/images/math/4/d/c/4dc731c6811ef2726fa549eeced2db7b.png)
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 parametrises linear
-sphere bundles over
where a pair
gives rise to a bundle with Euler number
and first Pontrjagin class
: here we orient
and so identify
. If we set
then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold
, the total space of the bundle
, is a homotopy sphere. Milnor first used a
-invariant, called the
-invariant, to show, e.g. that
is not diffeomorphic to
. A little later Kervaire and Milnor [Kervaire&Milnor1963] proved that
and Eells and Kuiper [Eells&Kuiper1962] defined a refinement of the
-invariant, now called the Eells-Kuiper
-invariant, which in particular gives
![\displaystyle \Sigma^7_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \in bP_8 \cong \Theta_7.](/images/math/7/f/2/7f2eff892e401fb5bba217507e9a81cb.png)
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 and the bundle
has Euler number
and second Pontrjagin class
. Moreover
where the
-summand is
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 \in bP_{16} \subset \Theta_{15}.](/images/math/5/0/a/50a7a550099fd08e7f123ede45ae698c.png)
- 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 for
where
is the group of isotopy classes of orientation preserving diffeomorphisms of
. 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}).](/images/math/d/0/7/d072f1885886fceaf69122f3fbd9b643.png)
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of 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, p.583, The group of diffeomorphisms of
].
Represent and
by smooth compactly supported functions
and
and define the following self-diffeomorphisms of
![\displaystyle F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),](/images/math/7/d/3/7d36dc9a32179e766170bb74ed21f890.png)
![\displaystyle F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),](/images/math/8/5/3/85389967919854e1c6b415093424fa3b.png)
![\displaystyle s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.](/images/math/c/0/e/c0e5ab892d04780eafb879215c80c85e.png)
If follows that is compactly supported and so extends uniquely to a diffeomorphism of
. 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)}](/images/math/8/7/9/87996aef0423e21e4408709348a3693d.png)
such that
![\displaystyle \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).](/images/math/e/6/f/e6f1117faaf1e63f4035d32aae480461.png)
In particular for we see that
generates
.
3 Invariants
Finding invariants of exotic sphere which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold
with
. 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 be a closed smooth manifold homeomorphic to the n-sphere. Then
- there is an isomorphism of tangent bundles
,
- the signature of
vanishes,
- the Kervaire invariant of
is zero for every framing of
.
(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to .)
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 and via a symmetric or quadratic form on
if
.
3.1 Bordism classes
As every homotopy sphere is stably parallelisable, homotopy spheres admit -structures for any
. If
is such that
for any stable framing
of
, then we obtain a well-defined homomorphism
![\displaystyle \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F].](/images/math/e/2/0/e20d20041da848dffc2059d82978962f.png)
- If
for
then
is isomorphic to almost framed bordism and the homomorphism
is the same thing as the
in Theorem 4.1.
- Perhaps surprisingly
for all
, as explained in the next subsection.
- In general determining
is a hard and interesting problem.
-
-coboundaries for elements of
are often used to define invariants of
-null bordant homotopy spheres.
3.2 The α-invariant
In dimensions , every exotic sphere
has a unique Spin structure and from above we have the homomorphism
. Recall the
-invariant homomorphism
and that there are isomorphisms
for all
.
Theorem 3.3 [Anderson&Brown&Peterson1967].
We have if and only if
and
if and only if
or
.
Remark 3.4.
Exotic spheres with
are often called Hitchin spheres, after [Hitchin1974]: see the discussion of curvature below.
3.3 The Eells-Kuiper invariant
3.4 The s-invariant
4 Classification
For and
,
. For
,
is unknown. We therefore concentrate on higher dimensions.
For , the group of exotic n-spheres
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~.](/images/math/5/5/e/55e335cf4828c6374f94b02f5b162c43.png)
Here is the n-th L-group of the the trivial group:
as n = 0, 1, 2 or 3 modulo 4 and the sequence ends at
. Also
is the stable orthogonal group and
is the stable group of homtopy self-equivalences of the sphere. There is a fibration
and the groups
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](/images/math/f/c/9/fc90f1024728e11ddfb9c6f6f77bf929.png)
of this fibration. The homomorphism is the stable J-homomorphism. In particular, by [Serre1951] the groups
are finite and by [Bott1959], [Adams1966] and [Quillen1971] the domain, image and kernel of
have been completely determined. An important result in [Kervaire&Milnor1963] is that the homomorphism
is nonzero. The above sequence then gives
Theorem 4.1 [Kervaire&Milnor1963].
For , the group
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](/images/math/5/b/9/5b9cb10bae76afbfc8e01182257212f9.png)
where , the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if
is even. Moreover
unless
when it is
or
.
The groups are known for
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
: an extension problem and the comptutation of the order of the groups
and
. We discuss these in turn.
Theorem 4.2 [Brumfiel1968], [Brumfiel1969], [Brumfiel1970].
If the Kervaire-Milnor extension splits:
![\displaystyle \Theta_n \cong bP_{n+1} \oplus Ker(K_n).](/images/math/f/5/c/f5c1b71e7670232730b9a680949774a5.png)
The map is the Kervaire invariant and by definition
. By the long exact sequence above we have
Theorem 4.3 [Kervaire&Milnor1963, Section 8].
The group is either
or
. Moreover the following are equivalent:
-
,
- the Kervaire sphere
is diffeomorphic to the standard sphere,
- there is a framed manifold with Kervaire invariant 1:
.
Conversely the following are equivalent:
-
,
- the Kervaire sphere
is not diffeomorphic to the standard sphere,
- there is no framed manifold with Kervaire invariant 1:
.
4.1 The order of bP4k
The group is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of
. 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 , let
be the k-th Bernoulli number (topologist indexing) and for
let
denote the numerator of
expressed in lowest form. Then for
, the order of
is
![\displaystyle t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).](/images/math/7/6/b/76b3ebf01f0d09f9cd91fb477d3225f8.png)
Remark 4.5.
Note that is odd so the 2-primary order of
is
while the odd part is
. 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 following table lists factorisations of for
.
4k 8 12 16 20 24 28 32 order bP4k 22.7 25.31 26.127 29.511 210.2047.691 213.8191 214.16384.3617
4.2 The order of bP4k+2
The situation for is now almost completely understood as well. References for the theorem are given in the remark which follows it.
Theorem 4.6.
The group is given as follows:
-
,
-
or
,
-
else.
Remark 4.7.
The following is a chronological list of determinations of :
-
, [Kervaire1960a].
-
[Kervaire&Milnor1963].
-
, [Brown&Peterson1965, Corollary 1.3]; for another proof, see also [Anderson&Brown&Peterson1966a, Theorem 2.5].
-
, [Mahowald&Tangora1967].
-
unless
[Browder1969].
-
, [Barratt&Jones&Mahowald1984].
-
for
, [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 thus by the O'Neill formula 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 \Theta_n \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}^S \\ & \Omega^{alm} \ar[dr] \ar[ur] && \Theta_{n-1}^{fr} \ar[dr] \ar[ur] \\ \pi_n^S \ar[ur] \ar@/d\curv/[rr] && L_n(\Z) \ar[ur] \ar@/d\curv/[rr] && \Theta_{n-1} }](/images/math/3/8/9/3896e9a6621ee5bf4f971ebb7480660d.png)
![\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) }](/images/math/6/3/f/63f5dfa150a44639fc0eddf34c0651ac.png)
6 Topological manifolds admitting no smooth structure
Let be a plumbing manifold as described above. By a simple version of the Alexander trick, there is a homemorphism
and so we can form the closed topological manifold
![\displaystyle \bar W : = W \cup_f D^{2n}.](/images/math/4/1/6/416841c01e875ea3585c98967abf6fc1.png)
If is exotic then it turns out that
is a topological manifold which admits no smooth structure.
[Kervaire1960a] shows that is non-smoothable and the arugments there work for all odd
so long as the Kervaire sphere is exotic.
When 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 Novikov's theorem, some weaker statements were known. For example, when
and
is the total space of a
-bundle over
as above and if
then by [Tamura1961]
is smoothable if and only if
mod
. [1]; Applying Novikov's theorem we now know that
is smoothable if and only if
mod
.
7 Footnotes
- ↑ Note that Tamura uses a different identification
from the one used above.
8 References
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Tex syntax error
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9 External links
- The Wikipedia page on exotic spheres
- The tabulation of the order of the group of exotic spheres in the On-Line Encyclopedia of Integer Sequences
- Bulletin of the AMS Volume 52 Number 4 Volume focusing on smooth structures on manifolds, in particular the work of Kervaire and Milnor
- Andrew Ranicki's exotic sphere home page, with many of the original papers: http://www.maths.ed.ac.uk/~aar/exotic.htm
- Including some original correspondence between Kervaire and Milnor
- An animation of exotic 7-spheres: slides from a presentation by Nile Johsnon at the Second Abel conference in honor of John Milnor
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![\Sigma^n](/images/math/e/d/a/edada90b024daa0ebe27575cac271477.png)
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![n \geq 5](/images/math/6/5/9/659420caf2511152bb9358e2218a6d9e.png)
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![\displaystyle \Theta_{n}:=\{[\Sigma^n]|\Sigma^n\simeq S^n \}](/images/math/c/b/7/cb71ea20d95b5de0597d392b267be3e6.png)
to be the set of oriented -cobordism classes of homotopy spheres. Connected sum makes
into an abelian group with inverse given by reversing orientation. An important subgroup of
is
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 , let
be pairs of positive integers such that
and let
be the clutching functions of
-bundles over
![\displaystyle D^{q_i+1}\to D(\alpha_i) \to S^{p_i+1}.](/images/math/4/7/0/470c81dd4044e9fc54c82a845fdb9320.png)
Let be a graph with vertices
such that the edge set between
and
, is non-empty only if
. We form the manifold
from the disjoint union of the
by identifying
and
for each edge in
. If
is simply connected then
![\displaystyle \Sigma(G, \{\alpha_i \}):= \partial W](/images/math/b/5/b/b5b30b7ab7757d59d96727727c327ddb.png)
is often a homotopy sphere. We establish some notation for graphs, bundles and define
- let
denote the graph with two vertices and one edge connecting them and define
,
- let
denote the
-graph,
- let
denote the tangent bundle of the
-sphere,
- let
,
, denote a generator,
- let
, denote a generator:
- let
be the suspension homomorphism,
for
and
for
,
- let
be essential.
Then we have the following exotic spheres.
-
, the Milnor sphere, generates
,
.
-
, the Kervaire sphere, generates
.
-
is the inverse of the Milnor sphere for
.
- For general
,
is exotic.
- For general
-
, generates
.
-
, generates
.
2.2 Brieskorn varieties
Let be a point in
and let
be a string of n+1 positive integers. Given the complex variety
and the
-sphere
for small
, following [Milnor1968]
we define the closed smooth oriented
-connected
-manifold
![\displaystyle W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.](/images/math/c/0/e/c0e3bb8916c52378e049004dd164baf8.png)
The manifolds are often called Brieskorn varieties. By construction, every
lies in
and so bounds a parallelisable manifold. In [Brieskorn1966, Korollar 2] (see also [Brieskorn1966a] and [Hirzebruch&Mayer1968]), it is shown that all homotopy spheres in
and
can be realised as
for some
. Let
be a string of 2k-1 2's in a row with
, then there are diffeomorphisms
![\displaystyle W^{4k-1}(3, 6r+1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},](/images/math/e/5/e/e5e80750c335535911ec4bab24ad44ed.png)
![\displaystyle W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.](/images/math/4/d/c/4dc731c6811ef2726fa549eeced2db7b.png)
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 parametrises linear
-sphere bundles over
where a pair
gives rise to a bundle with Euler number
and first Pontrjagin class
: here we orient
and so identify
. If we set
then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold
, the total space of the bundle
, is a homotopy sphere. Milnor first used a
-invariant, called the
-invariant, to show, e.g. that
is not diffeomorphic to
. A little later Kervaire and Milnor [Kervaire&Milnor1963] proved that
and Eells and Kuiper [Eells&Kuiper1962] defined a refinement of the
-invariant, now called the Eells-Kuiper
-invariant, which in particular gives
![\displaystyle \Sigma^7_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \in bP_8 \cong \Theta_7.](/images/math/7/f/2/7f2eff892e401fb5bba217507e9a81cb.png)
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 and the bundle
has Euler number
and second Pontrjagin class
. Moreover
where the
-summand is
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 \in bP_{16} \subset \Theta_{15}.](/images/math/5/0/a/50a7a550099fd08e7f123ede45ae698c.png)
- 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 for
where
is the group of isotopy classes of orientation preserving diffeomorphisms of
. 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}).](/images/math/d/0/7/d072f1885886fceaf69122f3fbd9b643.png)
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of 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, p.583, The group of diffeomorphisms of
].
Represent and
by smooth compactly supported functions
and
and define the following self-diffeomorphisms of
![\displaystyle F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),](/images/math/7/d/3/7d36dc9a32179e766170bb74ed21f890.png)
![\displaystyle F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),](/images/math/8/5/3/85389967919854e1c6b415093424fa3b.png)
![\displaystyle s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.](/images/math/c/0/e/c0e5ab892d04780eafb879215c80c85e.png)
If follows that is compactly supported and so extends uniquely to a diffeomorphism of
. 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)}](/images/math/8/7/9/87996aef0423e21e4408709348a3693d.png)
such that
![\displaystyle \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).](/images/math/e/6/f/e6f1117faaf1e63f4035d32aae480461.png)
In particular for we see that
generates
.
3 Invariants
Finding invariants of exotic sphere which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold
with
. 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 be a closed smooth manifold homeomorphic to the n-sphere. Then
- there is an isomorphism of tangent bundles
,
- the signature of
vanishes,
- the Kervaire invariant of
is zero for every framing of
.
(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to .)
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 and via a symmetric or quadratic form on
if
.
3.1 Bordism classes
As every homotopy sphere is stably parallelisable, homotopy spheres admit -structures for any
. If
is such that
for any stable framing
of
, then we obtain a well-defined homomorphism
![\displaystyle \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F].](/images/math/e/2/0/e20d20041da848dffc2059d82978962f.png)
- If
for
then
is isomorphic to almost framed bordism and the homomorphism
is the same thing as the
in Theorem 4.1.
- Perhaps surprisingly
for all
, as explained in the next subsection.
- In general determining
is a hard and interesting problem.
-
-coboundaries for elements of
are often used to define invariants of
-null bordant homotopy spheres.
3.2 The α-invariant
In dimensions , every exotic sphere
has a unique Spin structure and from above we have the homomorphism
. Recall the
-invariant homomorphism
and that there are isomorphisms
for all
.
Theorem 3.3 [Anderson&Brown&Peterson1967].
We have if and only if
and
if and only if
or
.
Remark 3.4.
Exotic spheres with
are often called Hitchin spheres, after [Hitchin1974]: see the discussion of curvature below.
3.3 The Eells-Kuiper invariant
3.4 The s-invariant
4 Classification
For and
,
. For
,
is unknown. We therefore concentrate on higher dimensions.
For , the group of exotic n-spheres
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~.](/images/math/5/5/e/55e335cf4828c6374f94b02f5b162c43.png)
Here is the n-th L-group of the the trivial group:
as n = 0, 1, 2 or 3 modulo 4 and the sequence ends at
. Also
is the stable orthogonal group and
is the stable group of homtopy self-equivalences of the sphere. There is a fibration
and the groups
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](/images/math/f/c/9/fc90f1024728e11ddfb9c6f6f77bf929.png)
of this fibration. The homomorphism is the stable J-homomorphism. In particular, by [Serre1951] the groups
are finite and by [Bott1959], [Adams1966] and [Quillen1971] the domain, image and kernel of
have been completely determined. An important result in [Kervaire&Milnor1963] is that the homomorphism
is nonzero. The above sequence then gives
Theorem 4.1 [Kervaire&Milnor1963].
For , the group
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](/images/math/5/b/9/5b9cb10bae76afbfc8e01182257212f9.png)
where , the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if
is even. Moreover
unless
when it is
or
.
The groups are known for
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
: an extension problem and the comptutation of the order of the groups
and
. We discuss these in turn.
Theorem 4.2 [Brumfiel1968], [Brumfiel1969], [Brumfiel1970].
If the Kervaire-Milnor extension splits:
![\displaystyle \Theta_n \cong bP_{n+1} \oplus Ker(K_n).](/images/math/f/5/c/f5c1b71e7670232730b9a680949774a5.png)
The map is the Kervaire invariant and by definition
. By the long exact sequence above we have
Theorem 4.3 [Kervaire&Milnor1963, Section 8].
The group is either
or
. Moreover the following are equivalent:
-
,
- the Kervaire sphere
is diffeomorphic to the standard sphere,
- there is a framed manifold with Kervaire invariant 1:
.
Conversely the following are equivalent:
-
,
- the Kervaire sphere
is not diffeomorphic to the standard sphere,
- there is no framed manifold with Kervaire invariant 1:
.
4.1 The order of bP4k
The group is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of
. 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 , let
be the k-th Bernoulli number (topologist indexing) and for
let
denote the numerator of
expressed in lowest form. Then for
, the order of
is
![\displaystyle t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).](/images/math/7/6/b/76b3ebf01f0d09f9cd91fb477d3225f8.png)
Remark 4.5.
Note that is odd so the 2-primary order of
is
while the odd part is
. 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 following table lists factorisations of for
.
4k 8 12 16 20 24 28 32 order bP4k 22.7 25.31 26.127 29.511 210.2047.691 213.8191 214.16384.3617
4.2 The order of bP4k+2
The situation for is now almost completely understood as well. References for the theorem are given in the remark which follows it.
Theorem 4.6.
The group is given as follows:
-
,
-
or
,
-
else.
Remark 4.7.
The following is a chronological list of determinations of :
-
, [Kervaire1960a].
-
[Kervaire&Milnor1963].
-
, [Brown&Peterson1965, Corollary 1.3]; for another proof, see also [Anderson&Brown&Peterson1966a, Theorem 2.5].
-
, [Mahowald&Tangora1967].
-
unless
[Browder1969].
-
, [Barratt&Jones&Mahowald1984].
-
for
, [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 thus by the O'Neill formula 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 \Theta_n \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}^S \\ & \Omega^{alm} \ar[dr] \ar[ur] && \Theta_{n-1}^{fr} \ar[dr] \ar[ur] \\ \pi_n^S \ar[ur] \ar@/d\curv/[rr] && L_n(\Z) \ar[ur] \ar@/d\curv/[rr] && \Theta_{n-1} }](/images/math/3/8/9/3896e9a6621ee5bf4f971ebb7480660d.png)
![\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) }](/images/math/6/3/f/63f5dfa150a44639fc0eddf34c0651ac.png)
6 Topological manifolds admitting no smooth structure
Let be a plumbing manifold as described above. By a simple version of the Alexander trick, there is a homemorphism
and so we can form the closed topological manifold
![\displaystyle \bar W : = W \cup_f D^{2n}.](/images/math/4/1/6/416841c01e875ea3585c98967abf6fc1.png)
If is exotic then it turns out that
is a topological manifold which admits no smooth structure.
[Kervaire1960a] shows that is non-smoothable and the arugments there work for all odd
so long as the Kervaire sphere is exotic.
When 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 Novikov's theorem, some weaker statements were known. For example, when
and
is the total space of a
-bundle over
as above and if
then by [Tamura1961]
is smoothable if and only if
mod
. [1]; Applying Novikov's theorem we now know that
is smoothable if and only if
mod
.
7 Footnotes
- ↑ Note that Tamura uses a different identification
from the one used above.
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9 External links
- The Wikipedia page on exotic spheres
- The tabulation of the order of the group of exotic spheres in the On-Line Encyclopedia of Integer Sequences
- Bulletin of the AMS Volume 52 Number 4 Volume focusing on smooth structures on manifolds, in particular the work of Kervaire and Milnor
- Andrew Ranicki's exotic sphere home page, with many of the original papers: http://www.maths.ed.ac.uk/~aar/exotic.htm
- Including some original correspondence between Kervaire and Milnor
- An animation of exotic 7-spheres: slides from a presentation by Nile Johsnon at the Second Abel conference in honor of John Milnor
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![\Sigma^n](/images/math/e/d/a/edada90b024daa0ebe27575cac271477.png)
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![n \geq 5](/images/math/6/5/9/659420caf2511152bb9358e2218a6d9e.png)
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![\displaystyle \Theta_{n}:=\{[\Sigma^n]|\Sigma^n\simeq S^n \}](/images/math/c/b/7/cb71ea20d95b5de0597d392b267be3e6.png)
to be the set of oriented -cobordism classes of homotopy spheres. Connected sum makes
into an abelian group with inverse given by reversing orientation. An important subgroup of
is
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 , let
be pairs of positive integers such that
and let
be the clutching functions of
-bundles over
![\displaystyle D^{q_i+1}\to D(\alpha_i) \to S^{p_i+1}.](/images/math/4/7/0/470c81dd4044e9fc54c82a845fdb9320.png)
Let be a graph with vertices
such that the edge set between
and
, is non-empty only if
. We form the manifold
from the disjoint union of the
by identifying
and
for each edge in
. If
is simply connected then
![\displaystyle \Sigma(G, \{\alpha_i \}):= \partial W](/images/math/b/5/b/b5b30b7ab7757d59d96727727c327ddb.png)
is often a homotopy sphere. We establish some notation for graphs, bundles and define
- let
denote the graph with two vertices and one edge connecting them and define
,
- let
denote the
-graph,
- let
denote the tangent bundle of the
-sphere,
- let
,
, denote a generator,
- let
, denote a generator:
- let
be the suspension homomorphism,
for
and
for
,
- let
be essential.
Then we have the following exotic spheres.
-
, the Milnor sphere, generates
,
.
-
, the Kervaire sphere, generates
.
-
is the inverse of the Milnor sphere for
.
- For general
,
is exotic.
- For general
-
, generates
.
-
, generates
.
2.2 Brieskorn varieties
Let be a point in
and let
be a string of n+1 positive integers. Given the complex variety
and the
-sphere
for small
, following [Milnor1968]
we define the closed smooth oriented
-connected
-manifold
![\displaystyle W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.](/images/math/c/0/e/c0e3bb8916c52378e049004dd164baf8.png)
The manifolds are often called Brieskorn varieties. By construction, every
lies in
and so bounds a parallelisable manifold. In [Brieskorn1966, Korollar 2] (see also [Brieskorn1966a] and [Hirzebruch&Mayer1968]), it is shown that all homotopy spheres in
and
can be realised as
for some
. Let
be a string of 2k-1 2's in a row with
, then there are diffeomorphisms
![\displaystyle W^{4k-1}(3, 6r+1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},](/images/math/e/5/e/e5e80750c335535911ec4bab24ad44ed.png)
![\displaystyle W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.](/images/math/4/d/c/4dc731c6811ef2726fa549eeced2db7b.png)
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 parametrises linear
-sphere bundles over
where a pair
gives rise to a bundle with Euler number
and first Pontrjagin class
: here we orient
and so identify
. If we set
then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold
, the total space of the bundle
, is a homotopy sphere. Milnor first used a
-invariant, called the
-invariant, to show, e.g. that
is not diffeomorphic to
. A little later Kervaire and Milnor [Kervaire&Milnor1963] proved that
and Eells and Kuiper [Eells&Kuiper1962] defined a refinement of the
-invariant, now called the Eells-Kuiper
-invariant, which in particular gives
![\displaystyle \Sigma^7_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \in bP_8 \cong \Theta_7.](/images/math/7/f/2/7f2eff892e401fb5bba217507e9a81cb.png)
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 and the bundle
has Euler number
and second Pontrjagin class
. Moreover
where the
-summand is
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 \in bP_{16} \subset \Theta_{15}.](/images/math/5/0/a/50a7a550099fd08e7f123ede45ae698c.png)
- 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 for
where
is the group of isotopy classes of orientation preserving diffeomorphisms of
. 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}).](/images/math/d/0/7/d072f1885886fceaf69122f3fbd9b643.png)
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of 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, p.583, The group of diffeomorphisms of
].
Represent and
by smooth compactly supported functions
and
and define the following self-diffeomorphisms of
![\displaystyle F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),](/images/math/7/d/3/7d36dc9a32179e766170bb74ed21f890.png)
![\displaystyle F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),](/images/math/8/5/3/85389967919854e1c6b415093424fa3b.png)
![\displaystyle s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.](/images/math/c/0/e/c0e5ab892d04780eafb879215c80c85e.png)
If follows that is compactly supported and so extends uniquely to a diffeomorphism of
. 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)}](/images/math/8/7/9/87996aef0423e21e4408709348a3693d.png)
such that
![\displaystyle \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).](/images/math/e/6/f/e6f1117faaf1e63f4035d32aae480461.png)
In particular for we see that
generates
.
3 Invariants
Finding invariants of exotic sphere which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold
with
. 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 be a closed smooth manifold homeomorphic to the n-sphere. Then
- there is an isomorphism of tangent bundles
,
- the signature of
vanishes,
- the Kervaire invariant of
is zero for every framing of
.
(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to .)
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 and via a symmetric or quadratic form on
if
.
3.1 Bordism classes
As every homotopy sphere is stably parallelisable, homotopy spheres admit -structures for any
. If
is such that
for any stable framing
of
, then we obtain a well-defined homomorphism
![\displaystyle \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F].](/images/math/e/2/0/e20d20041da848dffc2059d82978962f.png)
- If
for
then
is isomorphic to almost framed bordism and the homomorphism
is the same thing as the
in Theorem 4.1.
- Perhaps surprisingly
for all
, as explained in the next subsection.
- In general determining
is a hard and interesting problem.
-
-coboundaries for elements of
are often used to define invariants of
-null bordant homotopy spheres.
3.2 The α-invariant
In dimensions , every exotic sphere
has a unique Spin structure and from above we have the homomorphism
. Recall the
-invariant homomorphism
and that there are isomorphisms
for all
.
Theorem 3.3 [Anderson&Brown&Peterson1967].
We have if and only if
and
if and only if
or
.
Remark 3.4.
Exotic spheres with
are often called Hitchin spheres, after [Hitchin1974]: see the discussion of curvature below.
3.3 The Eells-Kuiper invariant
3.4 The s-invariant
4 Classification
For and
,
. For
,
is unknown. We therefore concentrate on higher dimensions.
For , the group of exotic n-spheres
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~.](/images/math/5/5/e/55e335cf4828c6374f94b02f5b162c43.png)
Here is the n-th L-group of the the trivial group:
as n = 0, 1, 2 or 3 modulo 4 and the sequence ends at
. Also
is the stable orthogonal group and
is the stable group of homtopy self-equivalences of the sphere. There is a fibration
and the groups
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](/images/math/f/c/9/fc90f1024728e11ddfb9c6f6f77bf929.png)
of this fibration. The homomorphism is the stable J-homomorphism. In particular, by [Serre1951] the groups
are finite and by [Bott1959], [Adams1966] and [Quillen1971] the domain, image and kernel of
have been completely determined. An important result in [Kervaire&Milnor1963] is that the homomorphism
is nonzero. The above sequence then gives
Theorem 4.1 [Kervaire&Milnor1963].
For , the group
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](/images/math/5/b/9/5b9cb10bae76afbfc8e01182257212f9.png)
where , the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if
is even. Moreover
unless
when it is
or
.
The groups are known for
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
: an extension problem and the comptutation of the order of the groups
and
. We discuss these in turn.
Theorem 4.2 [Brumfiel1968], [Brumfiel1969], [Brumfiel1970].
If the Kervaire-Milnor extension splits:
![\displaystyle \Theta_n \cong bP_{n+1} \oplus Ker(K_n).](/images/math/f/5/c/f5c1b71e7670232730b9a680949774a5.png)
The map is the Kervaire invariant and by definition
. By the long exact sequence above we have
Theorem 4.3 [Kervaire&Milnor1963, Section 8].
The group is either
or
. Moreover the following are equivalent:
-
,
- the Kervaire sphere
is diffeomorphic to the standard sphere,
- there is a framed manifold with Kervaire invariant 1:
.
Conversely the following are equivalent:
-
,
- the Kervaire sphere
is not diffeomorphic to the standard sphere,
- there is no framed manifold with Kervaire invariant 1:
.
4.1 The order of bP4k
The group is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of
. 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 , let
be the k-th Bernoulli number (topologist indexing) and for
let
denote the numerator of
expressed in lowest form. Then for
, the order of
is
![\displaystyle t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).](/images/math/7/6/b/76b3ebf01f0d09f9cd91fb477d3225f8.png)
Remark 4.5.
Note that is odd so the 2-primary order of
is
while the odd part is
. 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 following table lists factorisations of for
.
4k 8 12 16 20 24 28 32 order bP4k 22.7 25.31 26.127 29.511 210.2047.691 213.8191 214.16384.3617
4.2 The order of bP4k+2
The situation for is now almost completely understood as well. References for the theorem are given in the remark which follows it.
Theorem 4.6.
The group is given as follows:
-
,
-
or
,
-
else.
Remark 4.7.
The following is a chronological list of determinations of :
-
, [Kervaire1960a].
-
[Kervaire&Milnor1963].
-
, [Brown&Peterson1965, Corollary 1.3]; for another proof, see also [Anderson&Brown&Peterson1966a, Theorem 2.5].
-
, [Mahowald&Tangora1967].
-
unless
[Browder1969].
-
, [Barratt&Jones&Mahowald1984].
-
for
, [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 thus by the O'Neill formula 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 \Theta_n \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}^S \\ & \Omega^{alm} \ar[dr] \ar[ur] && \Theta_{n-1}^{fr} \ar[dr] \ar[ur] \\ \pi_n^S \ar[ur] \ar@/d\curv/[rr] && L_n(\Z) \ar[ur] \ar@/d\curv/[rr] && \Theta_{n-1} }](/images/math/3/8/9/3896e9a6621ee5bf4f971ebb7480660d.png)
![\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) }](/images/math/6/3/f/63f5dfa150a44639fc0eddf34c0651ac.png)
6 Topological manifolds admitting no smooth structure
Let be a plumbing manifold as described above. By a simple version of the Alexander trick, there is a homemorphism
and so we can form the closed topological manifold
![\displaystyle \bar W : = W \cup_f D^{2n}.](/images/math/4/1/6/416841c01e875ea3585c98967abf6fc1.png)
If is exotic then it turns out that
is a topological manifold which admits no smooth structure.
[Kervaire1960a] shows that is non-smoothable and the arugments there work for all odd
so long as the Kervaire sphere is exotic.
When 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 Novikov's theorem, some weaker statements were known. For example, when
and
is the total space of a
-bundle over
as above and if
then by [Tamura1961]
is smoothable if and only if
mod
. [1]; Applying Novikov's theorem we now know that
is smoothable if and only if
mod
.
7 Footnotes
- ↑ Note that Tamura uses a different identification
from the one used above.
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9 External links
- The Wikipedia page on exotic spheres
- The tabulation of the order of the group of exotic spheres in the On-Line Encyclopedia of Integer Sequences
- Bulletin of the AMS Volume 52 Number 4 Volume focusing on smooth structures on manifolds, in particular the work of Kervaire and Milnor
- Andrew Ranicki's exotic sphere home page, with many of the original papers: http://www.maths.ed.ac.uk/~aar/exotic.htm
- Including some original correspondence between Kervaire and Milnor
- An animation of exotic 7-spheres: slides from a presentation by Nile Johsnon at the Second Abel conference in honor of John Milnor