Exotic spheres
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1 Introduction
A homotopy sphere is a closed, smooth 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/0/8/b/08bc0e5cc03477080c3f7936eafb4724.png)
to be the set of oriented diffeomorphism classes of homotopy spheres. Connected sum makes 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
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] show that
and Eells and Kuiper [Eells&Kuiper1962] showed that
![\displaystyle \Sigma^7_{m, 1} = \frac{m(m-1)}{56} \in \Zz/28 \cong \Theta_7.](/images/math/5/9/c/59c5a9d817c6757ca7b6eac2eddcf316.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 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}.](/images/math/8/a/9/8a933b3b6f2b4960a276c48192246626.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.
2.3 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/c/2/b/c2ba1bc59b99c5ffb4937789425c7a9a.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/3/c/7/3c7d1b519d615f791f35d2b9a3d0650f.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
.
2.4 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].
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 diffeomrphism 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 generates
for
.
3 Invariants
Signature, Kervaire invariant, -invariant, Eels-Kuiper invariant,
-invariant.
4 Classification
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]):
![\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 .](/images/math/3/e/1/3e18dd5afc72a5e23a32d67b6e0bd6c1.png)
Here is the i-th L-group of the the trivial group:
as i = 0, 1, 2 or 3 modulo 4 and the sequence ends at
. The groups
are isomorphic to the cokernel of the J-homomorphism,
.
5 Further discussion
![\displaystyle \def\curv{1.5pc}% Adjust the curvature of the curved arrows here \xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here \pi_n(\Top/O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(G) \\ & \pi_n(G/O) \ar[dr] \ar[ur] && \pi_{n-1}(\Top) \ar[dr] \ar[ur] \\ \pi_n(G) \ar[ur] \ar@/d\curv/[rr] && \pi_n(G/\Top) \ar[ur] \ar@/d\curv/[rr] && \pi_{n-1}(\Top/O) }](/images/math/8/3/2/8329c4c4fce71895022bc3e0ed9b9b7b.png)
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
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- [Milnor1959] J. Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962–972. MR0110107 (22 #990) Zbl 0111.35501
- [Shimada1957] N. Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957), 59–69. MR0096223 (20 #2715) Zbl 0145.20303
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
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