Exotic spheres
Contents |
1 Introduction
Let denote the set of oriented diffeomorphism classes of closed, smooth n-manifolds homotopy equivalent to
.
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
...
A little later Shimada [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 Hopf-invariant 1 problem, [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.
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 maps:
- let
denote the graph with two vertices and one edge connecting them and
- write
,
- write
- let
denote the
-graph,
- let
denote the tangent bundle of the
-sphere,
- let
,
, denote a generator,
- let
be essential.
Then we have the following exotic spheres.
-
, the Milnor sphere, generates
,
.
-
, the Kervaire sphere, generates
.
-
, 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 do this now.
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)
Then is compactly suppored and so extends uniquely to a diffeomrphism of
. In this way we obtain a bilinear pairing
![\displaystyle \sigma : \pi_q(SO(q)) \to \pi_q(SO(p)) \longrightarrow \Gamma_{p+q+1}, ~~~~(\alpha, \beta) \longmapsto \Sigma_{s(\alpha, \beta)}](/images/math/e/7/0/e70b42339adca2c7e43fb92fbc38e588.png)
such that
![\displaystyle \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta)](/images/math/7/b/6/7b6c9616120b8f0f017473bb43d7b0ae.png)
where denotes the suspenion
. In particular if
generates
for
.
3 Invariants
Signature, Kervaire invaiant, -invariant, Eels-Kuiper invariant,
-invariant.
4 Classification
[Kervaire&Milnor1963], [Levine1983]
5 Further discussion
... is welcome
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
- [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
- [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
- [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
-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
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