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 $\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$-sphere bundles over $S^4$ ...
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], dimensions n = 1, 3, 7 and 15 are the only dimensions where an n-sphere can be fibre over an m-sphere for m<n. </wikitex>
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 bundles, graphs and maps:
- let
denote the tangent bundle of the
-sphere,
- let
,
, denote a generator,
- let
denote the
-graph,
- let
denote the graph with two vertices and one edge connecting them,
- 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 D^{n+1} \cup_f (-D^{n+1}).](/images/math/e/7/b/e7befcfe5bad19cfcc0035d5527a4fe8.png)
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of which are not isotopic to the identity.
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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