Exotic spheres
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/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. 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/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
.
2.2 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
, we define the closed smooth oriented (2n-1)-manifolds
![\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] and [Brieskorn1966a] (see also [Hirzebruch&Mayer1968]), it is shown in particular 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/5/c/d/5cda4365d8c19ee6b3e3f26a4e11adbf.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)/56)\cdot \Sigma_M \in bP_8 \cong \Theta_7.](/images/math/9/6/d/96daea26e6fb24fabcd61092c0221966.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)/16,256)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.](/images/math/f/3/8/f385cbefe0d3346f906d154629d4a7bf.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].
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 for we see that
generates
.
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
- [Brieskorn1966] E. Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2 (1966), 1–14. MR0206972 (34 #6788) Zbl 0145.17804
- [Brieskorn1966a] E. V. Brieskorn, Examples of singular normal complex spaces which are topological manifolds, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 1395–1397. MR0198497 (33 #6652) Zbl 0144.45001
- [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
- [Hirzebruch&Mayer1968] F. Hirzebruch and K. H. Mayer,
-Mannigfaltigkeiten, exotische Sphären und Singularitäten, Springer-Verlag, Berlin, 1968. MR0229251 (37 #4825) Zbl 0172.25304
- [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
-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
- [Wall1962a] C. T. C. Wall, Classification of
-connected
-manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR0145540 (26 #3071) Zbl 0218.57022