Exotic spheres
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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
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
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
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.
- , 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
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
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
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
- 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
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
If follows that is compactly supported and so extends uniquely to a diffeomrphism of . In this way we obtain a bilinear pairing
such that
In particular for we see that generates .
3 Invariants
Signature, Kervaire invariant, -invariant, Eels-Kuiper invariant, -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]):
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
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
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:
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 orders of bP4k and bP4k+2
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
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 next theorem describes the situation for which is now almost completely understood as well. References for the theorem are given in the remark which follows it.
Theorem 4.6. The order of is given as follows:
- ,
- or ,
- else.
Remark 4.7. The following is a chronological list of determinations of :
- , [Kervaire1960a].
- [Kervaire&Milnor1963].
- , [Anderson&Brown&Peterson1966a].
- , [Mahowald&Tangora1967].
- unless [Browder1969].
- , [Barratt&Jones&Mahowald1984].
- for , [Hill&Hopkins&Ravenel2009].
5 Further discussion
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
- [Adams1966] J. F. Adams, On the groups . IV, Topology 5 (1966), 21–71. MR0198470 (33 #6628) Zbl 0145.19902
- [Anderson&Brown&Peterson1966a] D. W. Anderson, E. H. Brown and F. P. Peterson, -cobordism, -characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54–67. MR0189043 (32 #6470) Zbl 0137.42802
- [Barratt&Jones&Mahowald1984] M. G. Barratt, J. D. S. Jones and M. E. Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension , J. London Math. Soc. (2) 30 (1984), no.3, 533–550. MR810962 (87g:55025) Zbl 0606.55010
- [Bott1959] R. Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR0110104 (22 #987) Zbl 0129.15601
- [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
- [Browder1969] W. Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. (2) 90 (1969), 157–186. MR0251736 (40 #4963) Zbl 0198.28501
- [Brumfiel1968] G. Brumfiel, On the homotopy groups of and , Ann. of Math. (2) 88 (1968), 291–311. MR0234458 (38 #2775) Zbl 0179.28601
- [Brumfiel1969] G. Brumfiel, On the homotopy groups of and . II, Topology 8 (1969), 305–311. MR0248830 (40 #2080) Zbl 0179.28601
- [Brumfiel1970] G. Brumfiel, The homotopy groups of and . III, Michigan Math. J. 17 (1970), 217–224. MR0271938 (42 #6819) Zbl 0201.55901
- [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
- [Hill&Hopkins&Ravenel2009] M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, (2009). Available at the arXiv:0908.3724.
- [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
- [Kervaire1960a] M. A. Kervaire, A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34 (1960), 257–270. MR0139172 (25 #2608) Zbl 0145.20304
- [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
- [Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020
- [Mahowald&Tangora1967] M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349–369. MR0214072 (35 #4924) Zbl 0213.24901
- [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
- [Quillen1971] D. Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR0279804 (43 #5525) Zbl 0219.55013
- [Serre1951] J. Serre, Homologie singulière des espaces fibrès. Applications, Ann. of Math. (2) 54 (1951), 425–505. MR0045386 (13,574g) Zbl 0045.26003
- [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
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
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
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.
- , 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
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
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
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
- 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
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
If follows that is compactly supported and so extends uniquely to a diffeomrphism of . In this way we obtain a bilinear pairing
such that
In particular for we see that generates .
3 Invariants
Signature, Kervaire invariant, -invariant, Eels-Kuiper invariant, -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]):
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
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
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:
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 orders of bP4k and bP4k+2
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
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 next theorem describes the situation for which is now almost completely understood as well. References for the theorem are given in the remark which follows it.
Theorem 4.6. The order of is given as follows:
- ,
- or ,
- else.
Remark 4.7. The following is a chronological list of determinations of :
- , [Kervaire1960a].
- [Kervaire&Milnor1963].
- , [Anderson&Brown&Peterson1966a].
- , [Mahowald&Tangora1967].
- unless [Browder1969].
- , [Barratt&Jones&Mahowald1984].
- for , [Hill&Hopkins&Ravenel2009].
5 Further discussion
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
- [Adams1966] J. F. Adams, On the groups . IV, Topology 5 (1966), 21–71. MR0198470 (33 #6628) Zbl 0145.19902
- [Anderson&Brown&Peterson1966a] D. W. Anderson, E. H. Brown and F. P. Peterson, -cobordism, -characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54–67. MR0189043 (32 #6470) Zbl 0137.42802
- [Barratt&Jones&Mahowald1984] M. G. Barratt, J. D. S. Jones and M. E. Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension , J. London Math. Soc. (2) 30 (1984), no.3, 533–550. MR810962 (87g:55025) Zbl 0606.55010
- [Bott1959] R. Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR0110104 (22 #987) Zbl 0129.15601
- [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
- [Browder1969] W. Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. (2) 90 (1969), 157–186. MR0251736 (40 #4963) Zbl 0198.28501
- [Brumfiel1968] G. Brumfiel, On the homotopy groups of and , Ann. of Math. (2) 88 (1968), 291–311. MR0234458 (38 #2775) Zbl 0179.28601
- [Brumfiel1969] G. Brumfiel, On the homotopy groups of and . II, Topology 8 (1969), 305–311. MR0248830 (40 #2080) Zbl 0179.28601
- [Brumfiel1970] G. Brumfiel, The homotopy groups of and . III, Michigan Math. J. 17 (1970), 217–224. MR0271938 (42 #6819) Zbl 0201.55901
- [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
- [Hill&Hopkins&Ravenel2009] M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, (2009). Available at the arXiv:0908.3724.
- [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
- [Kervaire1960a] M. A. Kervaire, A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34 (1960), 257–270. MR0139172 (25 #2608) Zbl 0145.20304
- [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
- [Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020
- [Mahowald&Tangora1967] M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349–369. MR0214072 (35 #4924) Zbl 0213.24901
- [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
- [Quillen1971] D. Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR0279804 (43 #5525) Zbl 0219.55013
- [Serre1951] J. Serre, Homologie singulière des espaces fibrès. Applications, Ann. of Math. (2) 54 (1951), 425–505. MR0045386 (13,574g) Zbl 0045.26003
- [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
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
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
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.
- , 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
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
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
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
- 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
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
If follows that is compactly supported and so extends uniquely to a diffeomrphism of . In this way we obtain a bilinear pairing
such that
In particular for we see that generates .
3 Invariants
Signature, Kervaire invariant, -invariant, Eels-Kuiper invariant, -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]):
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
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
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:
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 orders of bP4k and bP4k+2
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
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 next theorem describes the situation for which is now almost completely understood as well. References for the theorem are given in the remark which follows it.
Theorem 4.6. The order of is given as follows:
- ,
- or ,
- else.
Remark 4.7. The following is a chronological list of determinations of :
- , [Kervaire1960a].
- [Kervaire&Milnor1963].
- , [Anderson&Brown&Peterson1966a].
- , [Mahowald&Tangora1967].
- unless [Browder1969].
- , [Barratt&Jones&Mahowald1984].
- for , [Hill&Hopkins&Ravenel2009].
5 Further discussion
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
- [Adams1966] J. F. Adams, On the groups . IV, Topology 5 (1966), 21–71. MR0198470 (33 #6628) Zbl 0145.19902
- [Anderson&Brown&Peterson1966a] D. W. Anderson, E. H. Brown and F. P. Peterson, -cobordism, -characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54–67. MR0189043 (32 #6470) Zbl 0137.42802
- [Barratt&Jones&Mahowald1984] M. G. Barratt, J. D. S. Jones and M. E. Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension , J. London Math. Soc. (2) 30 (1984), no.3, 533–550. MR810962 (87g:55025) Zbl 0606.55010
- [Bott1959] R. Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR0110104 (22 #987) Zbl 0129.15601
- [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
- [Browder1969] W. Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. (2) 90 (1969), 157–186. MR0251736 (40 #4963) Zbl 0198.28501
- [Brumfiel1968] G. Brumfiel, On the homotopy groups of and , Ann. of Math. (2) 88 (1968), 291–311. MR0234458 (38 #2775) Zbl 0179.28601
- [Brumfiel1969] G. Brumfiel, On the homotopy groups of and . II, Topology 8 (1969), 305–311. MR0248830 (40 #2080) Zbl 0179.28601
- [Brumfiel1970] G. Brumfiel, The homotopy groups of and . III, Michigan Math. J. 17 (1970), 217–224. MR0271938 (42 #6819) Zbl 0201.55901
- [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
- [Hill&Hopkins&Ravenel2009] M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, (2009). Available at the arXiv:0908.3724.
- [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
- [Kervaire1960a] M. A. Kervaire, A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34 (1960), 257–270. MR0139172 (25 #2608) Zbl 0145.20304
- [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
- [Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020
- [Mahowald&Tangora1967] M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349–369. MR0214072 (35 #4924) Zbl 0213.24901
- [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
- [Quillen1971] D. Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR0279804 (43 #5525) Zbl 0219.55013
- [Serre1951] J. Serre, Homologie singulière des espaces fibrès. Applications, Ann. of Math. (2) 54 (1951), 425–505. MR0045386 (13,574g) Zbl 0045.26003
- [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