5-manifolds: 1-connected
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Contents |
1 Introduction
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of . Simply-connected
Tex syntax error-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected
Tex syntax error-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected
Tex syntax error-dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected
Tex syntax error
-manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
-
Tex syntax error
be the second integral homology group ofTex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite groupTex syntax erroris a bi-linear function such that
Tex syntax errorand
Tex syntax errorfor all
Tex syntax errorif and only if
Tex syntax error. By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover
Tex syntax errormust be isomorphic to
Tex syntax erroror
Tex syntax errorfor some finite group
Tex syntax errorwith
Tex syntax errorif generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold
Tex syntax errordetermines the isomorphism class of the linking form
Tex syntax errorand we see that the torsion subgroup of
Tex syntax erroris of the form
Tex syntax errorif
Tex syntax erroror
Tex syntax errorif
Tex syntax errorin which case the summand is an orthogonal summand of
Tex syntax error.
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into
Tex syntax error
.
- As the invariants for
Tex syntax error
are isomorphic to the invariants ofTex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Tex syntax errorpreserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above
we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [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
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Smale1962] S. Smale, On the structure of
Tex syntax error
-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103 - [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
- [Stöcker1982] R. Stöcker, On the structure of
Tex syntax error
-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012 - [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of . Simply-connected
Tex syntax error-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected
Tex syntax error-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected
Tex syntax error-dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected
Tex syntax error
-manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
-
Tex syntax error
be the second integral homology group ofTex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite groupTex syntax erroris a bi-linear function such that
Tex syntax errorand
Tex syntax errorfor all
Tex syntax errorif and only if
Tex syntax error. By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover
Tex syntax errormust be isomorphic to
Tex syntax erroror
Tex syntax errorfor some finite group
Tex syntax errorwith
Tex syntax errorif generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold
Tex syntax errordetermines the isomorphism class of the linking form
Tex syntax errorand we see that the torsion subgroup of
Tex syntax erroris of the form
Tex syntax errorif
Tex syntax erroror
Tex syntax errorif
Tex syntax errorin which case the summand is an orthogonal summand of
Tex syntax error.
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into
Tex syntax error
.
- As the invariants for
Tex syntax error
are isomorphic to the invariants ofTex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Tex syntax errorpreserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above
we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [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
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Smale1962] S. Smale, On the structure of
Tex syntax error
-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103 - [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
- [Stöcker1982] R. Stöcker, On the structure of
Tex syntax error
-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012 - [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of . Simply-connected
Tex syntax error-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected
Tex syntax error-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected
Tex syntax error-dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected
Tex syntax error
-manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
-
Tex syntax error
be the second integral homology group ofTex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite groupTex syntax erroris a bi-linear function such that
Tex syntax errorand
Tex syntax errorfor all
Tex syntax errorif and only if
Tex syntax error. By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover
Tex syntax errormust be isomorphic to
Tex syntax erroror
Tex syntax errorfor some finite group
Tex syntax errorwith
Tex syntax errorif generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold
Tex syntax errordetermines the isomorphism class of the linking form
Tex syntax errorand we see that the torsion subgroup of
Tex syntax erroris of the form
Tex syntax errorif
Tex syntax erroror
Tex syntax errorif
Tex syntax errorin which case the summand is an orthogonal summand of
Tex syntax error.
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into
Tex syntax error
.
- As the invariants for
Tex syntax error
are isomorphic to the invariants ofTex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Tex syntax errorpreserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above
we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [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
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Smale1962] S. Smale, On the structure of
Tex syntax error
-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103 - [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
- [Stöcker1982] R. Stöcker, On the structure of
Tex syntax error
-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012 - [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of . Simply-connected
Tex syntax error-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected
Tex syntax error-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected
Tex syntax error-dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected
Tex syntax error
-manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
-
Tex syntax error
be the second integral homology group ofTex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite groupTex syntax erroris a bi-linear function such that
Tex syntax errorand
Tex syntax errorfor all
Tex syntax errorif and only if
Tex syntax error. By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover
Tex syntax errormust be isomorphic to
Tex syntax erroror
Tex syntax errorfor some finite group
Tex syntax errorwith
Tex syntax errorif generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold
Tex syntax errordetermines the isomorphism class of the linking form
Tex syntax errorand we see that the torsion subgroup of
Tex syntax erroris of the form
Tex syntax errorif
Tex syntax erroror
Tex syntax errorif
Tex syntax errorin which case the summand is an orthogonal summand of
Tex syntax error.
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into
Tex syntax error
.
- As the invariants for
Tex syntax error
are isomorphic to the invariants ofTex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Tex syntax errorpreserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above
we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [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
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Smale1962] S. Smale, On the structure of
Tex syntax error
-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103 - [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
- [Stöcker1982] R. Stöcker, On the structure of
Tex syntax error
-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012 - [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of . Simply-connected
Tex syntax error-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected
Tex syntax error-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected
Tex syntax error-dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected
Tex syntax error
-manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
-
Tex syntax error
be the second integral homology group ofTex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite groupTex syntax erroris a bi-linear function such that
Tex syntax errorand
Tex syntax errorfor all
Tex syntax errorif and only if
Tex syntax error. By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover
Tex syntax errormust be isomorphic to
Tex syntax erroror
Tex syntax errorfor some finite group
Tex syntax errorwith
Tex syntax errorif generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold
Tex syntax errordetermines the isomorphism class of the linking form
Tex syntax errorand we see that the torsion subgroup of
Tex syntax erroris of the form
Tex syntax errorif
Tex syntax erroror
Tex syntax errorif
Tex syntax errorin which case the summand is an orthogonal summand of
Tex syntax error.
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into
Tex syntax error
.
- As the invariants for
Tex syntax error
are isomorphic to the invariants ofTex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Tex syntax errorpreserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above
we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [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
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Smale1962] S. Smale, On the structure of
Tex syntax error
-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103 - [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
- [Stöcker1982] R. Stöcker, On the structure of
Tex syntax error
-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012 - [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub