5-manifolds: 1-connected
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Let be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds and let be the subset of diffeomorphism classes of spinable manifolds. In this article we report the calculation of first obtained in [Smale1962] and of first obtained in general in [Barden1965].
Contents |
1 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.
- !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.
2 Invariants
Consider the following invariants of a closed simply-connected 5-manifold .
- be the second integral homology group of , with torsion subgroup .
- , the homomorphism defined by evaluation with the second Stiefel-Whitney class of , .
- , the smallest extended natural number such that and . If is Spin we set .
- , the linking form of 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 group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold determines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3 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 3.2 [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 3.3 [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 maps to and to the induced map on .
Theorem [Barden1965] 3.4. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
3.1 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
4 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.
- The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
4.1 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 and is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
- !!! Add references here.
4.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure [Thomas1986].
4.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving 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 .
- Open problem: as of writing there is no computation of for a general simply-connected 5-manifold.
- The isotopy group is known to be the trivial group [[[#|]]].
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of , has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
5 References
- [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
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [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
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [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 0215.39903MediaWiki:Stub
Contents |
1 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.
- !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.
2 Invariants
Consider the following invariants of a closed simply-connected 5-manifold .
- be the second integral homology group of , with torsion subgroup .
- , the homomorphism defined by evaluation with the second Stiefel-Whitney class of , .
- , the smallest extended natural number such that and . If is Spin we set .
- , the linking form of 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 group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold determines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3 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 3.2 [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 3.3 [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 maps to and to the induced map on .
Theorem [Barden1965] 3.4. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
3.1 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
4 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.
- The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
4.1 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 and is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
- !!! Add references here.
4.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure [Thomas1986].
4.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving 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 .
- Open problem: as of writing there is no computation of for a general simply-connected 5-manifold.
- The isotopy group is known to be the trivial group [[[#|]]].
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of , has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
5 References
- [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
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [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
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [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 0215.39903MediaWiki:Stub
Contents |
1 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.
- !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.
2 Invariants
Consider the following invariants of a closed simply-connected 5-manifold .
- be the second integral homology group of , with torsion subgroup .
- , the homomorphism defined by evaluation with the second Stiefel-Whitney class of , .
- , the smallest extended natural number such that and . If is Spin we set .
- , the linking form of 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 group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold determines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3 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 3.2 [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 3.3 [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 maps to and to the induced map on .
Theorem [Barden1965] 3.4. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
3.1 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
4 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.
- The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
4.1 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 and is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
- !!! Add references here.
4.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure [Thomas1986].
4.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving 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 .
- Open problem: as of writing there is no computation of for a general simply-connected 5-manifold.
- The isotopy group is known to be the trivial group [[[#|]]].
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of , has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
5 References
- [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
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [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
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [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 0215.39903MediaWiki:Stub
Contents |
1 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.
- !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.
2 Invariants
Consider the following invariants of a closed simply-connected 5-manifold .
- be the second integral homology group of , with torsion subgroup .
- , the homomorphism defined by evaluation with the second Stiefel-Whitney class of , .
- , the smallest extended natural number such that and . If is Spin we set .
- , the linking form of 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 group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold determines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3 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 3.2 [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 3.3 [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 maps to and to the induced map on .
Theorem [Barden1965] 3.4. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
3.1 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
4 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.
- The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
4.1 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 and is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
- !!! Add references here.
4.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure [Thomas1986].
4.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving 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 .
- Open problem: as of writing there is no computation of for a general simply-connected 5-manifold.
- The isotopy group is known to be the trivial group [[[#|]]].
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of , has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
5 References
- [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
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [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
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [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 0215.39903MediaWiki:Stub
Contents |
1 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.
- !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.
2 Invariants
Consider the following invariants of a closed simply-connected 5-manifold .
- be the second integral homology group of , with torsion subgroup .
- , the homomorphism defined by evaluation with the second Stiefel-Whitney class of , .
- , the smallest extended natural number such that and . If is Spin we set .
- , the linking form of 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 group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifold determines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3 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 3.2 [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 3.3 [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 maps to and to the induced map on .
Theorem [Barden1965] 3.4. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
3.1 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
4 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.
- The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
4.1 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 and is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
- !!! Add references here.
4.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure [Thomas1986].
4.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving 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 .
- Open problem: as of writing there is no computation of for a general simply-connected 5-manifold.
- The isotopy group is known to be the trivial group [[[#|]]].
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of , has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
5 References
- [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
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [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
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [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 0215.39903MediaWiki:Stub