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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. The calculation of was first obtained by Smale [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden [Barden1965] devised an elegant surgery argument and applied results of [Wall1964] on the diffeomorphism groups of 4-manifolds to give an explicit and complete classification of all of .
Simply-connected 5-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 5-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that not every simply-connected 5-dimensional Poincaré space is smoothable. The classification of simply-connected 5-dimensional Poincaré spaces was achieved by Stöcker [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 standard inclusion of .
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 -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.
2.1 The general spin case
Next we present a construction of simply-connected spin 5-manifolds. A priori the construction depends upon choices, but applying Theorem 4.3 below shows that the choices do not affect the diffeomorphism type of the manifold constructed. 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 with only -cells and -cells 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.
2.2 The general non-spin case
For the non-spin case we construct only those manifolds which are boundaries of -manifolds. As in the spin case, the construction depends a priori on choices, but Theorem 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.
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 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.
Consider the following invariants of a closed simply-connected -manifold :
- be the second integral homology group of ,
- , the homomorphism defined by evaluation with the second Stiefel-Whitney class of , ,
- , the smallest extended natural number such that for some . If is spinable we set .
For example, the manifold has invariants , non-trivial and . The Wu-manifold, , has invariants , non-trivial and .
The above list is the minimal list of invariants required to give the classification of closed simply-connected -manifolds: see Theorem 4.2 below.
In addition we mention two further invariants of :
- , the third Stiefel-Whitney class,
- , the linking form of which is a non-singular anti-symmetric bi-linear pairing on , the torsion subgroup of .
By [Milnor&Stasheff1974, Problem 8-A], , and so determines .
By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity where we regard as an element of . The classification of anti-symmetric linking forms is rather simple and this leads to the fact that one only needs to list the extended natural number in order to obtain a complete list of invariants of simply-connected -manifolds: This point is clarified in the following sub-section where we report on the classification of anti-symmetric linking forms.
3.1 Linking forms
An abstract non-singular anti-symmetric linking form on a finite abelian group is a bi-linear function
such that for all if and only if and for all pairs and . For example, if denotes the cyclic group of order , we have the following linking forms specified by their linking matricies,
If is the sum of cyclic groups we shall write for the sum .
By [Wall1963, Theorem 3], all non-singular anti-symmetric linking forms are isomorphic to a sum of the linking forms above and indeed 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 .
3.2 Values for constructions
The spin manifolds all have vanishing of course and so by Wall's classification of linking forms we see that the linking form of is the linking form .
As we mentioned above, the non-spin manifolds have given by projecting to and then applying :
If has height finite height then it follows from Wall's classification of linking forms that where and if has infinite height then .
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 of .
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 the groupoid with objects where is a finitely generated abelian group, is an 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 .
- Let be the groupoid with objects simply-connected closed smooth -manifolds embedded in some fixed for large and morphisms isotopy classes of diffeomorphisms.
- Consider the functor
Theorem 4.4 [Barden1965]. The functor is a detecting functor. That is, it induces a bijection on isomorphism classes of objects.
We first 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 simply-connected 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 spinable -manifold embeds 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.
- Barden's results have been nicely discussed and re-proven by Zhubr [Zhubr2001].
5.1 Bordism groups
As , and admits a unique spin structure which extends to we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group vanishes.
5.2 Curvature and contact structures
The following theorem is an immediate consequence of [Geiges1991, Theorem 1 & Lemma 7].
Theorem 5.1 [Geiges1991]. A simply connected -manifold admits a contact structure if and only if has an integral lift in . Hence admits a contact structure if and only if or ; equivalently admits a contact structure if and only if or where .
Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of prime to 3 was proven in [Thomas1986].
5.3 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 Theorem 4.3 above we obtain the following 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 , has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
- Open problem: as of writing there is no computation of for a general simply-connected 5-manifold in the literature.
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- The Wikipedia page on 1-connected 5-manifolds