5-manifolds: 1-connected
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Contents |
1 Introduction
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 in [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 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 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 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 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 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.
3 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
.
3.1 Linking forms
An abstract non-singular, anti-symmetric linking form on a finite group is a bi-linear function
![\displaystyle b \co H \times H \rightarrow \Qq/\Zz](/images/math/f/f/9/ff9001fa270e8fcd03cac63e8e2dd49b.png)
such that and
for all
if and only if
. For example, if
is the sum of cyclic groups of order
we have the following linking forms specified by their linking matricies
![\displaystyle b_0(C_k) : (C_k \oplus C_k) \times (C_k \oplus C_k) \to \Qq/\Zz, \quad \left( \begin{array}{cc} 0 & ~\frac{1}{k} \\-\frac{1}{k} & ~0 \end{array} \right),](/images/math/9/1/f/91f489c12e8aeb16f2814f088d3af9e3.png)
![\displaystyle b_j(C_{2^j}) : (C_{2^j} \oplus C_{2^j}) \times (C_{2^j} \oplus C_{2^j}) \to \Qq/\Zz, \quad \left( \begin{array}{cc} \frac{1}{2} & ~\frac{1}{2^j} \\ -\frac{1}{2^j} & ~0 \end{array} \right).](/images/math/8/6/1/8612e445c4982ca55ba60b2ab991d4dd.png)
If is the sum of cyclic groups we shall write
for the sum
.
By [Wall1963] all non-singular anti-symmetric linking forms are isomorphic to a sum of the linking forms above. Indeed such linking forms are classified up to isomorphism by the homomorphism
![\displaystyle w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).](/images/math/b/6/6/b66a937a62e234b7ed95115c9e203674.png)
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
:
![\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz/2.](/images/math/b/f/e/bfe9e7bb608a2e0cbcca54ffa2844ee2.png)
If has height finite height
then it follows from Wall's classification of linking forms that
where
and if
has infinite height then
.
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.
![\displaystyle \mathcal{M}_5^{\Spin} \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].](/images/math/4/e/e/4eeb35c40485d773fe9acb58d484eb16.png)
Theorem 4.2 [Barden1965]. The mapping
![\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))](/images/math/b/6/6/b66170d9d73814c75d7d944eca181a3a.png)
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
.
- Let
be a small groupoid with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy classes of diffeomorphisms.
- Consider the functor
![\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).](/images/math/9/a/2/9a22c460195b2e5423084b67fd6472e2.png)
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is bijective on isomorphism classes of objects.
4.1 Enumeration
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
![\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}](/images/math/c/8/d/c8dc36c624269e889ce0ea5e15bbfc68.png)
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
![\displaystyle \mathcal{M}_5 = \mathcal{M}_5^{\Spin} \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}](/images/math/1/7/8/1781e6fb953d69992b8eafcf98b91432.png)
![\displaystyle \mathcal{M}_5^{\Spin} = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.](/images/math/5/a/9/5a9e34a969aff7a144a224b6626d1e9d.png)
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
.
- 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 are nicely discussed and re-proven in [Zhubr2001].
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
![\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2](/images/math/6/6/c/66c4a5b8dccb0488c85686f7977a6287.png)
![w_2(X_{-1}) \neq 0](/images/math/b/a/c/bac1dfbbcc7a4415a803c207412a5913.png)
![w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0](/images/math/c/9/3/c93b2b1a029ed8343e9316bc013cacef.png)
![\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0](/images/math/8/2/1/821189f95965b9293938b9750b268abc.png)
![[X_{-1}]](/images/math/3/f/3/3f3bed6681734999df212ea7e3e6b26d.png)
![\Omega_5^{\SO}](/images/math/a/1/c/a1ccd7971eeb1822e97871ff927788f3.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![X_{-1} \sharp M_0](/images/math/d/b/1/db1319706421e10fe3dbf6c30eb97b4b.png)
![M_0](/images/math/7/7/9/779dee235baf1f57777af5cd125b9736.png)
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
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
![\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast)](/images/math/d/f/0/df015be610eaa218f9a45d256eccdcdf.png)
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.
- However if
has no
-torsion and no
-torsion then
was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
- Even the computation of
still leaves an unsolved extension problem in
above.
- However if
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
- [Fang1993] F. Fang, Diffeomorphism groups of simply connected 5-manifolds, unpublished pre-print (1993).
- [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
-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
-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
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-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101
- [Zhubr2001] A. V. Zhubr, On a paper of Barden, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001), Geom. i Topol. 6, 70–88, 247; translation in J. Math. Sci. (N. Y.) 119 (2004), no. 1, 35–44. MR1846073 (2002e:57040) Zbl 1072.57024