5-manifolds: 1-connected

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1 Introduction

Let \mathcal{M}_{5}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_I6yOdc be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_cYJlmn and let \mathcal{M}_5^{\text{Spin}}\subset \mathcal{M}_5/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_eOlE8y be the subset of diffeomorphism classes of spinable manifolds. The calculation of \mathcal{M}_5^\Spin/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Yx0FAL 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 \mathcal{M}_{5}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_IynXMY.

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:

  • X_0 \coloneq S^5.
  • M_\infty \coloneq S^2 \times S^3.
  • X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3, the total space of the non-trivial S^3-bundle over S^2.
  • X_{-1} \coloneq \SU_3/\SO_3, the Wu-manifold, is the homogeneous space obtained from the standard inclusion of \SO_3 \rightarrow SU_3.

In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles M \cong W \cup_f W where W is a certain simply connected 5-manifold with boundary \partial W a simply-connected 4-manifold and f: \partial W \cong \partial W is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of H_2(\partial W) 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 G, let X_G denote the degree 2 Moore space with H_2(X_G) = G. The space X_G may be realised as a finite CW-complex with only 2-cells and 3-cells and so there is an embedding X_G\to\Rr^6. Let N_G be a regular neighbourhood of X_G\subset\Rr^6 and let M_G be the boundary of N_G. Then M_G is a closed, smooth, simply-connected, spinable 5-manifold with H_2(M_G)\cong G \oplus TG where TG is the torsion subgroup of G. For example, M_{\Zz^r} \cong \sharp_r (S^2 \times S^3) where \sharp_r denotes the r-fold connected sum.

2.2 The general non-spin case

For the non-spin case we construct only those manifolds which are boundaries of 6-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 (G, w) be a pair with w\co G \to\Zz_2 a surjective homomorphism and G as above. We shall construct a non-spin 5-manifold M_{(G, w)} with H_2(M_{(G, w)}) \cong G \oplus TG and second Stiefel-Whitney class w_2 given by w composed with the projection G \oplus TG \to G.

If (G, w) = (\Zz, 1) let N_{(\Zz, 1)} be the non-trivial D^4-bundle over S^2 with boundary \partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}. If (G, w) = (\Zz, 1) \oplus (\Zz^r, 0) let N_{(G, w)} be the boundary connected sum N_{(\Zz, 1)} \natural_r (S^2 \times D^4) with boundary M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3).

In the general case, present G = F/i(R) where i \co R \to F is an injective homomorphism between free abelian groups. Lift (G, w) to (F, \bar w) and observe that there is a canonical identification F = H_2(M_{(F, \bar w)}). If \{r_1, \dots, r_n \} is a basis for R note that each i(r_i) \in H_2(M_{(F, \bar w)}) is represented by an embedded 2-sphere with trivial normal bundle. Let N_{(G, w)} be the manifold obtained by attaching 3-handles to N_{(F, \bar w)} along spheres representing i(r_i) and let M_{(G, w)} = \partial N_{(G, w)}. One may check that M_{(G, w)} is a non-spin manifold as described above.

3 Invariants

Consider the following invariants of a closed simply-connected 5-manifold M:

  • H_2(M) be the second integral homology group of M,
  • w_2 \co H_2(M) \rightarrow \Zz_2, the homomorphism defined by evaluation with the second Stiefel-Whitney class of M, w_2 \in H^2(M; \Zz_2),
  • h(M) \in \Nn \cup \{\infty\}, the smallest extended natural number r such that 2^r \cdot x = 0 for some x \in w_2^{-1}(1). If M is spinable we set h(M) = 0.

For example, the manifold X_{\infty} has invariants H_2(X_{\infty}) \cong \Z, non-trivial w_2 and h(X_{\infty}) = \infty. The Wu-manifold, X_{-1}, has invariants H_2(X_{-1}) = \Zz_2, non-trivial w_2 and h(X_{-1}) = 1.

The above list is the minimal list of invariants required to give the classification of closed simply-connected 5-manifolds: see Theorem 4.2 below.

In addition we mention two further invariants of M:

  • w_3 \in H^3(M; \Zz_2), the third Stiefel-Whitney class,
  • b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz, the linking form of M which is a non-singular anti-symmetric bi-linear pairing on TH_2(M), the torsion subgroup of H_2(M).

By [Milnor&Stasheff1974, Problem 8-A], w_3 = Sq^1(w_2), and so w_2 determines w_3.

By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity b_M(x, x) = w_2(x) where we regard w_2(x) as an element of \{0, 1/2\} \subset \Qq/\Zz. 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 h(M) in order to obtain a complete list of invariants of simply-connected 5-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 H is a bi-linear function

\displaystyle  b \co H \times H \rightarrow \Qq/\Zz

such that b(x, y) = 0 for all y \in H if and only if x = 0 and b(x, y) = -b(y, x) for all pairs x and y. For example, if C_k denotes the cyclic group of order k, we have the following linking forms specified by their linking matricies,

\displaystyle  b_{-1} : C_2 \times C_2 \to \Qq/\Zz, \quad \left( \frac{1}{2} \right),
\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),
\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).

If T = \oplus_{i=1}^r C_{k_r} is the sum of cyclic groups we shall write b_0(T) for the sum \oplus_{i=1}^r b_0(C_{k_r}).

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

\displaystyle  w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).

Moreover H must be isomorphic to T \oplus T or T \oplus T \oplus \Zz_2 for some finite group T with b(x,x) = 1/2 if x generates the \Zz_2 summand. In particular the second Stiefel-Whitney class of a 5-manifold M determines the isomorphism class of the linking form b_M and we see that the torsion subgroup of H_2(M) is of the form TH_2(M) \cong T \oplus T if h(M) \neq 1 or TH_2(M) \cong T \oplus T \oplus \Zz_2 if h(M) = 1 in which case the \Zz_2 summand is an orthogonal summand of b_M.

3.2 Values for constructions

The spin manifolds M_G all have vanishing w_2 of course and so by Wall's classification of linking forms we see that the linking form of M_G is the linking form b_0(TG).

As we mentioned above, the non-spin manifolds M_{(G, \omega)} have w_2 given by projecting to G and then applying \omega:

\displaystyle  w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.

If M_{(G, \omega)} has height finite height h(M_{(G, \omega)}) = j then it follows from Wall's classification of linking forms that b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T) where TG \cong C_{2^j} \oplus T and if M_{(G, \omega)} has infinite height then b_{M_{(G, \omega)}} = b_0(TG).

4 Classification

We first present the most economical classifications of \mathcal{M}^\Spin_5 and \mathcal{M}_5. Let {\mathcal Ab}^{T \oplus T \oplus *} be the set of isomorphism classes finitely generated abelian groups G with torsion subgroup TG \cong H \oplus H \oplus C where C is trivial or C \cong \Zz_2 and write {\mathcal Ab}^{T \oplus T} and {\mathcal Ab}^{T \oplus T \oplus \Zz_2} for the obvious subsets of {\mathcal Ab}^{T \oplus T \oplus *}.

Theorem 4.1 [Smale1962, Theorem p.38]. There is a bijective correspondence
\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].

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))

is an injection onto the subset of pairs ([G], n) where [G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2} if and only if n = 1.

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 M_0 and M_1 be simply-connected, closed, smooth 5-manifolds and let A\co H_2(M_0) \cong H_2(M_1) be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then A is realised by a diffeomorphism f_A : M_0 \cong M_1.

This theorem can re-phrased in categorical language as follows.

  • Let \mathcal{Q}_5 be the groupoid with objects (G, b, w) where G is a finitely generated abelian group, b \co TG \times TG \to \Qq/\Zz is an anti-symmetric non-singular linking form and w\co G \to \Zz_2 is a homomorphism such that w(x) = b(x, x) for all x \in TG. The morphisms of \mathcal{Q}_5 are isomorphisms of abelian groups commuting with both w and b.
  • Let \widetilde{\mathcal{M}}_5 be the groupoid with objects simply-connected closed smooth 5-manifolds embedded in some fixed \Rr^N for N large 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).

Theorem 4.4 [Barden1965]. The functor (b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5 is a detecting functor. That is, it induces a bijection on isomorphism classes of objects.

4.1 Enumeration

We first give Barden's enumeration of the set \mathcal{M}^{}_5, [Barden1965, Theorem 2.3].

  • X_0 \coloneq S^5, M_\infty\coloneq S^2 \times S^3, X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3, X_{-1} \coloneq \SU_3/\SO_3.
  • For 1 < k < \infty, M_k = M_{\Zz_k} is the spin manifold with H_2(M) = \Zz_k \oplus \Zz_k constructed above.
  • For 1 < j < \infty let X_j = M_{(\Zz_{2^j}, 1)} constructed above be the non-spin manifold with H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}.

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}

where -1 \leq j \leq \infty, 1 < k_i, k_i divides k_{i+1} or k_{i+1} = \infty and \sharp denotes the connected sum of oriented manifolds. The manifold X_{j', k_1', \dots k_n'} is diffeomorphic to X_{j, k_1, \dots, k_n} if and only if (j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n).

An alternative complete enumeration is obtained by writing \mathcal{M}_5 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}
where the last two sets denote the diffeomorphism classes of non-spinable 5-manifolds which are respectively boundaries and not boundaries. Then
\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] \}.

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 5-manifold embeds into \Rr^6.
  • As the invariants for -M are isomorphic to the invariants of M 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 \mathcal{M}_5^\Spin = \{[M_G]\}, M_G = \partial N_G and M_G admits a unique spin structure which extends to N_G we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group \Omega_5^\Spin vanishes.

The bordism group \Omega_5^{\SO} is isomorphic to \Zz_2, see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number \langle w_2(M)w_3(M), [M] \rangle \in \Zz_2. The Wu-manifold has cohomology groups

\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2, \quad \ast = 0, 1, 2, 3, 4, 5,
and w_2(X_{-1}) \neq 0. It follows that w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0 and so we have that \langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0. We see that [X_{-1}] is the generator of \Omega_5^{\SO} and that a closed, smooth simply-connected 5-manifold M is not a boundary if and only if it is diffeomorphic to X_{-1} \sharp M_0 where M_0 is a Spin manifold.

5.2 Curvature and contact structures

Every manifold \sharp_r(S^2 \times S^3) admits a metric of positive Ricci curvature by [Boyer&Galicki2006].

The following theorem is an immediate consequence of [Geiges1991, Theorem 1 & Lemma 7].

Theorem 5.1 [Geiges1991]. A simply connected 5-manifold M admits a contact structure if and only if w_2(M) \in H^2(M; \Zz_2) has an integral lift in H^2(M; \Zz). Hence M admits a contact structure if and only if h(M) = 0 or \infty; equivalently M admits a contact structure if and only if M \in \mathcal{M}_5^\Spin or M \cong M_0 \sharp X_{\infty} where M_0 \in \mathcal{M}_5^\Spin.

Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of TH_2(M) prime to 3 was proven in [Thomas1986].

5.3 Mapping class groups

Let \pi_0\Diff_{+}(M) denote the group of isotopy classes of orientation preserving diffeomorphisms f\co M \cong M and let \Aut(H_2(M)) be the group of isomorphisms of H_2(M) preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence

(1)0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad

where \pi_0\SDiff(M) is the group of isotopy classes of diffeomorphisms inducing the identity on H_*(M).

  • There is an isomphorism \pi_0\Diff_{+}(S^5) \cong 0. By [Cerf1970] and [Smale1962a], \pi_0\Diff_{+}(S^5) \cong \Theta_6, the group of homotopy 6-spheres. But by [Kervaire&Milnor1963], \Theta_6 \cong 0.
  • In the homotopy category, \mathcal{E}_{+}(M), the group of homotopy classes of orientation preserving homotopy equivalences of M, 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 \pi_0\SDiff(M) for a general simply-connected 5-manifold in the literature.
    • However if TH_2(M) has no 2-torsion and no 3-torsion then \pi_0\SDiff(M) was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
    • Even the computation of \pi_0\SDiff(M) still leaves an unsolved extension problem in (1) above.

6 References

7 External links

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