6-manifolds: 1-connected

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1 Introduction

Let \mathcal{M}_6^\Cat be the set of \Cat isomorphism classes of closed oriented simply connected 6-dimensional \Cat-manifolds, where \Cat stands for \Diff (smooth manifolds), \PL (piecewise linear manifolds) or \Top (topological manifolds). On this page we describe the results of calculation of the sets \mathcal{M}_6^{\textup{Diff}} and \mathcal{M}_6^{\textup{Top}} begun by [Smale1962], extended in [Wall1966], [Jupp1973] and [Zhubr1975], and finally completed in [Zhubr2000]. An excellent summary for the torsion-free case (for those M\in\mathcal{M}_6^{\textup{Diff}} with \textup{Tors} H_2(M)=0) may be found in [Okonek&Van de Ven1995, Section 1]. For the case H_2(M)=0 see 6-manifolds:2-connected.

Remark 1.1.

  1. The sets \mathcal{M}_6^{\textup{Diff}} and \mathcal{M}_6^\PL are actually the same (as Wall points out in [Wall1966]): by Whitehead triangulation theorem we have the canonical forgetting map \mathcal{M}_6^{\textup{Diff}}\to\mathcal{M}_6^\PL; by smoothing theory and the fact that \PL/O is 6-connected, this is a bijection.
  2. The forgetting map \mathcal{M}_6^{\textup{Diff}}\to\mathcal{M}_6^{\textup{Top}} is injective: this follows from classification results below. Thus \mathcal{M}_6^{\textup{Diff}} can be viewed as a subset of \mathcal{M}_6^{\textup{Top}} (determined by the equation \Delta=0, where \Delta is the Kirby-Siebenmann triangulation class). In what follows, we abbreviate \mathcal{M}_6^{\textup{Top}} to just \mathcal{M}_6.

2 Classification

2.1 Notation

The standard projections \Zz/{mn}\to\Zz/n are denoted by \rho_n, the standard injections \Zz/m\to\Zz/{mn} by i_n (here m may equal \infty, in which case \rho_n:\Zz\to\Zz/n is the reduction modulo n, while i_n:\Zz\to\Zz is the multiplication by n). By P we denote the (non-stable) cohomology operation ``Pontryagin square´´

\displaystyle  H^{2i}(X;\,\Zz/2^m)\to H^{4i}(X;\,\Zz/2^{m+1}) \text{ with } i,m\ge1.

It is known that (with the same m) the following equalities hold: \rho_{2^m}P(x)=x^2 and P(x+y)=P(x)+P(y)+i_2xy. There exists also the ``Pontryagin cube´´

\displaystyle  P_3:H^{2i}(X;\,\Zz/3^m)\to H^{6i}(X;\,\Zz/3^{m+1})

and generally the ``Pontryagin p-th power´´ for every prime p [Thomas1956] (here we only need p=2 and p=3).

2.2 Classical invariants

Let M be a closed oriented simply connected 6-manifold (M\in\mathcal{M}_6 for short). Our first two invariants determine the (additive) homology structure of M:

  • b=b_2(M)\in2\Zz - the 3-dimensional Betty number,
  • G=H_2(M) - the 2-dimensional homology group.

Next, we consider the characteristic classes:

  • w=w_2(M)\in H^2(M;\,\Zz/2)=\Hom(G,\Zz/2) - the second Stiefel-Whitney class;
  • p=p_1(M)\in H^4(M)=G - the first Pontryagin class (in view of Poincaré duality, we will freely use such identifications as H^4(M)=H_2(M)=G etc.);
  • \Delta=\KS(M)\in H^4(M;\,\Zz/2)=G/2G -the Kirby-Siebenmann class (obstruction to smoothing the \Top-manifold M).

Note that all the other Stiefel-Whitney classes are uniquely determined as w_3=\Sq^1w, w_4=w^2 by the well-known Wu formulas and w_1=w_5=w_6=0 by trivial reasons. Also, the class p_1(M) may need some comment. In general, Pontryagin classes for \Top manifolds (or \Top microbundles) are defined as rational cohomology classes. The class p_1 is an exception, due to the equality H^4(B\Top)=H^4(BO), see [Jupp1973] or [Kirby&Siebenmann1977]. We denote by k_M the canonical mapping (or rather homotopy class of mappings) M\to K(G,2), inducing the identity H_2(M)\to G. Now, the last invariant

  • \mu=(k_M)_*[M]\in H_6(G,2),
the ``abstract fundamental class´´, incorporates all possible information about multiplication and other cohomology operations on H^2(M;\,\Zz/n) for every n. First, the class \mu determines the family of trilinear functions
\displaystyle  H^2(G,2;\,\Zz/n)^3\to\Zz/n, \quad 2\le n\le\infty
by the rule x,y,z\mapsto\langle xyz,\mu\rangle; note that H^2(G,2;\,\Zz/n) is the same as H^2(M;\,\Zz/n). In what follows, instead of \langle xyz,\mu\rangle we may write \langle xyz,[M]\rangle or sometimes simply xyz. Second, there are two more families:
\displaystyle  x,y\mapsto\langle P(x)y,\mu \rangle \in\Zz/2^{m+1}
for x\in H^2(G,2;\,\Zz/2^m), y\in H^2(G,2;\,\Zz/2^{m+1}), and
\displaystyle  x\mapsto\langle P_3(x),\mu \rangle \in\Zz/3^{m+1}
for x\in H^2(G,2;\,\Zz/3^m). From the structure of the groups H_6(G,2) one can easily deduce that, conversely, \mu is uniquely determined by all these values.

2.3 Relations for classical invariants

There are two evident restrictions for invariants b and G:

  • b\equiv0\mod2;
  • G is a finitely generated abelian group

(where the first one follows from the existence of non-singular skew-symmetric form on H_3(M)/\textup{Tors}). There are two more restrictions (Wu relations):

  • (R_1) \langle xy(x+y+w),\mu \rangle =0 for all x,y\in H^2(G,2;\,\Zz/2),
  • (R_2) \langle x,p \rangle=\langle x^3,\mu \rangle for all x\in H^2(G,2;\,\Zz/3).

These relations are given in [Wall1966, Theorem 3]. Wall formulates them for torsion-free case and in integral form (and for smooth category), but the argument for general case is the same. We call them ``Wu relations´´ because (as Wall points out) they are easily deduced from the well-known Wu formula \langle (\mathrm{Sq}+v)x,[M] \rangle =0 for \Zz/2-coefficients, and its certain analogue (due to Wu as well) for \Zz/3. Note that (R_2) could be also written as px=x^3, having in mind multiplication ``on M´´.

2.4 Further notation

For any m, 1\le m\le\infty, we denote by \mathcal{E}_m(w) the set of all \omega\in H^2(G,2;\,\Zz/2^m) satisfying \rho_2(\omega)=w. Note that \mathcal{E}_1(w)=\{w\}, and that \mathcal{E}_m(w) may become empty for m sufficiently large. We write \mathrm{h}(w) for \sup\{m\mid\mathcal{E}_m(w)\ne\varnothing\}\in\mathbb{N}\cup\infty. Let m be any integer in \{2,\ldots,\mathrm{h}(w)\}. For any \omega\in\mathcal{E}_m(w) and any x\in H^2(G,2;\,\Zz/2^{m-1}), and assuming relation (R_1) satisfied, we have
\displaystyle  \rho_2\langle \omega P(x),\mu \rangle=\langle wx^2, \mu \rangle = 0.
Hence, \langle \omega P(x),\mu \rangle is in the image of the inclusion i_2:\Zz/2^{m-1}\to\Zz/2^m, and we set
\displaystyle  R_\mu(\omega,x)=\langle \omega^2x+3i_2^{-1}\omega P(x)+x^3,\mu \rangle \in \Zz/2^{m-1}.
One can check that
\displaystyle  R_\mu(\omega,x+y)-R_\mu(\omega,x)-R_\mu(\omega,y)=3\langle xy(x+y+\omega),\mu \rangle,
so that R_\mu(\omega,x) is linear in x for m=2 (by (R_1) again).

2.5 Special invariants

There is a detailed treatment in [Zhubr2000]. Here we only give a formal description. For any M\in\mathcal{M}_6, and each m\in\{2,\ldots,\mathrm{h}(w)\}, there are functions

  • \Gamma: \mathcal{E}_m(w) \to \Zz/2^{m-1},
  • \gamma: \mathcal{E}_m(w) \to G/2^{m-1}G,

satisfying the following set of identities (which are considered to be a part of definition, whereas relations that go further on define the range):

  • (I_1) \rho_{2^{m-1}}\Gamma(\omega)=\Gamma(\rho_{2^m}\omega) (first coefficient formula),
  • (I_2) \rho_{2^{m-1}}\gamma(\omega)=\gamma(\rho_{2^m}\omega) (second coefficient formula)

for 2\le m< \mathrm{h}(w), and

  • (I_3) \Gamma (\omega+i_2x)-\Gamma (\omega) = \langle x,\gamma(\omega) \rangle - R_\mu(\omega,x) (first difference formula),
  • (I_4) \gamma (\omega+i_2x)-\gamma (\omega) = \mu\cap(\omega x + x^2) (second difference formula)

for \omega\in\mathcal{E}_m(w), x\in H^2(G,2;\,\Zz/2^{m-1}). In what follows, the values of \Gamma,\gamma at \omega\in\mathcal{E}_m(w) may be also written as \Gamma_\omega,\gamma_\omega for convenience.

Remark 2.1.

  1. In view of these identities, one easily sees that the functions \Gamma and \gamma are completely determined by their values at some fixed \omega\in\mathcal{E}_{h(w)}(w). Thus, if we could make a canonical choice, then our couple of invariants would trivialize to just \Gamma_0\in\Zz/2^{m-1}, \gamma_0\in G/2^{m-1}G. Evidently, such canonical choice is impossible in general, however in the spin case one can take \omega=0 with m=\infty (see below).
  2. From (I_3) it easily follows that \gamma is in fact determined by \Gamma, so our list of invariants could be reduced by 1, at the cost of reduced convenience.

2.6 Relations for special invariants

  • (R_3) i_4\gamma_\omega=\rho_{2^{m+1}}p + \mu\cap P(\omega) for \omega\in\mathcal{E}_m(w),
  • (R_4) \langle\omega,\gamma_\omega\rangle=2\Gamma_\omega + i_2^{-1}\langle \omega^3,\mu \rangle for \omega\in\mathcal{E}_m(w),
  • (R_5) \Gamma_\omega=\langle w,\Delta \rangle for \omega\in\mathcal{E}_2(w).

2.7 The splitting theorem

Wall in [Wall1966] proves the following

Theorem 2.2. Let M be a closed, smooth, 1-connected 6-manifold. Then we can write M as a connected sum M_1\#M_2, where H_3(M_1) is finite and M_2 is a connected sum of copies of S^3\times S^3.

This theorem allows to restrict the classification problem to the case where b_3(M)=0. The proof is rather easy and basically reduces to realizing the standard ``symplectic´´ basis of H_3(M) with embedded 3-spheres (and applying ``Whitney trick´´ where necessary). As is pointed out in [Jupp1973], the same argument works for \Top category. Wall does not state the uniqueness of M_1 in this theorem, however uniqueness follows from his classification [Wall1966, Theorem 5] for smooth, spin, torsion-free manifolds. Likewize, uniqueness of the above splitting follows for all torsion-free manifolds (both in \Diff and \Top) from the results of [Jupp1973], and in full generality from the general classification theorem of [Zhubr2000] (see below). Note that the invariants (except b of course) are ``insensitive´´ to connected summing with S^3\times S^3 (this is evident for classical invariants, while for \Gamma,\gamma we refer to their definition in [Zhubr2000]). It should be also noted that the uniqueness statement for Theorem 2.2 was proved directly (independent of classification) in [Zhubr1973].

2.8 Functorial behaviour of invariants

Consider the set of invariants G, w, p, \Delta, \mu, \Gamma, \gamma (with b left out). We divide these into two subsets: (G,w) and (p,\Delta,\mu,\Gamma,\gamma). We say that the set s=(p,\Delta,\mu,\Gamma,\gamma) is admissible for (G,w) if p\in G, \Delta\in G/2G etc. satisfy all the identities and relations given above (these invariants are now regarded in ``abstract´´ way, irrelative to any manifold). Let \mathcal{S}(G,w) denote the collection of all admissible sets of invariants for (G,w). Consider now the category \mathcal{A} of finitely generated abelian groups, and the category \Hom(\mathcal{A},\Zz/2), whose objects are homomorphisms w:G\to\Zz/2 with G\in\mathcal{A}, and whose morphisms are commutative diagrams of the form

\displaystyle  \xymatrix{G\ar[rr]\ar[dr]^w && G'\ar[dl]_{w'} \\ & \Zz/2}

For each morphism \varphi:(G,w)\to(G',w'), we can define the induced map \varphi_*:\mathcal{S}(G,w)\to\mathcal{S}(G',w') in a natural way: if s=(p,\Delta,\mu,\Gamma,\gamma)\in\mathcal{S}(G,w), then we set \varphi_*(s)=(p',\Delta',\mu',\Gamma',\gamma') with \Gamma'(\omega)=\Gamma(\varphi^*\omega), \gamma'(\omega)=\varphi(\gamma(\varphi^*\omega)), and so on (the rest is quite evident). One easily verifies that the new invariant set is admissible again. Hence we have a functor \mathcal{S}:\Hom(\mathcal{A},\Zz/2)\to\textup{Sets}.

2.9 Classification theorem (the general case)

We use the notation \mathcal{M}_6^r for the subset of \mathcal{M}_6, defined by the equation b_3(M)=r. For any M\in\mathcal{M}_6 and (G,w)=(H_2(M),w_2(M)), let I(M)\in\mathcal{S}(G,w) be the set (p(M),\Delta(M),\mu(M),\Gamma(M),\gamma(M)). The following theorem [Zhubr2000, Theorem 6.3] gives the topological and differential classification of all closed oriented simply connected 6-manifolds.

Theorem 2.3. (1) Let (G,w)=(H_2(M),w_2(M)) and (G',w')=(H_2(M'),w_2(M')), where M,M'\in\mathcal{M}_6^r. An isomorphism \varphi:(G,w)\to(G',w') is induced by orientation-preserving homeomorphism f:M\to M' if and only if \varphi_*I(M)=I(M') (completeness of the set of invariants). (2) For each s\in\mathcal{S}(G,w) there exists M\in\mathcal{M}_6^0 with (H_2(M),w_2(M))=(G,w) and I(M)=s (completeness of the set of relations). (3) If manifolds M and M' (statement (1)) are given smooth structures, then homeomorphism f can be chosen smooth also.

Remark 2.4.

  1. The clause ``only if´´ of statement (1) is tautological (it just says that our invariants are invariants indeed).
  2. For any (G,w)\in\Hom(\mathcal{A},\Zz/2) let \mathcal{M}_6^r(G,w) denote the set of homeomorphism classes of pairs (M,\psi), where M\in\mathcal{M}_6^r and \psi:H_2(M)\to G is an isomorphism with \psi^*(w)=w_2(M). One can say that \mathcal{M}_6^r(G,w) is the set of (homeomorphism classes of) manifolds with prescribed homology and second Stiefel-Whitney class. We can write (taking some liberty in notations):
    \displaystyle \mathcal{M}_6^r=\bigcup_{(G,w)} \mathcal{M}_6^r(G,w).
    Now we have the natural maps I:\mathcal{M}_6^r(G,w)\to\mathcal{S}(G,w), and from the above theorem it follows that all these maps are bijections.
  3. From the statement (3) it evidently follows that a closed simply connected 6-manifold has at most one (up to homeomorphism) smooth structure (Hauptvermutung).

2.10 The spin case

For w=0 we have the canonical choice \omega=0\in\mathcal{E}_\infty(0), as was noted above. Applying relations (R_3)--(R_5) to \Gamma_0\in\Zz and \gamma_0\in G, we obtain the equalities 4\gamma_0=p, 0=2\Gamma_0 and 0=0, respectively. Thus we can ``cross out´´ the invariants p and \Gamma, which leaves us with (G,\Delta,\mu,\gamma) (we remind that one can suppose b_3(M)=0 by Theorem 2.2).

It should be noted that one cannot simply drop the relations (R_3)--(R_5) after this ``crossing out´´: to preserve all information the relations may contain, we still have to apply them to entire families \{\Gamma_\omega\} and \{\gamma_\omega\}. Any \omega\in\mathcal{E}_m(0) can be written in the form \omega=i_2x for x\in H^2(G,2;\,\Zz/2^{m-1}); applying the identities (I_1)--(I_4), we may write \gamma_\omega = \rho_{2^{m-1}}\gamma_0 + \mu\cap x^2 and \Gamma_\omega = \langle x,\gamma_0 \rangle - \langle x^3,\mu \rangle. Applying this to (R_3)--(R_5) again, one can see by straightforward checking that (R_3) and (R_4) are tautologically true, while (R_5) turns into

  • (\bar R_5) \langle x,\gamma\rangle=\langle x^3,\mu\rangle for any x\in H^2(G,2;\,\Zz/2),

using \gamma as abbreviation for \gamma_0.

Now the relation (R_1) immediately follows from (\bar R_5), while (R_2) can be combined with (\bar R_5) in the form

  • (R_s) \langle x,\gamma \rangle = \langle x^3,\mu \rangle for any x\in H^2(G,2;\,\Zz/6).

Hence, for the spin case we have the complete set of invariants (G,\Delta,\mu,\gamma) with the only relation (R_s), which gives (for smooth category) the main result of [Zhubr1975].

2.11 The torsion-free case

It is convenient here to represent the additive homology information about some M\in\mathcal{M}_6^0 by its cohomology group H=H^2(M), and interpret our previous group G as \Hom(H,\Zz). Likewise, the elements of H_6(G,2) can be considered as symmetric trilinear forms (or cubic forms) on H. We thus have the following set of invariants:

  • (H,\ w{\in}H/2H,\ p{:}H{\to}\Zz,\ \Delta{:}H{\to}\Zz/2,\ \mu{:}H^3{\to}\Zz,\ \Gamma_\omega{\in}\Zz,\ \gamma_\omega{:}H{\to}\Zz)

(for any \omega\in H with \rho_2\omega=w). We rewrite (R_1),(R_2) in the form:

  • (R_1') \mu(x,y,x+y+\omega)\equiv0{\mod2}\quad for any x,y\in H,
  • (R_2') p(x)\equiv\mu(x,x,x){\mod3}\quad for any x\in H.

Next consider the relations (R_3) and (R_4). We can now ``solve´´ them for \gamma_\omega and \Gamma_\omega:

  • (R_3') \displaystyle\gamma_\omega(x)=\frac{p(x)+\mu(\omega,\omega,x)}{4},
  • (R_4') \displaystyle\Gamma_\omega=\frac{2\gamma_\omega(\omega) - \mu(\omega,\omega,\omega)}{4}=\frac{p(\omega) - \mu(\omega,\omega,\omega)}{8}.

This shows that \Gamma and \gamma are expressible in terms of ``classical´´ invariants and should be dropped.

It only remains to consider the last relation (R_5). We denote by K some (arbitrary) element of G with \rho_2K=\Delta. In view of the above equalities, relation (R_5) can now be written in the form

  • (R_5') \displaystyle\frac{p(\omega) - \mu(\omega,\omega,\omega)}{8}\equiv K(\omega) \mod2

or, equivalently,

  • (R_5'') (p+8K)(\omega)\equiv\mu(\omega,\omega,\omega)\mod16.

Now, quite similar to the spin case above, it is an easy exercise to see that (R_1') follows from (R_5''), while (R_2') and (R_5'') together are equivalent to

  • (R_f) (p+24K)(\omega)\equiv\mu(\omega,\omega,\omega)\mod48.

We have, therefore, the complete set of invariants (H,w,p,\Delta,\mu) satisfying the only relation (R_f), which gives Theorem 1 of [Jupp1973]; restricting to \Cat=\Diff and w=0, we obtain Theorem 5 of [Wall1966].

Remark 2.5.

  1. Relations (R_3') and (R_4') can be written, for a manifold M, in the form \gamma_\omega=\frac14D(p_1(M)+\omega^2) and \Gamma_\omega=\frac18\langle p_1(M)\omega-\omega^3,[M]\rangle, respectively. The fact that p_1(M)+\omega^2 is divisible by 4 is due to Wu formulas ``modulo 2´´ (for Stiefel-Whitney classes) and ``modulo 4´´ (for Pontryagin classes). The divisibility of the second expression by 8 can be explained as follows: there exists a 4-submanifold N\subset M dual to the cohomology class \omega; any such N is spin, and its first Pontryagin class (=3~times its signature) is equal to (p_1(M)-\omega^2)|_N.
  2. The fact that relation (R_1)/(R_1') is excessive in the spin case went unnoticed in both [Wall1966] and [Zhubr1975].
  3. Relation (R_f) in the case of Wall, i.e. for \omega=2x and K even, simplifies to
    • (R_w) p(x)\equiv4\mu(x,x,x)\mod24.
  4. The proof of (R_w) given by Wall relies on construction of 6-manifolds of the type considered by surgery on S^6 along framed 3-dimensional links, and on relations between the invariants of such links (studied by Haefliger) and the invariants of the resulting manifolds (i.e. p and \mu). On the other hand, the proof of the more complicated relation (R_f) in [Jupp1973] is based on integrality theorem for \hat A-genus. In fact, for smooth case this follows immediately; regretfully, the argument Jupp gives to extend this to \Top category is incorrect, as it uses the erroneous homotopy classification theorem of [Wall1966] (see [Zhubr2000, Subsection 5.14]).

3 Examples and constructions

  • The r-fold connected sum \sharp_r S^3\times S^3 gives the only element of \mathcal{M}^{2r}_6(0).
  • The r-fold connected sum M=\sharp_r S^2\times S^4 gives the ``primary´´ element of \mathcal{M}^0_6(\mathbb{Z}^r,0) --- a manifold with I(M)=(p,\mu)=0.
  • In the same way, the non-trivial SO(5)-bundle S^2\tilde\times S^4 is the ``primary´´ element of \mathcal{M}^0_6(\mathbb{Z},1).
  • The two examples above can be easily generalized: let w:\mathbb{Z}^r\to\mathbb{Z}/2 be arbitrary homomorphism. Let M be the connected sum \sharp M_i, where each M_i is either S^2\times S^4 or S^2\tilde\times S^4, depending on the value w takes at the i-th basis vector of \mathbb{Z}^r. Then we have M\in\mathcal{M}^0_6(\mathbb{Z}^r,w) and I(M)=0.
  • Surgery lets us extend the above construction to arbitrary (G,w)\in\Hom(\mathcal{A},\mathbb{Z}/2). Take any epimorphism f:\mathbb{Z}^r\to G and build a manifold M_0\in\mathcal{M}^0_6(\mathbb{Z}^r,w\circ f) as above. Now represent a free basis of \Ker f by embedded spheres S^2\to M_0 and do surgery on M_0 along these spheres. As may be checked, the result is a manifold M\in\mathcal{M}^0_6(G,w) with I(M)=0 (therefore, uniquely defined).
  • The SO(3)-bundles M_\beta=S^4\tilde\times_\beta S^2, with \beta\in\pi_3(SO(3))=\mathbb{Z}, give us manifolds in \mathcal{M}^0_6(\mathbb{Z},0) with I(M_\beta)=(4\beta,\beta); in particular, M_1=\mathbb{C} P^3.
  • Complex algebraic geometry makes (potentially) a very powerful source of examples. In particular, any regular complete intersection M_d of complex dimension 3 represents an element of \mathcal{M}^r_6(\mathbb{Z},w), where r,w and I(M_d)=(p,\mu) can be directly calculated from the multidegree d=(d_1,\ldots,d_m) (see ``Complete intersections´´). In the easiest case --- when M_d\subset\mathbb{C} P^4 is a non-singular hypersurface of degree d --- we have M_d\in\mathcal{M}^r_6(\mathbb{Z},\rho_2(d-1)), I(M_d)=(d(5-d^2),d), and r=d^4+\ldots (a polynome in d of degree 4).

4 References

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