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Let be the set of isomorphism classes of closed oriented simply connected 6-dimensional -manifolds, where stands for (smooth manifolds), (piecewise linear manifolds) or (topological manifolds). On this page we describe the results of calculation of the sets and begun by [Smale1962], extended in [Wall1966], [Jupp1973] and [Zhubr1975], and finally completed in [Zhubr2000]. An excellent summary for the torsion-free case (for those with ) may be found in [Okonek&Van de Ven1995, Section 1]. For the case see 6-manifolds:2-connected.
- The sets and are actually the same (as Wall points out in [Wall1966]): by Whitehead triangulation theorem we have the canonical forgetting map ; by smoothing theory and the fact that is -connected, this is a bijection.
- The forgetting map is injective: this follows from classification results below. Thus can be viewed as a subset of (determined by the equation , where is the Kirby-Siebenmann triangulation class). In what follows, we abbreviate to just .
The standard projections are denoted by , the standard injections by (here may equal , in which case is the reduction modulo , while is the multiplication by ). By we denote the (non-stable) cohomology operation ``Pontryagin square´´
It is known that (with the same ) the following equalities hold: and . There exists also the ``Pontryagin cube´´
and generally the ``Pontryagin -th power´´ for every prime [Thomas1956] (here we only need and ).
2.2 Classical invariants
Let be a closed oriented simply connected 6-manifold ( for short). Our first two invariants determine the (additive) homology structure of :
- - the 3-dimensional Betty number,
- - the 2-dimensional homology group.
Next, we consider the characteristic classes:
- - the second Stiefel-Whitney class;
- - the first Pontryagin class (in view of Poincaré duality, we will freely use such identifications as etc.);
- -the Kirby-Siebenmann class (obstruction to smoothing the -manifold ).
Note that all the other Stiefel-Whitney classes are uniquely determined as , by the well-known Wu formulas and by trivial reasons. Also, the class may need some comment. In general, Pontryagin classes for manifolds (or microbundles) are defined as rational cohomology classes. The class is an exception, due to the equality , see [Jupp1973] or [Kirby&Siebenmann1977]. We denote by the canonical mapping (or rather homotopy class of mappings) , inducing the identity . Now, the last invariant
2.3 Relations for classical invariants
There are two evident restrictions for invariants and :
- is a finitely generated abelian group
(where the first one follows from the existence of non-singular skew-symmetric form on ). There are two more restrictions (Wu relations):
- () for all ,
- () = for all .
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 for -coefficients, and its certain analogue (due to Wu as well) for . Note that () could be also written as , having in mind multiplication ``on ´´.
2.4 Further notation
2.5 Special invariants
There is a detailed treatment in [Zhubr2000]. Here we only give a formal description. For any , and each , there are functions
satisfying the following set of identities (which are considered to be a part of definition, whereas relations that go further on define the range):
- () (first coefficient formula),
- () (second coefficient formula)
for , and
- () (first difference formula),
- () (second difference formula)
for . In what follows, the values of at may be also written as for convenience.
- In view of these identities, one easily sees that the functions and are completely determined by their values at some fixed . Thus, if we could make a canonical choice, then our couple of invariants would trivialize to just . Evidently, such canonical choice is impossible in general, however in the spin case one can take with (see below).
- From () it easily follows that is in fact determined by , so our list of invariants could be reduced by 1, at the cost of reduced convenience.
2.6 Relations for special invariants
- () for ,
- () for ,
- () for .
2.7 The splitting theorem
Wall in [Wall1966] proves the following
Theorem 2.2. Let be a closed, smooth, 1-connected 6-manifold. Then we can write as a connected sum , where is finite and is a connected sum of copies of .
This theorem allows to restrict the classification problem to the case where . The proof is rather easy and basically reduces to realizing the standard ``symplectic´´ basis of with embedded 3-spheres (and applying ``Whitney trick´´ where necessary). As is pointed out in [Jupp1973], the same argument works for category. Wall does not state the uniqueness of 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 and ) from the results of [Jupp1973], and in full generality from the general classification theorem of [Zhubr2000] (see below). Note that the invariants (except of course) are ``insensitive´´ to connected summing with (this is evident for classical invariants, while for 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 , , , , , , (with left out). We divide these into two subsets: and . We say that the set is admissible for if , etc. satisfy all the identities and relations given above (these invariants are now regarded in ``abstract´´ way, irrelative to any manifold). Let denote the collection of all admissible sets of invariants for . Consider now the category of finitely generated abelian groups, and the category , whose objects are homomorphisms with , and whose morphisms are commutative diagrams of the form
For each morphism , we can define the induced map in a natural way: if , then we set with , , and so on (the rest is quite evident). One easily verifies that the new invariant set is admissible again. Hence we have a functor .
2.9 Classification theorem (the general case)
We use the notation for the subset of , defined by the equation . For any and , let be the set . 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 and , where . An isomorphism is induced by orientation-preserving homeomorphism if and only if (completeness of the set of invariants). (2) For each there exists with and (completeness of the set of relations). (3) If manifolds and (statement (1)) are given smooth structures, then homeomorphism can be chosen smooth also.
- The clause ``only if´´ of statement (1) is tautological (it just says that our invariants are invariants indeed).
- For any let denote the set of homeomorphism classes of pairs , where and is an isomorphism with . One can say that is the set of (homeomorphism classes of) manifolds with prescribed homology and second Stiefel-Whitney class. We can write (taking some liberty in notations): Now we have the natural maps , and from the above theorem it follows that all these maps are bijections.
- 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 we have the canonical choice , as was noted above. Applying relations -- to and , we obtain the equalities , and , respectively. Thus we can ``cross out´´ the invariants and , which leaves us with (we remind that one can suppose by Theorem 2.2).
It should be noted that one cannot simply drop the relations -- after this ``crossing out´´: to preserve all information the relations may contain, we still have to apply them to entire families and . Any can be written in the form for ; applying the identities --, we may write and . Applying this to -- again, one can see by straightforward checking that and are tautologically true, while turns into
- () for any ,
using as abbreviation for .
Now the relation immediately follows from , while can be combined with in the form
- () for any .
Hence, for the spin case we have the complete set of invariants with the only relation , 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 by its cohomology group , and interpret our previous group as . Likewise, the elements of can be considered as symmetric trilinear forms (or cubic forms) on . We thus have the following set of invariants:
(for any with ). We rewrite in the form:
- for any ,
- for any .
Next consider the relations and . We can now ``solve´´ them for and :
This shows that and are expressible in terms of ``classical´´ invariants and should be dropped.
It only remains to consider the last relation . We denote by some (arbitrary) element of with . In view of the above equalities, relation can now be written in the form
Now, quite similar to the spin case above, it is an easy exercise to see that follows from , while and together are equivalent to
- Relations and can be written, for a manifold , in the form and , respectively. The fact that 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 dual to the cohomology class ; any such is spin, and its first Pontryagin class (~times its signature) is equal to .
- The fact that relation is excessive in the spin case went unnoticed in both [Wall1966] and [Zhubr1975].
- Relation in the case of Wall, i.e. for and even, simplifies to
- The proof of given by Wall relies on construction of 6-manifolds of the type considered by surgery on 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. and ). On the other hand, the proof of the more complicated relation in [Jupp1973] is based on integrality theorem for -genus. In fact, for smooth case this follows immediately; regretfully, the argument Jupp gives to extend this to category is incorrect, as it uses the erroneous homotopy classification theorem of [Wall1966] (see [Zhubr2000, Subsection 5.14]).
3 Examples and constructions
- The -fold connected sum gives the only element of .
- The -fold connected sum gives the ``primary´´ element of --- a manifold with .
- In the same way, the non-trivial -bundle is the ``primary´´ element of .
- The two examples above can be easily generalized: let be arbitrary homomorphism. Let be the connected sum , where each is either or , depending on the value takes at the -th basis vector of . Then we have and .
- Surgery lets us extend the above construction to arbitrary . Take any epimorphism and build a manifold as above. Now represent a free basis of by embedded spheres and do surgery on along these spheres. As may be checked, the result is a manifold with (therefore, uniquely defined).
- The -bundles , with , give us manifolds in with ; in particular, .
- Complex algebraic geometry makes (potentially) a very powerful source of examples. In particular, any regular complete intersection of complex dimension 3 represents an element of , where and can be directly calculated from the multidegree (see ``Complete intersections´´). In the easiest case --- when is a non-singular hypersurface of degree --- we have , , and (a polynome in of degree 4).
- [Jupp1973] P. E. Jupp, Classification of certain -manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
- [Okonek&Van de Ven1995] C. Okonek and A. Van de Ven, Cubic forms and complex -folds, Enseign. Math. (2) 41 (1995), no.3-4, 297–333. MR1365849 (97b:32035) Zbl 0869.14018
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [Thomas1956] E. Thomas, A generalization of the Pontrjagin square cohomology operation, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 266–269. MR0079254 (18,57b) Zbl 0071.16302
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain -manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zhubr1973] A. V. Zhubr, A decomposition theorem for simply connected 6-manifolds, LOMI seminar notes 36 (1973), 40–49. (Russian)
- [Zhubr1975] A. V. Zhubr, Classification of simply connected six-dimensional spinor manifolds, (English) Math. USSR, Izv. 9 (1975), (1976), 793–812 . Zbl 0337.57004
- [Zhubr2000] A. V. Zhubr, Closed simply connected six-dimensional manifolds: proofs of classification theorems, Algebra i Analiz 12 (2000), no.4, 126–230. MR1793619 (2001j:57041)