6-manifolds: 1-connected
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* Surgery lets us to 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). | * Surgery lets us to 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 $M_\alpha=S^4\tilde\times_\alpha S^2$, with $\alpha\in\pi_3(SO(3))=\mathbb{Z}$, give us manifolds in $\mathcal{M}^0_6(\mathbb{Z},0)$ with $I(M_\alpha)=(4\alpha,\alpha)$; in particular, $M_1=\mathbb{C} P^3$. | * The $SO(3)$-bundles $M_\alpha=S^4\tilde\times_\alpha S^2$, with $\alpha\in\pi_3(SO(3))=\mathbb{Z}$, give us manifolds in $\mathcal{M}^0_6(\mathbb{Z},0)$ with $I(M_\alpha)=(4\alpha,\alpha)$; 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 $p,\mu$ can be directly calculated from the multidegree $d=(d_1,\ldots,d_m)$ (see [[Complete intersections|complete intersections]). In the easiest case --- when $M_d\subset\mathbb{C} P^4$ is a non-singular hypersurface of degree~$d$ --- we have $M_d\in\mathcal{M}^r_6(\mathbb{Z},\rho_2(d-1))$, $I(M_d)=(d(5-d),d)$, and $r=d^4+\ldots$ (a polynome in $d$ of degree~4). | + | * 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 $p,\mu$ can be directly calculated from the multidegree $d=(d_1,\ldots,d_m)$ (see [[Complete intersections|complete intersections]]). In the easiest case --- when $M_d\subset\mathbb{C} P^4$ is a non-singular hypersurface of degree~$d$ --- we have $M_d\in\mathcal{M}^r_6(\mathbb{Z},\rho_2(d-1))$, $I(M_d)=(d(5-d),d)$, and $r=d^4+\ldots$ (a polynome in $d$ of degree~4). |
** in particular, every [[Complete intersections|complete intersection]] of complex dimension 3 is a 1-connected 6-manifold. | ** in particular, every [[Complete intersections|complete intersection]] of complex dimension 3 is a 1-connected 6-manifold. | ||
* Let $\phi \co \sqcup_{i=1}^r (D^3 \times S^3) \to S^6$ be an $r$-component framed link and let denote by $M^6_\phi$ the outcome of surgery on $\phi$. Then $M_\phi$ is a simply connected spinable 6-manifold with $H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$ and $H_3(M_\phi) = 0$. | * Let $\phi \co \sqcup_{i=1}^r (D^3 \times S^3) \to S^6$ be an $r$-component framed link and let denote by $M^6_\phi$ the outcome of surgery on $\phi$. Then $M_\phi$ is a simply connected spinable 6-manifold with $H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$ and $H_3(M_\phi) = 0$. |
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1 Introduction
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.
Remark 1.1.
- 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 .
2 Classification
2.1 Notation
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 ``Pontrjagin square´´
It is known that (with the same ) the following equalities hold: and . There exists also the ``Pontrjagin cube´´
and generally the ``Pontrjagin -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 Pontrjagin 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, Pontrjagin 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
Tex syntax errorand :
- ;
- 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 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 each there are functions
- ,
- .
In what follows, the values of at will be written either or , depending on convenience. These functions satisfy the following set of identities (which are considered to be part of the definition, whereas the relations define the range):
- [] (first coefficient formula),
- [] (second coefficient formula)
for , and
- [] (first difference formula),
- [] (second difference formula)
for .
Remark 2.1.
- 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 ).
- 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 .
Tex syntax errorof 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
Tex syntax errorleft 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.
Remark 2.4.
- The clause ``only if´´ of the 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
...
2.11 The torsion-free case
...
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 to 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).
- in particular, every complete intersection of complex dimension 3 is a 1-connected 6-manifold.
- Let be an -component framed link and let denote by the outcome of surgery on . Then is a simply connected spinable 6-manifold with and .
4 References
- [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)
- 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 $m=\infty$).
- 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.
- The clause ``only if´´ of the statement (1) is tautological (it just says that our invariants are invariants indeed).
- 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): $$\mathcal{M}_6^r=\bigcup_{(G,w)} \mathcal{M}_6^r(G,w).$$ 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.
- From the statement (3) it evidently follows that a closed simply connected 6-manifold has at most one (up to homeomorphism) smooth structure (Hauptvermutung).
Remark 1.1.
- 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 .
2 Classification
2.1 Notation
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 ``Pontrjagin square´´
It is known that (with the same ) the following equalities hold: and . There exists also the ``Pontrjagin cube´´
and generally the ``Pontrjagin -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 Pontrjagin 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, Pontrjagin 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
Tex syntax errorand :
- ;
- 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 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 each there are functions
- ,
- .
In what follows, the values of at will be written either or , depending on convenience. These functions satisfy the following set of identities (which are considered to be part of the definition, whereas the relations define the range):
- [] (first coefficient formula),
- [] (second coefficient formula)
for , and
- [] (first difference formula),
- [] (second difference formula)
for .
Remark 2.1.
- 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 ).
- 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 .
Tex syntax errorof 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
Tex syntax errorleft 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.
Remark 2.4.
- The clause ``only if´´ of the 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
...
2.11 The torsion-free case
...
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 to 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).
- in particular, every complete intersection of complex dimension 3 is a 1-connected 6-manifold.
- Let be an -component framed link and let denote by the outcome of surgery on . Then is a simply connected spinable 6-manifold with and .
4 References
- [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)