Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification
aa r X i v : . [ m a t h . G T ] D ec Embeddings of non-simply-connected 4-manifolds in 7-spaceII. On the smooth classification
D. Crowley and A. Skopenkov ∗ Abstract
We work in the smooth category. Let N be a closed connected orientable 4-manifold withtorsion free H , where H q := H q ( N ; Z ). Our main result is a readily calculable classificationof embeddings N → R up to isotopy , with an indeterminancy. Such a classification was onlyknown before for H = 0 by our earlier work from 2008. Our classification is complete when H = 0 or when the signature of N is divisible neither by 64 nor by 9.The group of knots S → R acts on the set of embeddings N → R up to isotopy byembedded connected sum. In Part I we classified the quotient of this action. The mainnovelty of this paper is the description of this action for H = 0, with an indeterminancy.Besides the invariants of Part I, the classification involves a refinement of the Kreckinvariant from our work of 2008 which detects the action of knots.For N = S × S we give a geometrically defined 1–1 correspondence between the set ofisotopy classes of embeddings and a quotient of the set Z ⊕ Z ⊕ Z . Motivation and background for this paper may be found in Part I [CSI, § § smooth manifolds, embeddings andisotopies and recall that: • N is a closed connected orientable 4-manifold; • E m ( N ) denotes the set of isotopy classes of embeddings f : N → S m .In this paper we classify E ( N ) when H ( N ; Z ) it torsion free (up to an indeterminancy incertain cases). See Theorems 1.1, 1.4 and Corollaries 1.3, 1.6 below. Our classification is completewhen H = 0 (see Theorem 1.4 and Corollary 1.6.b) or when the signature of N is divisible neitherby 64 nor by 9 (see Theorem 1.4 and Corollary 1.3). The classification requires finding a completeset of invariants and constructing embeddings realizing particular values of these invariants. Theinvariants we use are described in [CSI, Lemma 1.3, § § § § E ( S ) ∼ = Z acts on E ( N ) by embedded connected sum. This action was investi-gated in [Sk10] and determined when H ( N ; Z ) = 0 in [CS11], which also classified E ( N ) in this ∗ Supported in part by the Russian Foundation for Basic Research Grant No. 15-01-06302, by Simons-IUMFellowship and by the D. Zimin Dynasty Foundation. E ( N ) := E ( N ) /E ( S ) when H ( N ; Z ) = 0. Thus themain novelty of this paper is the description of this action for H ( N ; Z ) = 0. Cf. Remark 1.9.Denote by q : E ( N ) → E ( N ) the quotient map.In order to state our main result for N = S × S consider the following diagram (where theleft triangle is not commutative): Z × Z / / × τ (cid:15) (cid:15) Z τ := q τ (cid:15) (cid:15) τ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ E ( S × S ) q / / E ( S × S ) . The maps τ, q are defined in [CSI, § × τ by ( × τ )( a, l, b ) := a τ ( l, b ) and τ := q τ. Theorem 1.1.
The map × τ : Z × Z → E ( S × S ) is a surjection such that(a) for different l, b the sets P l,b := ( × τ )( Z × ( l, b )) either are disjoint or coincide; ( b ) P l,b = P l ′ ,b ′ ⇔ ( l = l ′ and b ≡ b ′ mod 2 l );( c ) | P l,b | = ( l = 02 gcd( b, l = 0 . In Theorem 1.1 the surjectivity of τ and (a) and (b) follow from [CSI, Theorem 1.1]. The newpart of Theorem 1.1 is (c); this part follows from Corollary 1.6.b below (because for l = 0 thegroup coker l is finite, so div b = 0). Cf. Addendum 2.8. Example 1.2.
There is an embedding f : S × S → S with f ( N ) ⊂ S and a pair of non-isotopicembeddings g , g : S → S such that f g and f g are isotopic.This example follows because there is a representative of τ (0 ,
1) whose image is in S ⊂ S [CSI, Lemma 2.21] and | P , | = 2 by Theorem 1.1.Example 1.2 shows necessity of the assumption of simple-connectivity the following result(which is [Sk10, The Effectiveness Theorem 1.2]): If f : N → S is an embedding of a spin simply-connected closed 4-manifold N , f ( N ) ⊂ S and embeddings g , g : S → S are not isotopic, then f g and f g are not isotopic. Before stating our main result in the general case we state the following corollary of it.
Corollary 1.3 (of Theorem 1.4.c) . (a) If κ ( f ) is neither divisible by nor by , then for eachembedding g : S → S the embeddings f g and f are isotopic.(b) If κ ( f ) is divisible by 4 but neither by 8 nor by 3, then there is a non-trivial embedding g : S → S such that for every embedding g : S → S the embedding f g is isotopic to either f or f g . Corollary 1.3 follows from Corollary 1.6.bc or from Theorem 1.4.c (because 4 Z gcd( κ ( f ) , = 0under the assumptions of Corollary 1.3). The assumption of Corollary 1.3.a is automaticallysatisfied when the signature of N is divisible neither by 16 nor by 9. Cf. Remark 1.8.a. It follows that the Effectiveness Theorem 1.2 of the earlier versions of [Sk10] was false. § b n := gcd( n, . If H = 0 , then the map κ (which is the Bo´echat-Haefliger invariant defined in [CSI, § § κ = κ q is surjective: E ( N ) q / / E ( N ) κ / / H DIF F := { u ∈ H | ρ u = w ∗ ( N ) , u ∩ N u = σ ( N ) } ⊂ H . For each u ∈ H DIF F we have | κ − ( u ) | = b u/ gcd( u, . Our second main result is a generalization of the above statement to non-simply-connected4-manifolds. The maps κ , λ , β u,l , and η u,l,b , θ u,l,b of Theorem 1.4 below are defined in [CSI, § § § Definition of ∩ d . For a symmetric pair ( u, l ) and d := div u ∈ Z a bilinear map ∩ d : coker(2 ρ d l ) × ker(2 ρ d l ) → Z d is well-defined by [ c ] ∩ d y := c ∩ N y. Theorem 1.4.
Let N be a closed connected orientable 4-manifold with torsion free H .(a) The product κ × λ : E ( N ) → H DIF F × B ( H ) has non-empty image consisting of symmetric pairs.(b) For each ( u, l ) ∈ im( κ × λ ) denote d := div u . Each map β u,l : ( κ × λ ) − ( u, l ) → coker(2 ρ d l ) is surjective (see the remark immediately below the Theorem).(c) For each b ∈ coker(2 ρ d l ) the map θ u,l,b : ker(2 ρ d l ) → Z b d is a homomorphism and each map η u,l,b : β − u,l ( b ) → Z b d im θ u,l,b is an injection whose image consists of all even elements (see the remark immediately below theTheorem). Moreover, θ u,l,b ( y ) − θ u,l,b ′ ( y ) = 4 ρ b d ( b − b ′ ) ∩ d y for each y ∈ ker(2 ρ d l ) ⊂ H . (d) | β − u,l ( b ) | = b u gcd( u, · | im θ u,l,b | . This is proven in [Sk10, CS11] building on [BH70]. Indeed, for each x ∈ H and y ∈ ker(2 ρ d l ) we have 2 lx ∩ N y = 2 l ( x, y ) ≡ d l ( y, x ) = 2 ly ∩ N x ≡ d
0. Henceim(2 ρ d l ) ∩ N ker(2 ρ d l ) = { } ⊂ Z d . emark on relative invariants. We call geometrically defined maps invariants (this isinformal, formally an invariant is the same as a map). The maps λ and κ are invariants. Themaps β u,l and η u,l,b are relative invariants . For η u,l,b this means that for [ f ] , [ f ] ∈ β − u,l ( b ) there isan invariant ([ f ] , [ f ]) η ( f , f ) (defined in § η u,l,b ( f ) := η ( f, f ′ ) for a fixed choiceof [ f ′ ] ∈ β − u,l ( b ). We suppress the choice of [ f ′ ] from the notation. For β u,l the situation is similar.Parts (a) and (b) of Theorem 1.4 follow from [CSI, Theorem 1.2]. The new part of Theorem1.4 is (c), which is proven in in § η u,l,b = 2 Z b d / im θ u,l,b .We remark that Theorem 1.1 is not an immediate corollary of Theorem 1.4, cf. [CSI, Remarks1.4.a and 2.16].Identify E ( S ) and Z by any isomorphism. Addendum 1.5.
In the notation of Theorem 1.4, for each a ∈ Z and f ∈ β − u,l ( b ) η u,l,b ( f a ) = η u,l,b ( f ) + [2 a ] ∈ Z b d im θ u,l,b . This follows from the definition of η u,l,b ( § Corollary 1.6.
For each ( u, l ) ∈ im( κ × λ ) let d := div u . There is f u,l ∈ ( κ × λ ) − ( u, l ) suchthat for each f ∈ ( κ × λ ) − ( u, l ) and a, a ′ ∈ Z , denoting b := β ( f, f u,l ) ∈ coker(2 ρ d l ) we have(a) f a = f a ′ ⇔ a = a ′ , provided • u = 0 and div b is divisible by 6, or • u = 0 , ρ d l = 0 and u is divisible by
24 ord(4 b ) ;(b) f a = f a ′ ⇔ a ≡ a ′ mod 2 gcd(div b, , provided u = 0 ;(c) f a = f a ′ ⇔ a ≡ a ′ mod \ u ord(4 b ) gcd( u, , provided u = 0 and ρ d l = 0 . The class u is divisible by d and hence by the order ord(4 b ) of d in the d -group coker(2 ρ d l ).Part (a) follows from Parts (b,c). Parts (b,c) are proven in § Corollary 1.7.
Theorem 1.4 has the following restatement analogous to Theorem 1.1 and to [CSI,Corollary 2.14.b]. There is a surjection τ : Z × H × H DIF F × B ( H ) → E ( N ) such that τ ( a, b, u, l ) = τ ( a ′ , b ′ , u ′ , l ′ ) ⇔ u = u ′ , l = l ′ , b − b ′ ∈ ker(2 ρ div u l u ) and a − a ′ ∈ im η u,l u ,b , where l u := l + λτ (0 , , u, . Remark 1.8 (The action of knots in Theorem 1.4) . (a) Take any [ f ] ∈ E ( N ). Let O ( f ) = O ([ f ]) := { [ f g ] : [ g ] ∈ E ( S ) } be the orbit of [ f ] under the action of E ( S ). We have O ( f ) = β − u,l ( b ) when [ f ] ∈ β − u,l ( b ) by [CSI,Theorem 1.2] and the additivity of κ , λ and β [CSI, Lemmas 2.3 and 2.9].4efine the inertia group of f , I ( f ) ⊂ E ( S ) = Z , to be the subgroup of isotopy classes in E ( S ) which do not change [ f ] after embedded connected sum: I ( f ) = I ([ f ]) := { [ g ] ∈ E ( S ) : [ f g ] = [ f ] } For some cases this orbit and group are found in terms of u, l, b in Corollaries 1.3 and 1.6.(b) Problem: characterize those f for which | O ( f ) | = 12 (i.e. | I ( f ) | = 1), and those f forwhich | O ( f ) | = 1 (i.e. | I ( f ) | = 12).(c) The indeterminancy in the classification of Theorem 1.4.c corresponds to the fact that wedo not always know im θ u,l,b . Thus determining im θ u,l,b becomes a key problem. This image isfound in this paper when either u = 0 or 2 ρ d l = 0 (Corollary 1.6) or in the cases (1,2,3) of Remark2.6 below. For general u, l (and even simple enough N ) there are some b for which the methodsof this paper do not completely determine im θ u,l,b . Remark 1.9 (The action of knots in general) . If the quotient E m ( P ) is known for a closed n -manifold P , the description of E m ( P ) is reduced to the determination of the orbits of the embeddedconnected sum action of E m ( S n ) on E m ( P ). For a general closed n -manifold P describing theaction by a non-zero group of knots E m ( S n ) on E m ( P ) is a non-trivial task. For the cases whenthe quotient E ( N ) coincides with the set of PL embeddings up to PL isotopy, the quotient hasbeen known since 1960s [MAE, MAF, MAT]. However, until recently no description of the action(or, equivalently, no classification of E m ( P )) was known for E m ( S n ) = 0 and P not a disjointunion of homology spheres. For recent results see [Sk08’] and [Sk10, CS11] mentioned above. Onthe other hand, the description of the action in [CRS07, Sk11, CRS12, Sk15] is not hard, the hardpart of the cases considered there is rather the description of the quotient E ( N ).There are non-isotopic embeddings g , g : S → S and an embedding f : R P → S such that[ f g ] = [ f g ] [Vi73]. I.e. the action of the monoid E ( S ) on E ( R P ) is not free.Various authors have studied the analogous connected sum action of the group of homotopy n -spheres on the set of smooth n -manifolds homeomorphic to given manifold; see for example[Sc73, Wi74] and references there. Remark 1.10 (An approach to the action of knots) . Let us explain the ideas required to movefrom the classification modulo knots in [CSI] to the main results of this paper. We briefly recalland continue the discussion in [CSI, 1.4].Suppose that f , f : N → S are embeddings. Assume that f is isotopic to f g for someembedding g : S → S . By [CSI, Isotopy Classification Modulo Knots Theorem 2.8] this isequivalent to λ ( f ) = λ ( f ), κ ( f ) = κ ( f ) and β ( f , f ) = 0. The complements C and C may be glued together along a bundle isomorphism ϕ : ∂C → ∂C to form a parallelizable closed7-manifold M = C ∪ ϕ ( − C ). Recall that d := div κ ( f ) is the divisibility of κ ( f ) ∈ H . By theassumption on f , f there is a joint Seifert class Y ∈ H ( M ) such that ρ d Y = 0, i.e. a d -class [CSI, Lemma 4.1]. There is a spin null-bordism ( W, z ) of ( M ϕ , Y ), since Ω Spin ( C P ∞ ) = 0. Since ρ d Y = 0, the class ρ d z ∈ H ( W, ∂ ; Z d ) lifts to z ∈ H ( W ; Z d ). Recall that p ∗ W ∈ H ( W, ∂ ) is thePoincar´e dual of p W , the spin Pontrjagin class of W . We then verified that the Kreck invariant , η ( ϕ, Y ) := z ∩ W ρ b d ( z − p ∗ W ) ∈ Z b d , The inertia group of f is just the stabilizer of [ f ] under the action of E ( S ). We use the word ‘inertia’ followingits use for the action of the group homotopy spheres on the diffeomorphism classes of smooth manifolds: see thesecond paragraph of Remark 1.9. W to be spin diffeomorphic to the product C × I [CSI].We proved that η ( ϕ, Y ) is independent of the choices of W, z, z for a fixed bundle isomorphism ϕ and d -class Y [CSI, § η ( ϕ, Y ) is independent of the choice of ϕ : forthe precise statement, see [CSI, Lemma 4.3.c]. So we need to know the various values of η ( ϕ, Y )arising from the different possible choices of Y . These choices are described in [CSI, Descriptionof d -classes for M f Lemma 4.7]. The achievement of this paper is showing that the change of η ( ϕ, Y ) under a change of Y is precisely determined by θ u,l,b , and proving the properties of θ u,l,b (Lemma 2.1). Contents η - and θ -invariants and proof of Theorem 1.4.c . . . . . . . . . . 62.2 Proof of Corollary 1.6.bc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 θ -invariant 9 η - and θ -invariants and proof of Theorem 1.4.c In this paper we use the notation and definitions of [CSI, §§ H q := H q ( N ; Z ) are torsion free. Denote θ ( f, y ) := η (id ∂C f , Y f,y ) ∈ Z b d . Lemma 2.1 (proved in § § . (a) θ ( f, y ) is divisible by 4 for each f, y .(b) The map θ ( f, · ) : ker(2 ρ d λ ( f )) → Z b d is a homomorphism, where d := div κ ( f ) .(c) For each ( u, l ) ∈ im( κ × λ ) , d := div u , representatives f , f of two isotopy classes in ( κ × λ ) − ( u, l ) and y ∈ ker(2 ρ d l ) we have θ ( f , y ) − θ ( f , y ) = 4 ρ b d ( β ( f , f ) ∩ N y ) . Definition of θ u,l,b . Take any ( u, l ) ∈ im( κ × λ ) and b ∈ K u,l . Let d := div u . Define θ u,l,b : ker(2 ρ d l ) → Z b d by θ u,l,b ( y ) := θ ( f, y ) , where [ f ] ∈ β − u,l ( b ) . The map θ u,l,b is well-defined (i.e. is independent of the choice of f ) and is a homomorphism byLemma 2.1.ab and the transitivity of β [CSI, Lemma 2.10]. Definition of η ( f , f ) . Take representatives f , f of two isotopy classes in ( κ × λ ) − ( u, l )such that β ( f , f ) = 0. By [CSI, Lemma 2.5] there is a π -isomorphism ϕ : ∂C → ∂C . By [CSI,Lemma 4.1] there is a d -class Y ∈ H ( M ϕ ). Define η ( f , f ) := [ η ( ϕ, Y )] ∈ Z b d im θ u,l,b . Lemma 2.2.
Let f , f , f : N → S be embeddings, ϕ : ∂C → ∂C and ϕ : ∂C → ∂C π -isomorphisms, Y ∈ H ( M ϕ ) and Y ∈ H ( M ϕ ) d -classes. Then ϕ := ϕ ϕ is a π -isomorphism and there is a d -class Y ∈ H ( M ϕ ) such that η ( ϕ , Y ) = η ( ϕ , Y )+ η ( ϕ , Y ) . This is proved analogously to [CS11, Lemma 2.10], cf. [Sk08’, §
2, Additivity Lemma] (theproperty that Y is a d -class is achieved analogously to [CSI, proof of Lemma 4.6]). Lemma 2.3.
Take representatives f , f of two isotopy classes in ( κ × λ ) − ( u, l ) such that β ( f , f ) =0 . Denote d := div u .(a) For each π -isomorphism ϕ : ∂C → ∂C the residue η ( f , f ) is independent of the choiceof a d -class Y ∈ H ( M ϕ ) .(b) If η ( f , f ) = 0 , then for each π -isomorphism ϕ : ∂C → ∂C there is a d -class Y ∈ H ( M ϕ ) such that η ( ϕ, Y ) = 0 ∈ Z b d .Proof of (a). Take any d -classes Y ′ , Y ′′ ∈ H ( M ϕ ). Part (a) follows because η ( ϕ, Y ′ ) − η ( ϕ, Y ′′ ) (1) = η (id ∂C , Y ) (2) = θ ( f , y ) = θ u,l,β ( f ,f ′ ) ( y ) ∈ Z b d , where • equality (1) holds for some d -class Y ∈ H ( M f ) by Lemma 2.2; • equality (2) holds for some y ∈ ker(2 ρ d l ) by description of d -classes [CSI, Lemma 4.7]. Proof of (b).
Take any π -isomorphism ϕ : ∂C → ∂C . Part (b) follows because0 (1) = η ( ϕ, Y ′ ) − θ u,l,β u,l ( f ) ( y ) = η ( ϕ, Y ′ ) − θ ( f , y ) (3) = η ( ϕ, Y ) ∈ Z b d , where • equality (1) holds for some d -class Y ′ ∈ H ( M ϕ ) and y ∈ ker(2 ρ d l ) because η ( f , f ) = 0; • equality (3) holds for some d -class Y ∈ H ( M ϕ ) by Lemma 2.2. Lemma 2.4 (Transitivity of η ) . For each three embeddings f , f , f : N → S having the samevalues of κ - and λ -invariants and such that β ( f , f ) = β ( f , f ) = 0 we have η ( f , f ) = η ( f , f )+ η ( f , f ) . This follows by Lemma 2.2.
Theorem 2.5 (Isotopy classification) . If λ ( f ) = λ ( f ) , κ ( f ) = κ ( f ) , β ( f , f ) = 0 and η ( f , f ) = 0 , then f is isotopic to f .Proof. Analogously to the proof of [CSI, Isotopy Classification Modulo Knots Theorem 2.8]. Onlyreplace the second paragraph by ‘Since η ( f , f ) = 0, by Lemma 2.3.b we can change Y and assumeadditionally that η ( ϕ, Y ) = 0.’ Definition of η u,l,b . Take any [ f ] ∈ β − u,l ( b ). Define a map η u,l,b : β − u,l ( b ) → Z b d im θ u,l,b by η u,l,b [ f ] := η ( f, f ) . The map η u,l,b depends on f but we do not indicate this in the notation. Proof of Theorem 1.4.c.
The property on θ u,l,b − θ u,l,b ′ holds by Lemma 2.1.c. The map η u,l,b isinjective by the Isotopy Classification Theorem 2.5. The image of this map consists of all evenelements by [CSI, Lemma 4.3.a] and Addendum 1.5.7 .2 Proof of Corollary 1.6.bc Remark 2.6.
In Corollary 1.6.bc the assumption ‘ d = 0 or 2 ρ d l = 0’ can be replaced by each ofthe following successively weaker assumptions(1) ρ b d ker(2 ρ d l ) ⊂ ρ b d H is a direct summand, or(2) each homomorphism ρ b d ker(2 ρ d l ) → Z b d extends to ρ b d H , or(3) there is an element e b ∈ coker(2 ρ d l ) such that θ u,l, e b = 0.Clearly, ‘either d = 0 or 2 ρ d l = 0 ′ ⇒ (1) ⇒ (2) . Denote X u,l := ker(2 ρ div( u ) l ) ⊂ H . Proof of (2) ⇒ (3) . Take any b ′ ∈ coker(2 ρ d l ). We have θ u,l,b ′ = θ + u,l,b ′ ρ b d for some homomorphism θ + u,l,b ′ : ρ b d X u,l → Z b d . Extend θ + u,l,b ′ to a homomorphism ρ b d H → Z b d . Since H is free, ρ b d H isa free Z b d -module. Hence the latter homomorphism is divisible by 4. Then by Poincar´e dualitythere is a class x ∈ ρ b d H such that θ + u,l,b ′ ( z ) = 4 x ∩ N z for each z ∈ ρ b d X u,l . Let e b := b ′ + [ e x ], where e x ∈ ρ d H is a lifting of x . Then by Theorem 1.4 θ u,l, e b ( y ) = θ u,l,b ′ ( y ) − ρ b d ([ e x ] ∩ N y ) = θ + u,l,b ′ ( ρ b d y ) − x ∩ N ρ b d y = 0 for each y ∈ X u,l . Proof of the formula of Corollary 1.6.bc under the assumption (3) of Remark 2.6.
Define β ′ u,l ( f ) := e b − β u,l ( f ). Then θ ′ u,l,b = θ u,l, e b − b for each b ∈ coker(2 ρ d l ), hence θ ′ u,l, = 0. Therefore we may assumethat β u,l is chosen so that θ u,l, = 0.Take any b ∈ coker(2 ρ d l ) and denote X b,u,l := 4 b ∩ d X u,l . Sogcd( d, · | β − u,l ( b ) | (1) = b d | im θ u,l,b | (2) = b d | ρ b d X b,u,l | = [ Z b d : ρ b d X b,u,l ] (4) = gcd( b d, [ Z d : X b,u,l ]) = \ [ Z d : X b,u,l ] , where equalities (1) and (2) hold by Theorem 1.4. Now the formula of Corollary 1.6.bc is impliedby the following Lemma 2.7. Lemma 2.7.
Let V be a free Z -module, d an integer and m : V → V ∗ a homomorphism whosepolarization V × V → Z has a symmetric mod d reduction. Then for each c ∈ coker( ρ d m )[ Z d : c (ker( ρ d m ))] = div c d = 0 d ord c d = 0 ( c (ker( ρ d m )) ⊂ Z d is defined analogously to definition of ∩ d before Theorem 1.4).Proof for d = 0 . We need to prove the following.
Let V be a free Z -module and m : V → V ∗ a homomorphism whose polarization V × V → Z is symmetric. Then for each c ∈ V ∗ we have [ Z : c (ker m )] = div c . Let us prove the ‘ ⊂ ’ part. Assume that q is a divisor of c + Tors coker m . Then there exist s, l , . . . , l s ∈ Z and c , t , . . . , t s ∈ V ∗ such that l n t n ∈ im m for each n = 1 , . . . , s and c = qc + t + . . . + t s . Take any y ∈ ker m . Since the polarization of m is symmetric, we have l n t n ( y ) = 0 ∈ Z , thus t n ( y ) = 0. Hence c ( y ) = qc ( y ) + ( t + . . . + t s )( y ) = qc ( y ) is divisible by q .8et us prove the ‘ ⊃ ’ part. Assume that q is a divisor of c | ker m . The subgroup ker m ⊂ V is a direct summand. Take a decomposition V = ker m ⊕ V ′ . Since m | V ′ : V ′ → im m is anisomorphism, there is an element x ∈ V ′ such that c | V ′ = m ( x ) | V ′ . Since the polarization of m is symmetric, m ( x ) | ker m = 0. Then c − m ( x ) coincides with c on ker m and is zero on V ′ . So c − m ( x ) = qc for some c ∈ V ∗ . Hence d ( c + Tors coker m ) is divisible by q . Proof for d = 0 . We need to prove that | c (ker( ρ d m )) | = ord c for each c ∈ coker( ρ d m ) . (Weremark that this is obvious for ρ d m = 0, which case is sufficient for Corollary 1.6.c.)Denote K := ρ d ker( ρ d m ) ⊂ V /dV . Since the polarization V × V → Z of m has a symmetricmod d reduction, im( ρ d m ) ⊂ K ⊥ ⊂ V ∗ /dV ∗ . Since | im( ρ d m ) | = | V /dV || K | = | K ⊥ | , it follows thatim( ρ d m ) = K ⊥ . Now the required assertion follows because for each c ′ ∈ V ∗ /dV ∗ | c ′ ( K ) | = d div c ′ ( K ) = min { r | rc ′ ( K ) = 0 } = min { r | rc ′ ∈ K ⊥ } = ord ( V ∗ /dV ∗ ) /K ⊥ ( c ′ + K ⊥ ) . Addendum 2.8.
For each l ∈ Z − { } there is a map ψ l : Z × Z l → Z such that for each a, a ′ ∈ Z and l, b, l ′ , b ′ ∈ Z we have a τ ( l, b ) = a ′ τ ( l ′ , b ′ ) if and only if (cid:20) either l = l ′ = 0 , b = b ′ and a ≡ a ′ mod 2 gcd( b, l = l ′ = 0 , b ≡ b ′ mod 2 l and ρ ( a − a ′ ) = ψ l ([ b/ l ] , ρ l b ) − ψ l ([ b ′ / l ] , ρ l b ) Proof.
By Theorem 1.1.b if either l = l ′ or b b ′ mod 2 l , then the equivalence is clear becausenone of the two assertions holds.Assume that l = l ′ and b ≡ b ′ mod 2 l . Let τ := a τ ( l, b ) and τ ′ = a ′ τ ( l, b ′ ). By the IsotopyClassification Theorem 2.5 and Theorem 1.1.b τ = τ ′ ⇔ η ( τ, τ ′ ) = 0. Since div( κ ( τ )) = 0,we may use Corollary 1.6.b.If l = 0, then b = b ′ . By Theorem 1.1.b and Corollary 1.6.b im θ , ,b is formed by elementsof Z divisible by 4 gcd( b, η (Lemma 2.4) η ( τ, τ ′ ) = ρ b, (2 a − a ′ ) ∈ Z b, .If l = 0, then by Theorem 1.1.b and Corollary 1.6.b im θ ,l,b = 0 and η ( τ, τ ′ ) ∈ Z . For each x ∈ { , , . . . , l − } and k ∈ Z define ψ l ( k, ρ l x ) := η ( τ ( l, x ) , τ ( l, x + 2 kl )) ∈ Z . Then byAddendum 1.5 and the transitivity of η (Lemma 2.4) η ( τ, τ ′ ) = ρ (2 a − a ′ ) − η ( τ ( l, b ) , τ ( l, b ))+ η ( τ ( l, b ) , τ ( l, b ′ )) = ρ (2 a − a ′ ) − ψ l ([ b l ] , ρ l b )+ ψ l ([ b ′ l ] , ρ l b ) , where b ∈ { , , , . . . , l − } is uniquely defined by b ≡ b mod 2 l .These two formulas for η ( τ, τ ′ ) imply the equivalence describing preimages of τ . θ -invariant In this section we use homological Alexander duality and the restriction homomorphism definedin [CSI, § .1 Idea of proof of Lemma 2.1 We start with a lemma allowing to simplify the proof of the main result for N = S × S (Theorem1.1). The simplified proof is not presented, so the lemma is not used in the sequel. The standardembedding τ : S × S → S is defined in [CSI, § Lemma 3.1. η ( τ , y ) = 0 for each y ∈ H ( S × S ) .Proof. Define an extensioni : D × D → S of τ by i( x, y ) := ( y p − | x | , , , x ) / √ . Take an embedding v : S → S − i( S × D ) whose linking coefficient with i( S × D ) is y ∩ S × S [ S × ]. We omit subscript τ in this proof. Since lk( b Ay, τ ( S × )) = y ∩ S × S [ S × ],we have b Ay = [ v ( S )] ∈ H ( C ) ∼ = Z . We also have b Ay = i C ν ! y . Take a representative P of y anda chain V ∈ C ( C ) such that ∂V = ν − P − v ( S ) . Since C is parallelizable, v extends to an orientation-preserving embedding v : S × D → Int C =Int C × transversal to V and such that im v ∩ V = v ( S ). Extend v to an orientation-preservingembedding v : S × D → Int( C × I ). Let W − := C × I − Int im v and W := W − ∪ v | S × S D × S . Consider the cohomology exact sequence of pair (
W, W − ) in the following Poincar´e dual form(analogous to the sequence (*) in [CSI, Proof of Lemma 4.8]): H ( D × S ) / / H ( W, ∂ ) r W − / / H ( W − , ∂ ) / / H ( D × S ) H ( W, W − ) ∼ = P D ◦ ex O O H ( W, W − ) ∼ = P D ◦ ex O O . Since H ( D × S ) = 0, the map r W − is an epimorphism. Take any Z ∈ r − W − ( A [ N ] × I ∩ W − ) ⊂ H ( W, ∂ ) . Denote b V := V ∪ D × and z := Z + [ b V ] ∈ H ( W, ∂ ) . Since H ( D × S ) = 0, the spin structure on W − coming from S × I extends to W . Clearly, ∂W = spin ∂ ( C × I ) = spin M (for the ‘boundary’ spin structure on M coming from C × I ). Since ∂ W Z = ∂ C × I ( A [ N ] × I ) = Y and ∂ W [ b V ] = [ ν − P ×
12 ] = i M b Ay, we have ∂ W z = Y y . By [CSI, Lemma 4.7] ∂ W z = Y y = 0. So z ∈ im j W . Analogously to (*) we obtain an isomor-phism H ( C × I ) ∼ = H ( W ) commuting with i C × I : H ( M ) → H ( C × I ) and i W : H ( M ) → H ( W ). Since i C × I is onto, i W is onto. Hence j W = 0. Thus z = 0. So take z := 0 andobtain η ( τ , y ) = z ∩ W ( z − p ∗ W ) = 0. (We essentially proved that if b A f y is spherical, then η ( f, y ) = 0.) 10 .2 Proof of Lemma 2.1.b In this and the following subsection f, f , f : N → S are embeddings representing any elementsof ( κ × λ ) − ( u, l ); we denote d := div( u ). Definition of W ′ , W ′− and i ′ : W ′ → W . Let W ′− := C f − Int im v , W ′ := W ′− ∪ v | S × S × S S × D × S , (Manifold W ′ may be called the result of an S -parametric surgery along v .) Define an embedding W ′− → W − by x x × /
2. We assume that this embedding and the standard embedding S × D × S → S × D × S (that is the product of the identity and the equatorial inclusion S → S ) fit together to give an embedding i ′ : W ′ → W. Observe that ∆ , b V ⊂ W ′ . Lemma 3.2.
For each y ∈ H and W − , z, Z, V defined in [CSI, Proof of Lemma 4.8] we have z ∩ W W − ≡ d i V,W − ( Z ∩ V ) ∈ H ( W − , ∂ ) (since ∂V ⊂ ∂W − , the inclusion induces a map i V,W − : H ( V, ∂ ) → H ( W − , ∂ ) ).Proof. Since b V ⊂ W ′ , we have [ b V ] = 0 ∈ H ( W, ∂ ). Also Z ∩ W − = ( A f [ N ] × I ) ∩ W − = A f κ ( f ) × I ∩ W − ≡ d ∈ H ( W − , ∂ ) . Hence z ∩ W − = ( Z + [ b V ]) ∩ W − ≡ d Z ∩ W [ b V ]) ∩ W − = 2 i V,W − ( Z ∩ b V ∩ W − ) = 2 i V,W − ( Z ∩ V ) . Proof of Lemma 2.1.b.
In this proof a statement or a construction involving k holds or is madefor each k = 0 ,
1. Given y k ∈ ker(2 ρ d l ) construct manifold W k as W of [CSI, Proof of Lemma 4.8]by parametric surgery in C f × [ k − , k ]. We add subscript k to W − , W ′− , t, ∆ , Z, b V , z constructedin [CSI, Proof of Lemma 4.8]. (So unlike in other parts of this paper, subscript 0 of a manifolddoes not mean deletion of a codimension 0 ball from the manifold.) Define W := W ∪ C f × W and W − := C f × [ − , − Int im( v , ⊔ v , ) = W − ∪ C f × W − . This W should not be confused with what were previously denoted W but now is denoted W and W . Same remark should be done for W − and for Z, V, b V , z constructed below.The spin structure on W − coming from S × [ − ,
1] extends to W . Clearly, ∂W = spin = ∂ ( C f × [ − , ∼ = spin M f (for the ‘boundary’ spin structure on ∂ ( C f × [ − , M f ).Since H ( t k × ∆ k ) = 0, by the cohomological exact sequence of the pair ( W, W − ) (cf. diagram(*) in [CSI, Proof of Lemma 4.8]), r W − : H ( W, ∂ ) → H ( W − , ∂ ) is an epimorphism. Take any Z ∈ r − W − ( A f [ N ] × [ − , ∩ W − ) ⊂ H ( W, ∂ ) . V := V ⊔ V , b V := b V ⊔ b V and z := Z + [ b V ] ∈ H ( W, ∂ ) . Since ∂ W Z = Y f, and ∂ W [ b V k ] = i ∂W b A f y k , we have ∂z = Y f,y + y . Thus the pair ( W, z ) is a spinnull-bordism of ( M f , Y f,y + y ).Since y k ∈ ker(2 ρ d l ), we have ∂z k ≡ d
0. Take any z k ∈ j − W k ρ d z k . Let z := i W ,W z + i W ,W z . Then z ∩ W k = z k . Also j W z ∩ W − = X k =0 j W k z k ∩ W − = ρ d X k =0 z k ∩ W k − and X k =0 z k ∩ W k − (1) ≡ X k =0 i V k ,W k − ( Z k ∩ V k ) = 2 i V,W − ( Z ∩ V ) (3) ≡ z ∩ W − . Here congruences (1) and (3) modulo d hold by Lemma 3.2 and analogously to Lemma 3.2,respectively.Hence by the cohomological exact sequence of the pair ( W, W − ) with coefficients Z d (cf. diagram(*) in [CSI, Proof of Lemma 4.8]) j W z − ρ d z = n [ t ] + n [ t ] for some n , n ∈ Z d . We have n k [ t k ] = ( j W z − ρ d z ) ∩ W k = j W k z k − ρ d z k = 0 ∈ H ( W k , ∂ ; Z d ) . Therefore n = n = 0. So j W z = ρ d z .Since f W k := W k − C f × [0 , (2 k − /
3) is a deformation retract of W k , the inclusion f W k → W k induces an isomorphism on H . Clearly, z ∩ W k = z k , so z ∩ W k = z k . Hence z ∩ W ( z − p W ) = X k =0 ( z ∩ W k ) ∩ W k (( z − p W ) ∩ W k ) = X k =0 z k ∩ W k ( z k − p W k ) . So η ( f, · ) is a homomorphism. Lemma 3.3.
For each y ∈ H and W, Z, V, t, ∆ defined in [CSI, Proof of Lemma 4.8] we have(a) ∂ ( Z ∩ V ) = [ ∂ ∆] − i ∂C f ,∂V ξy ∈ H ( ∂V ) , where ξ : N → ∂C f is a weakly unlinked sectionfor f (see definition in [CSI, § p W = 2 m [ t ] ∈ H ( W, ∂ ) for some m ∈ Z . Lemma 3.3.b is essentially proved in the proof of [CSI, Lemma 4.8].
Proof of (a).
The equality follows because Z ∩ V = ( Z ∩ W − ) ∩ V = ( A f [ N ] × I ) ∩ V = A f [ N ] ∩ V ∈ H ( V, ∂ ) and ∂ ( A f [ N ] ∩ V ) = A f [ N ] ∩ ∂V = i ∂V ( A f [ N ] ∩ im v ) − i ∂V ( A f [ N ] ∩ ν − f P ) (3) = [ ∂ ∆] − [ ξP ] . P and v are defined in [CSI, Proof of Lemma 4.8]. Equality (3) follows because • A f [ N ] ∩ ν − f P = [ ξP ] by [CSI, Lemma 3.2.a]. • A f [ N ] ∩ im v = [ v (1 × S )] = [ ∂ ∆] since( A f [ N ] ∩ im v ) ∩ im v [ v ( S × )] = A f [ N ] ∩ C f v ( S × ) (2) = A f [ N ] ∩ C f S f = 1 . Here equality (2) holds because v ( S × ) is homologous to S f in C f . Lemma 3.4.
For each y ∈ ker(2 ρ d l ) and W, W − , z, t defined in [CSI, Proof of Lemma 4.8] thereis a class b z ∈ H ( W ; Z d ) such that(a) z := b z + n [ t ] ∈ j − W ρ d z ⊂ H ( W, ∂ ; Z d ) for some n ∈ Z d .(b) [ t ] = ( b z ) = 0 ∈ Z d and [ t ] ∩ W b z = 2 ∈ Z d . The proof is given later in this section. Proof of Lemma 2.1.a.
The Lemma follows by [CSI, Lemma 4.8] and Lemmas 3.3.b, 3.4. Indeed, z ∩ W ( z − p ∗ W ) = z ∩ W z − z ∩ W p ∗ W (2) = ( b z + n [ t ]) − ( b z + n [ t ]) ∩ W m [ t ] (3) = 4 n − m. Here • equality (2) holds by Lemma 3.3.b and property (a) of Lemma 3.4, • equality (3) holds by property (b) of Lemma 3.4.In the proof of Lemma 2.1.c we will use not only the statement of Lemma 3.4 but also thefollowing definition, which is also used in the proof of Lemma 3.4. Definition of a, s, b z for y ∈ ker(2 ρ d l ) . By Lemma 3.3.a there is a representative a ∈ C ( V ) of Z ∩ V ∈ H ( V, ∂ ) such that ∂a = ∂ ∆ − ξP. (Such a representative is obtained from a representative a ′ ∈ C ( V ) of Z ∩ V ∈ H ( V, ∂ ) such that ∂a ′ = ∂ ∆ − ξP + ∂a ′′ for some a ′′ ∈ C ( ∂V ) by the formula a := a ′ − a ′′ .)Since y ∈ ker(2 ρ d l ), by [CSI, Lemma 3.2. λ ] there is a chain s ∈ C ( C f × Z d ) such that ∂s = 2 ξP × . Define b z := [2 a − − ξP × [0 ,
12 ] + s ] ∈ H ( W ; Z d ) . Proof of Lemma 3.4.
We have ρ d z ∩ W − (1) = 2 ρ d i V,W − ( Z ∩ V ) = [2 a ] W − ,∂ = [2 a − ξP × [0 ,
12 ] + s ] W − ,∂ = b z ∩ W − = j W b z ∩ W − , where equality (1) follows by Lemma 3.2. Hence by the cohomology exact sequence of pair ( W, W − )(cf. diagram (*) in [CSI, Proof of Lemma 4.8]) ρ d z = j W ( b z + n [ t ]) for some n ∈ Z d . Thus property(a) holds. Equality (3) from [CSI, Proof of Lemma 4.3.a] also holds by [CSI, Proof of Lemma 4.7] and 3.4 because byLemma 3.4.a for d = 2 we have η ′ (id ∂C f , Y f ,y ) = z ∩ W z = z ∩ W z = 2[ t ] ∩ W b z = 0 ∈ Z . t ] = [ S × × S ] ∩ S × D × S [ S × × S ] = 0. Sincethe support of b z is in W ′ ∪ ∂C f × [0 , ] ∪ C f × W − W ′ , we have ( b z ) = 0. Also[ t ] ∩ W b z = [ t ] ∩ W − ( b z ∩ W − ) = [ t ] ∩ W − [2 a ] W − ,∂ = 2[ t ] ∩ ∂W − [ ∂a ] = 2[ t ] ∩ t × ∂ ∆ [ ∂ ∆] = 2 . Here the homology classes are taken in the space indicated under ‘ ∩ ’ (so [ t ] has different meaningin different parts of the formula), and b z ∩ W − = [2 a ] W − ,∂ is proved in the proof of (a). Proof of Lemma 2.1.c.
Take any bundle isomorphism ϕ : ∂C → ∂C given by [CSI, Lemma 2.5].Take a closed oriented 3-submanifold P ⊂ N realizing y ∈ H f = H f . For k = 0 , v jk , j = 0 , , ,
3, manifolds V k ⊂ C k , b V k , W ′ k and W k , chains a k , s k and classes Z k , z k , b z k asin [CSI, Proof of Lemma 4.8] and above. (So unlike in other parts of this paper, subscript 0 of amanifold does not mean deletion of a codimension 0 ball from the manifold.) Define W := W ∪ ϕ × id I : ∂C × I → ∂C × I W . This W should not be confused with what was previously denoted W but now is denoted W and W . Same remark should be done for z, Z, b V constructed below.Consider the following segment of the (‘cohomological’) Mayer-Vietoris sequence: H ( W, ∂ ) r W ⊕ r W → H ( W , ∂ ) ⊕ H ( W , ∂ ) r ⊕ ( − r ) → H ( ∂C ) . Here r k is the composition H ( W k , ∂ ) ∂ → H ( ∂W k ) r ∂C → H ( ∂C ). We have r k Z k = ( ∂Z k ) ∩ ∂C = Y f k ∩ ∂C = ∂ ( Y f k ∩ C k ) (4) = ∂A k [ N ] (5) = ∂A − k [ N ] (6) = r − k Z − k ∈ H ( ∂C ) . Here • equality (4) holds by descriptions of of joint Seifert classes [CSI, Lemma 3.13.a]. • equality (5) holds by agreement of Seifert classes [CSI, Lemma 3.5.a] • equality (6) holds analogously to the previous set of equalities.Hence there exists Z ∈ H ( W, ∂ ) such that Z ∩ W k = Z k . Denote b V := b V [ ϕ : ν − P → ν − P b V ⊂ W ′ and z := Z + [ b V ] ∈ H ( W, ∂ ) . Clearly, z ∩ W k = z k . Take b z k ∈ H ( W ; Z d ) given by Lemma 3.4. Then by Lemmas 3.3.b and 3.4 z k ∩ W p ∗ W = 4 m k = b z k ∩ W p ∗ W and z k ∩ W z k = z k ∩ W z k = 4 n k = 2 b z k ∩ W z k = 2 b z k ∩ W z k . Hence η ( f k , y ) = ρ b d ( b z k ∩ W k (2 z k − p ∗ W k )) = ρ b d ( b z k ∩ W (2 z − p ∗ W )) . Take a weakly unlinked section ξ : N → ∂C of f . By [CSI, Lemma 3.4] ξ := ϕξ is an unlinkedsection of f . Hence ∂a − ∂ ∆ = − ξ P = − ξ P = ∂a − ∂ ∆ and ∂s = 2 ξ P = 2 ξ P = ∂s . Note that (
W, z ) is not assumed to be spin bordism of anything and possibly ρ d ∂z = 0. M ϕ and its subsets with M ϕ × ⊂ ∂W and with corresponding subsets. Denote b a := [∆ − a + a − ∆ ] ∈ H ( b V ; Z d ) and s := [ s − s ] ∈ H ( M ϕ ; Z d ) . Then by definition of b z k b z − b z = i ϕ s − i b a, where i ϕ := i M ϕ ,W and i := i b V ,W . We have i ϕ s ∩ W p ∗ W = s ∩ M ϕ p ∗ M ϕ = 0.Since ( z ∩ M ϕ ) ∩ M ϕ S f = ( ∂z ∩ C ) ∩ C S f = Y f ∩ C S f = 1 ,z ∩ M ϕ is a joint Seifert class for ϕ . Then i ϕ s ∩ W z = ( s ∩ ∂C ) ∩ ∂C ( z ∩ ∂C ) (2) = 2 ξ y ∩ ∂C ν !0 β = 2 β ∩ N y, where • β ∈ H ( N ; Z d ) is a lifting of β ( f , f ); • equality (2) follows because s ∩ ∂C = 2[ ξ P ] = 2 ξ y and because z ∩ ∂C = ( z ∩ M ϕ ) ∩ ∂C = ν !0 β by definition of β ( f , f ).We have z ∩ W i b a (1) = ( Z + [ b V ]) ∩ W i b a (2) = Z ∩ W i b a + 2 Z ∩ W [ b V ] ∩ W i b a (3) == ( Z ∩ b V ) b V ∩ b V b a + 2 i ( Z ∩ b V ) ∩ W i b a (4) = ( b a ) b V + 2( i b a ) = ( b a ) b V , where • equality (1) follows by definition of z ; • equalities (2) and (5) follow because b V ⊂ W ′ , so [ b V ] = 0 and ( i b a ) = 0; • equality (3) is obvious; • equality (4) follows because Z ∩ b V = b a by definition of a , a , b a .Therefore i b a ∩ W (2 z − p ∗ W ) = 2( b a ) b V − b a ∩ b V p ∗ b V ≡ b z − b z ) ∩ W (2 z − p ∗ W ) = 2 i ϕ s ∩ W z − i ϕ s ∩ W p ∗ W − i b a ∩ W (2 z − p ∗ W ) ≡ β ∩ N y. References [BH70] J. Boechat and A. Haefliger,
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