The Arf-Kervaire invariant of framed manifolds as an obstruction to embeddability
aa r X i v : . [ m a t h . A T ] A ug The Arf–Kervaire invariant of framed manifoldsas obstruction to embeddability
Peter M. Akhmetiev, Matija Cencelj and Duˇsan RepovˇsNovember 13, 2018
Abstract
We prove that an arbitrary -connected (resp. -connected) stablyparallelizable manifold N (resp. N ) of dimension (resp. ) withthe Arf-Kervaire invariant 1 cannot be smoothly embedded into R (resp. R ). Let us consider a closed stably framed n –dimensional manifold. Such a manifoldis presented by the pair ( N n , Ξ) , where N n is a closed manifold of the dimen-sion dim( N ) = n , and Ξ is an isomorphism of bundles Ξ : ν ( N ) = R k × N n ,where ν ( N ) is the k –dimensional normal bundle of N n , k > n + 1 . A stablyparallelizable manifold is a stably framed manifold with the forgotten stableframing.Suppose that n = 2 ℓ − l + 2 and that N n is l –connected. Then N n is diffeomorphic to the connected sum of manifolds of the following three types(see [ K ] for the proof and further references):— closed manifold Σ n , homotopically equivalent to the standard n –dimensional sphere;— a product of two standard spheres S l +1 × S l +1 ;— a standard Arf-Kervaire manifold (constructed later).A connected sum of two third type manifolds is diffeomorphic to the stan-dard sum of some number of first and second type manifolds. The dimension dim( H l +1 ( N n ; Z / is always even and equals p , where p is the number ofsummands of second and third type. Following the theorem of Hill, Hopkins andRavenel, third type manifold can be constructed only for ℓ = 5 , and eventually (see [H-H-R]). Definition 1.1. A l –connected manifold N n , n = 4 l + 2 is said to have non-trivial Arf–Kervaire invariant if n = 2 ℓ − , ℓ = 5 , or , and N n is diffeomorphicto the connected sum of one third type manifold and of some number of firstand second type manifolds. 1 tandard Arf–Kervaire manifold The third type manifold is based on the manifold M l +20 , l + 2 = n . Thismanifold is constructed by "plumbing" in the book [Br1, Ch. V, TheoremV.2.11.]The manifold M l +20 is well-defined for all nonnegative l , but the condition ∂M l +20 = S l +1 is fulfilled only for l + 2 = 2 , , , , and eventually .For l + 2 = 2 , , , , and eventually = 126 , the boundary ∂M l +20 is P L –homeomorphic but not diffeomorphic to the standard (4 l + 1) –dimensionalsphere. In these exceptional dimensions, the manifold N l +2 is defined as M l +20 with the standard (4 l + 2) -dimensional disk glued along the boundary ∂M l +20 .In the case n = 2 , , a second type manifold is obtained. In the case n = 30 , and eventually , the obtained manifold is of third type.A simplified proof of existence of M l +20 for n = 62 appears in [L]. In [J-R]there is a remark, that the manifold N n of third type is P L –embeddable into R n +2 .The main result is formulated in the following theorem. Theorem 1.2. (1) Let N (resp. N ) be an arbitrary closed -connected(resp. -connected) stably parallelizable manifold with the nontrivial Arf–Kervaire invariant. Then the product N × I (resp. N × I ) with the in-terval I = [0 , is not smoothly embeddable into the Euclidean space R (resp. R ) provided that the corresponding embedding is equipped by a nondegeneratenormal field of –frames (resp. –frames) on the complement of the Cartesianproduct of the interval I and a point N × I \{ pt }× I (resp. on N × I \{ pt }× I ).(2) No stably parallelizable manifold N (resp. N ) is smoothly embeddableinto the Euclidean space R (resp. R ). Remark 1.3.
Obviously the assertion 2 of Theorem 1.2 follows from the as-sertion 1. Indeed, the composition N ⊂ R ⊂ R provides the embedding N × I × D ⊂ R . The restriction of this embedding to the submanifold N × I ⊂ N × I × D ensures the condition of stable parallelizability in asser-tion 2. Analogically, the composition N ⊂ R ⊂ R provides the embedding N × I × D ⊂ R , and the restriction of this embedding to the submanifold N × I ⊂ N × I × D ensures the condition of stable parallelizability inassertion 2. Nevertheless we give an independent proof of assertion 2 since thisproof is simpler than the proof of assertion 1. Remark 1.4.
Eccles constructed in [E2][E3] a stably parallelizable -dimensional (resp. -dimensional) manifold N (resp. N ) with Arf–Kervaireinvariant 1, which is embeddable in R (resp. R ). Remark 1.5.
Other geometric applications of Arf–Kervaire invariant, relatedto the problem of embeddability, can be found in [Ra].
Let us denote the cobordism group of immersions of oriented n -manifolds in thecodimension 1 by Imm fr ( n, . The class of the regular cobordism of immersion2 : N n R n +1 represents an element of this cobordism group. The set of theseelements is equipped with an equivalence relation of cobordance.It follows by the Pontryagin-Thom construction [P], in the form proposed in[W], that the group Imm fr ( n, is isomorphic to the stable n -homotopy groupof spheres.First, we describe the Pontryagin-Thom construction. A stably framed man-ifold is a pair ( N n , Ξ) where N n is a smooth manifold and Ξ is a trivializationof the normal bundle ν N . Namely, N n is diffeomorphic to a submanifold M in R n + k . Then the normal bundle ν N is isomorphic to the trivial normal bundle ν M and Ξ is the chosen trivialization. The word "stably" means that k >> n (infact k ≥ n + 2 suffices). It is convenient to introduce the direct limit k → + ∞ .The Pontryagin-Thom construction [P, Ch. 6], gives the map F : S n + k → S k as a composition of the standard projection S n + k → M E ( k )( N n ) and the stan-dard map M E ( k )( N n )) → S k . Here M E ( k )( N n ) (or M ( N n ) ) denotes theThom space of the trivial k –dimensional bundle. This space is homeomorphicto the k –fold suspension of N n + = N n ∪ { x } , where x is a point. The base point pt ∈ S k is a regular value of the map F and preimage of a small neighbourhoodof that point defined the framed manifold ( N n , Ξ) , corresponding to the sub-space of zero section of M ( N n ) . The homotopy class [ F ] ∈ Π n is well-defined.Moreover, if F ′ : S n + k → S n is another map, homotopical to F , and the basepoint pt is also a regular value of F ′ , then can be constructed a framed mani-fold ( N ′ , Ξ ′ ) analogously, with a framed cobordism ( W, Ξ W ) , connecting ( N, Ξ) and ( N ′ , Ξ ′ ) . Therefore the mapping [ F ] [( N n , Ξ)] defined the isomorfismbetween the stable homotopical group of spheres and the cobordism group ofstably framed manifolds.By the Smale-Hirsch theorem [Hi] a stably framed manifold ( N n , Ξ) definesan immersion f : N n R n . The immersion f is not defined uniquely; if f ′ isanother immersion, corresponding to ( N n , Ξ) , then f and f ′ are regularly cobor-dant. If ( N ′ n , Ξ ′ ) is cobordant to ( N n , Ξ) , then the corresponding immersion f ′ : N ′ n R n , is element of the same regular cobordism class as the immer-sion f . The mapping [( N n , Ξ)] [ f ] , constructed by Hirsch and the mapping [ F ] [( N n , Ξ)] , constructed by Pontryagin, define the isomorphism betweenthe cobordism group of immersions
Imm fr ( n, , and the stable homotopicalgroup of spheres Π n , constructed by Wells.Consider the case n = 4 l + 2 . Definition of Z / -quadratic form of an immersion and of its Arf-invariant. Let f : N l +2 R l +3 be the immersion, representing an elementin the group Imm fr (4 l + 2 , . The homology group H l +1 ( N k +2 ; Z / denoteshortly by H . By the Poincar e duality the bilinear nondegenerate form b : H × H → Z / is well-defined.Take the map S l +2+ k → M ( N l +2+ ) , defined by the Pontryagin–Thom con-struction. Obviously, the map fulfills the conditions of Theorem 1.4 [Br], hencea quadratic form q : H → Z / , associated with the form b , can be defined.Define the Arf -invariant
Arf ( H, q ) as the equivalence class of q in the Wittgroup of quadratic forms [Br1, Sect. 4]. It turns out that Arf ( H, q ) is an in-variant of the regular cobordism class of the immersion f . It is said to be theArf–Kervaire invariant of f . Hence, by the Wells theorem, that invariant definesa homomorfism of groups Θ :
Imm fr (4 l + 2 , −→ Z / , q can be constructed in a different way. Take a cycle x ∈ H . Bythe Thom theorem there exist a (possibly nonoriented) manifold X l +1 and amap i X : X l +1 → N l +2 such that i X, ∗ ([ X ]) = x where [ X ] is the fundamentalclass of manifold X . Because of general position it can be taken w.l.o.g. thatthe map f ◦ i X : X l +1 → N l +2 is an immersion with only finitely manytransversal self-intersections. Denote the self-intersection points of immersion i X by { x , . . . , x s } . For each point x i there exists a neighbourhood consistingof two l + 1 –disks intersecting in x i . Perform a surgery to obtain a manifold Y l +1 ⊂ N l +2 such that also i Y, ∗ ([ Y ]) = x , where i Y : Y l +1 ⊂ N l +2 isthe inclusion. To this aim remove both disks and glue their boundaries by a1-handle. The idea of such surgery in the case l = 0 is known from [P, Ch.15, Theorem 22]. Take the immersion f : N l +2 R n and consider the map j Y = f ◦ i Y : Y l +1 ⊂ R n . By the general position argument j Y is also anembedding and is equipped with a cross-section ξ Y of the normal bundle ν j Y .This cross-section is defined by the oriented normal of immersion f along thecycle.The linking number of the framed embedding ( i Y , ξ Y ) denote by lk ( i Y , ξ Y ) (it is defined as the linking number of i Y ( Y ) and of its copy along ξ Y ). Define q ( x ) = lk ( i Y , ξ Y )( mod . (1) Lemma 2.1.
The function q : H → Z / , given by q ( x ) = lk ( i Y , ξ Y )( mod , is well defined. It coincides with the Browderfunction as constructed in Lemma 1.2 [Br]. Corollary 2.2.
The function q is the quadratic form, associated to the bilinearform b . Proof of Corollary 2.2
The proof is in [Br, Theorem 1.4].
Proof of Lemma 2.1
Consider a stably framed cobordism ( W l +3 , Ξ W ) , connecting given pair ( N l +2 , Ξ) and ( N l +21 , Ξ ) of stably framed manifolds. First suppose that thefollowing conditions are true:1. the manifold N l +21 is l –connected;2. the cobordism W consists of i –handles, ≤ i ≤ l .The construction of cobordism W is based on spherical surgery as describedin the section 5 of [K-M]. In the section 1 of [N] the spherical surgery is developedfor a more general situation. It follows from the condition 2 that the mapping H l +1 ( N l +2 ; Z / → H l +1 ( W l +3 ; Z / , which is induced by the inclusion N l +2 ⊂ W l +3 is an isomorphism whence the mapping H l +1 ( N l +21 ; Z / → H l +1 ( W l +3 ; Z / , which is induced by the inclusion N l +21 ⊂ W l +3 , is anepimorphism.Let q ′ be the function, constructed by Browder. Under the condition 1 thefunction q ′ has the following geometric meaning (see [Br, the last paragraphof proof of Theorem 3.2 and the corresponding reference]). By the Hurewicz4heorem, an element x ∈ H l +1 ( N l +21 ; Z / can be realized as a map of spheres: ϕ : S l +1 → N l +21 . Furthermore, ϕ can be realized in its homotopy class byan embedding ϕ : S l +1 ⊂ N l +21 . Consider the embedding I N ◦ ϕ : S l +1 → N l +21 ⊂ R l +2+ k where I N is the inclusion which parametrizes the manifold N l +21 . The embedding I N ◦ ϕ is equipped by the normal vector field of k –frames. Then by the Hirsch theorem the immersion I N ◦ ϕ is regularlyhomotopic to the immersion into standard space R l +2 ⊂ R l +2+ k . Hence theframing vectors are parallel complements to the subspace of coordinate axes.This immersion is denoted by ¯ ϕ : S l +1 R l +2 . The stable Hopf invariantof immersion ¯ ϕ is defined as the number of transversal self-intersection points.This number denote by q ′ ( x ) . It depends neither on the choice of embedding ϕ in the homotopy class of ϕ nor on the choice of map ϕ realizing the homologyclass x .Apply the Hirsch theorem to the embedding i Y : S l +1 ⊂ R l +3 , equippedwith the cross-section ξ Y , to construct the immersion i ′ Y : S l +1 R l +2 . Thevalue of lk ( i Y , ξ Y ) in the right part of formula ( ) coincides with the parity ofnumber of self-intersection points of the immersion i ′ Y . This proves that undercondition 1 the function q , which is defined in 1, coincides with the function q ′ ,which was constructed by Browder.Now we prove Lemma 2.1 in the general case. Let us consider the cobordism W under the condition 2. Take an arbitrary element x ∈ H l +1 ( N l +2 ; Z / andsuch an element x ∈ H l +1 ( N l +21 ; Z / that the homological class x + x istrivial in H l +1 ( W l +3 ; Z / . It follows by [Br, Lemma 3.1.] that q ′ ( x ) = q ′ ( x ) .We have proved that q ′ ( x ) = q ( x ) . Let us prove: q ( x ) = q ( x ) . (2)Let the homology class x be equal to the image of the fundamental classby the embedding i Y : Y l +1 ⊂ N l +2 and let the homology class x be equalto the image of the fundamental class by the embedding i Y : Y l +11 ⊂ N l +21 .Suppose that the mapping of polyhedron i Z : Z l +2 → W , which realizes thesingular boundary of homology classes x , x , is represented by the submanifolds Y l +1 and Y l +11 . It is well known that the polyhedron W can be chosen to bea manifold in the complement of some subpolyhedron of codimension 2. Con-sider the singular points and the self-intersection curve of polyhedron i Z ( Z l +2 .The self-intersection curve of polyhedron i Z ( Z l +2 ) lies outside the consideredcodimension 2 subpolyhedron of the polyhedron Z l +2 . The boundary of self-intersection curve is the set of critical points of the map i Z , and the numberof these points is even. Modify the polyhedron Z l +2 on its regular part andmodify the map i Z by surgery in 1-handles in such a way that the map i Z hasno critical points.Consider the immersion f : N l +2 R l +3 × { } , the immersion f : N l +21 R l +3 × { } and the immersion F : W l +3 R l +3 × [0 , , suchthat its restriction on the upper and the lower components of boundary co-incides with the immersions f and f , respectively. Consider the pairs ofembeddings and corresponding normal sections ( i Y : Y ⊂ R l +3 × { } ; ξ Y ) , ( i Y : Y ⊂ R l +3 × { } ; ξ Y ) . Take the pair of embedding and the correspond-ing normal section ( i Z : Z l +2 R l +3 × [0 , , ξ Z ) , such that its restrictionto both components of boundary coincides with the pairs ( i Y , ξ Y ) , ( i Y ; ξ Y ) ,5espectively. Obviously, the self-linking numbers of boundary embeddings withgiven normal sections are equal modulo 2. Move i Y ( Y ) along ξ Y ; the obtainedmanifold is denoted by ( i Y ( Y )) ′ . Analogously, denote by ( i Y ( Y )) ′ the man-ifold obtained from i Y ( Y ) by sliding along ξ Y and by ( i Z ( Z )) ′ the manifoldobtained from i Z ( Z ) by sliding along ξ Z . The self-linking number of framed em-bedding ( i Y , ξ Y ) is defined as the parity of number of points of self-intersectionof the manifold ( i Y ( Y )) ′ with the manifold i Y ( Y ) by homotoping ( i Y ( Y )) ′ toinfinity. The self-linking number of ( i Y , ξ Y ) is defined analogically. Both self-linking numbers are congruent modulo 2 since ( i Z ( Z )) ′ intersects i Z ( Z ) in aneven number of points and when homotoping ( i Z ( Z )) ′ to infinity, the intersec-tion of ( i Z ( Z )) ′ ( t ) and ( i Z ( Z )) is a collection of curves lying completely in theregular part of polyhedra Z ′ and Z . Therefore, the boundary of this 1-manifoldconsists of an even number of points and these points are intersection points oftwo families of the boundary polyhedra. The formula ( ) and the Lemma 2.2are thus proved. Cobordism group of stably skew-framed immersions.
Let ( ϕ, Ψ L ) be a pair consisting of a l + 1 -dimensional closed manifold ϕ : L l +1 R l +2 and of a skew-framing Ψ L of the normal bundle ν ϕ , i.e. an isomorphism Ψ L : ν ϕ = (2 l + 1) κ , where κ is the orientation line bundle L l +1 . It means that w ( κ ) = w ( L l +1 ) . The cobordism relation of pairs is the standard one. Theset of all such pairs forms an abelian group Imm sf (2 l + 1 , l + 1) with respectto the operation of disjoint union. The Pontryagin-Thom construction in theform of Wells can be applied to this cobordism group. It induces the followingisomorphism Imm sf (2 l + 1 , l + 1) ≡ Π l +2 ( P l +1 ) , where P l +1 = R P ∞ / R P l is the skew projective space and Π l +2 ( P l +1 ) =lim t → + ∞ π l +2+ t (Σ t P l +1 ) is the stable homotopical group of the l + 2 -dimensional space P l +1 . [A-E2]. The connecting homomorphism δ . Define the homomorphism δ : Imm fr (4 l + 2) → Imm sf (2 l + 1) , which is called the connecting homomorphism. It is a modification of the tran-spher homomorphism of Kahn-Priddy.( [E1]).Let the immersion f : N l +2 R k +3 represents an element in thegroup Imm fr (4 l + 2 , . Construct a skew-framed immersion ( ϕ, Ψ L ) , where ϕ : L l +1 R l +2 . Consider the immersion I ◦ f , where I : R k +3 ⊂ R l +3 denotes the standard embedding. The immersion I ◦ f is equipped with thestandard framing. Let g : N l +2 R l +3 be an immersion, obtained from f by a small deformation which assures general position. Hence the immersion g self-intersects transversally. The double point manifold of immersion g is de-noted by by L l +1 . Let h : L l +1 R l +3 be the parametrizing immersion of L l +1 . The normal bundle ν h of immersion h is naturally isomorphic to thebundle lε ⊕ lκ , where κ is the line bundle over L l +1 , which is associated to thecanonical 2-fold covering of a double point manifold. By the Hirsch theoremthere exists an immersion h regularly homotopic to h and having its image inthe subspace R l +2 ⊂ R l +3 . The regular homotopy between immersions h and h can be extended to the regular homotopy of normal bundles, hence the directsummands of normal bundle are parallel to the complementary coordinate axesof the subspace R l +2 ⊂ R l +3 . 6he immersion h : L l +1 R l +2 is equipped with a skew-framing ofthe normal bundle, defined by the bundle isomorphism Ψ L : kκ ≡ ν h . Startingfrom the immersion f , we have constructed the skew-framed immersion ( h , Ψ L ) .Define the element δ ([ f ]) ∈ Imm sf (2 l + 1 , l + 1) to be the regular skew-framedcobordism class [( h , Ψ L )] . The Browder-Eccles invariant.
An alternative definition of the Arf-Kervaire invariant of framed immersions was given by Eccles in [E1]. It usesthe characteristic numbers of double point manifolds and is based on a theoremof Browder ([Br]). In this theorem the Arf-Kervaire invariant is constructedby means of the Adams spectral sequence. The following simplest version ofdefinition was given by Eccles in [A-E1].
Definition 2.3.
Define the Browder-Eccles invariant ¯Θ( f ) of a framed immer-sion f by the formula ¯Θ( f, Ξ N ) = h ◦ δ ( f, Ξ N ) (mod 2) , where h ◦ δ ( f, Ξ N ) = h ( I ◦ f ) is the number of self-intrsection points of theimmersion I ◦ f . In this section we define new variants of cobordism groups, namely the cobor-dism groups of stably framed immersions (stably skew-framed immersions, re-spectively), i.e. the immersions which are not framed in their image–Euclideanspace but are framed only in ambiental Euclidean spaces of sufficiently big di-mensions. Such a framing (skew-framing, respectively) is said to be the stableframing (stable skew-framing, respectively). Cobordism groups of stably framedand of stably skew-framed immersions generalize intermediate cobordism groups ,as introduced in [E3]. The Arf-Kervaire and the Browder-Eccles invariant canbe generalized to the invariants defined on the cobordism group of stably framedimmersions.The new invariants are called the twisted Arf-Kervaire invariant and thetwisted Browder-Eccles invariant , respectively. The definition of the twistedArf-Kervaire invariant is closely connected to the definition of
Arf-changeableinvariant of framed immersions in the sense of Jones and Rees [J-R].
Definition of stably framed cobordism groups
Imm stfr (4 l + 2 , l + 1) . Let ( f, Ξ N ) be a pair, where f : N l +2 R l +3 is an immersion in the codi-mension l + 1 , Ξ N be a stable framing of the manifold N l +2 , i.e. a framing ofthe normal bundle of the composition I ◦ f : N l +2 R l +3 ⊂ R r , r ≥ l + 6 .The set of pairs described above is equipped by equivalence relation, whichis given by the standard relation of cobordism. Up to the cobordism relationthe set of pairs generates an Abelian group with the operation determined bythe disjoint union. This group is denoted by Imm stfr (4 l + 2 , l + 1) .7 efinition of stably skew-framed cobordism groups Imm stsf (2 l + 1 , l +1) . Let ( ϕ, Ψ) be a pair, where ϕ : L l +1 R l +2 is an immersion in thecodimension l + 1 , Ψ L be a stable skew-framing of the manifold L l +1 inthe codimension l + 1 , i.e. a skew-framing of the normal bundle of thecomposition I ◦ f : L l +1 R l +2 ⊂ R r , r ≥ l + 4 with the bundle (2 l + 1) κ ⊕ ( r − l − ε , where ε is a trivial line bundle on L l +1 , κ is a given linebundle over L l +1 , which coincides with the orienting line bundle over L l +1 ,since w ( L l +1 ) = w ( ν ϕ ) = w ((2 l + 1) κ ) = w ( κ ) .The set of pairs described above is equipped by an equivalence relation,which is defined as the standard regular cobordism. The set of equivalenceclasses generates an Abelian group with the operation determined by the disjointunion. This group is denoted by Imm stsf (2 l + 1 , l + 1) . Homomorphisms A : Imm fr (4 l + 2 , −→ Imm stfr (4 l + 2 , l + 1) , B : Imm stfr (4 l + 2 , l + 1) → Imm fr (4 l + 2 , . An arbitrary immersion f : N l +2 R l +3 determined the immersion ( I ◦ f, Ξ N ) in the codimension l + 1 , I : R l +3 ⊂ R l +3 , which is stably framed in the ambiental space R l +3 ⊂ R n . Thehomomorphism A is defined.An arbitrary stably framed immersion ( f : N l +2 R l +3 , Ψ N ) obviouslyinduces the immersion into the space R r , r > l + 6 .The Hirsch theorem, applied to this immersion of the codimension r − l − ,assures the existence of framed immersion ( ϕ, Ξ N ) , where ϕ : N l +2 R l +3 .Now define B ([( f, Ψ N )]) = [( ϕ, Ξ N )] ∈ Imm fr (4 l + 2 , . Obviously, by thisconstruction B ◦ A = Id : Imm fr (4 l + 2 , → Imm fr (4 l + 2 , . Twisted Arf-Kervaire invariant Θ st : Imm stfr (4 l + 2 , l + 1) −→ Z / . We generalize the Arf-Kervaire homomorphism
Θ :
Imm fr (4 l + 1 , → Z / and definea homomorphism Θ st : Imm stfr (4 l + 2 , l + 1) → Z / provided that l + 1 = 1 , , , called the twisted Arf-Kervaire invariant such that the followingdiagram commutes: Imm fr (4 l + 2 , A −→ Imm stfr (4 l + 2 , l + 1) ց Θ ւ Θ st Z / . (1) Auxiliary homomorphism π : H l +1 ( N l +2 ; Z / → Z / , l + 1 = 1 , , . It is known that for l + 1 = 1 , , there exist exactly two stably trivial S l +1 -dimensional vector SO –bundles (this fact is applied in the proof of Lemma 8.3[K-M]). One of these bundles is the trivial; we denote it by E (2 l + 1) . Theother bundle is nontrivial, it coincides with the tangent bundle T ( S l +1 ) overthe sphere S l +1 . We need a generalization of this fact to the case of a l + 1 –dimensional stably trivial bundle over an arbitrary closed l + 2 –dimensionalmanifold, possibly nonoriented.Let M l +1 be a closed, possibly nonoriented manifold, ξ be a SO (2 l + 1) –bundle over M l +1 such that it is trivial as a stable SO –bundle. Let M l +11 ξ be another manifold and a SO (2 l + 1) –bundle as above, respectively.Let W l +2 be a l + 2 –dimensional polyhedron, such that it is a manifold inthe exterior of the codimension 2 skeleton. Let the polyhedron W realize ahomology between the fundamental classes of manifolds M l +1 and M l +11 , i.e. ∂W l +2 = M l +1 ∪ M l +11 . Additionally suppose that there exists a stably trivial SO (2 l + 1) –bundle Ξ over W l +2 such that the restrictions of Ξ on M l +1 andon M l +11 coincides with the bundles ξ and ξ , respectively. Lemma 3.1.
For arbitrary above described pair ( M l +1 , ξ ) there exists anobstruction c ( M, ξ ) ∈ Z / to the trivialization of the bundle ξ . Moreover, c ( M l +1 , ξ ) = c ( M l +11 , ξ ) . Corollary 3.2.
Let f : M l +1 → S l +1 be a map of a closed manifold tothe standard sphere, such that deg( f ) = 1 (mod 2) . Let ξ be a stably trivial SO (2 l + 1) –bundle over S l +1 , such that c ( S l +1 , ξ ) = 1 . Then f ∗ ( ξ ) is a stablytrivial SO (2 l + 1) –bundle over M l +1 , and c ( M l +1 , f ∗ ( ξ )) = 1 (mod 2) . Proof of Corollary 3.2.
The corollary follows from the fact that there exists a homology between f ∗ ([ M ]) и [ S ] , where [ M ] and [ S ] are the fundamental classes of M l +1 and S l +1 . Proof of Lemma 3.1.
Let ( M l +1 , ξ ) be the pair described in the preamble of Lemma. Denote by M [2 l ] ⊂ M l +1 the complement of the highest l + 1 –dimensional cell in theskeleton of cellular decomposition of M l +1 . By the dimensionality argument,the restriction of ξ on M [2 l ] is a trivial bundle. Hence there exists a map f : ( M l +1 , M [2 l ] ) → ( S l +1 , pt ) , such that f ∗ ( ψ ) = ξ , where ( S l +1 , ψ ) isa bundle over S l +1 , satisfying the conditions of Lemma. In the case when M l +1 = S l +1 , Lemma is true since in the proof of Theorem 3.2 in [Br] theobstruction ( S l +1 , ξ ) is constructed by the corresponding functional cohomo-logical operation. Define that ( M l +1 , ξ ) = ( S l +1 , ψ ) . Let ( M l +11 , ξ ) bethe second pair described above and ( W l +2 , Ξ) be the homology, connecting ( M l +1 , ξ ) and ( M l +11 , ξ ) . If c equals to zero for both pairs, Lemma is proved.Suppose that for at least one pair – say ( M l +1 , ξ ) – the value of obstruction ( M l +1 , ξ ) is 1. Consider the handle (cell) decomposition of the cobordism ( W l +2 , M l +1 ) . The index of handles can be restricted to ≤ l (the dimensionof handles to l + 1 ) as in the case when W l +2 is a smooth manifold. Indeed,the handle (cell) decomposition of ( W l +2 , M l +1 ) can be chosen so that all l + 2 –dimensional cells retract to the l + 1 –skeleton of the polyhedron W l +2 without the l + 1 –dimensional cells of the upper base M l +11 .Define the map F : ( W l +2 , M l +1 ) → ( S l +1 × I, S l +1 × { } ) such that F ∗ π ∗ ( ψ ) = Ξ , (3)where π : S l +1 × I → S l +1 is a projection onto the lower base. The map F can be extended to the l –dimensional handles uniquely up to homotopy. Themap F can be extended also to the l + 1 –dimensional handles, but possiblynonuniquely. By assumption, c ( S l +1 , ψ ) = 1 . Hence the extension of the map F to the l + 1 –dimensional handles can be realized in such a way that the9ondition 3 be true. Now on the upper base of the cobordism f ∗ ( ψ ) = ξ ,therefore by the definition c ( M l +11 , ξ ) = 1 . Lemma 3.1 is proved.Let ( f : N l +2 R l +3 , Ξ N ) be a pair, defining an element of the group Imm stfr (4 l + 2 , l + 1) .Consider an arbitrary cycle x ∈ H = H l +1 ( N l +2 ; Z / . It is representedby an embedding i Y : Y l +1 → N l +2 . Denote shortly by ξ the bundle i ∗ Y ( ν f ) ,where ν f is the normal bundle of an immersion f . Since ν f is a stably trivialbundle (because the manifold N l +2 is stably framed by Ξ ), for the pair i ∗ Y ( ν f ) is defined the obstruction ( Y l +1 , ξ ) provided that l + 1 = 1 , , . Define themapping π : H l +1 ( N l +2 ; Z / → Z / (4)given by the formula π ( x ) = c ( Y l +1 , ξ ) , y ∈ H = H l +1 ( N l +2 ; Z / . Lemma3.1 implies that the value of π ( x ) does not depend on the choice of manifold Y l +1 and on the choice of embedding l x , which realizes given cycle x . It canbe easily verified that the mapping 4 is a homomorphism. Definition 3.3.
Let q : H → Z / , H = H l +1 ( N l +2 ; Z / be the quadraticform defined in 1 for a stably framed manifold ( N l +2 , Ξ) . For l + 1 = 1 , , define the twisted quadratic form q tw by the formula q tw = q + π : H → Z / .The Arf invariant of this twisted quadratic form defined a homomorphism Θ st : Imm stfr (4 l + 2 , l + 1) −→ Z / , which is said to be the twisted Arf-Kervaireinvariant. Twisted Browder-Eccles invariant . Define the invariant ¯Θ st : Imm stfr (4 l + 2 , l + 1) → Z / , which is said to be the twisted Browder-Ecclesinvariant, starting by the construction of homomorphism δ st : Imm stsf (4 l + 2 , l + 1) → Imm stsf (2 l + 1 , l + 1) . (5)Suppose that an element of the group Imm stfr (4 l + 2 , l + 1) is representedby the pair ( f : N l +2 R l +3 , Ξ) . The double point manifold of immer-sion f is denoted by L l +1 . This manifold is equipped by the parametrizingimmersion ϕ ′ : L l +1 R l +3 . The corresponding normal bundle ν ϕ ′ admits(for a sufficiently big natural k ) a stable isomorphism Ψ ′ : ν ϕ ′ ⊕ kε ⊕ kκ =(2 l + 1 + k ) ε ⊕ (2 l + 1 + k ) κ .The stable isomorphism Ψ ′ defines a stable isomorphism Ψ L : ν L ⊕ kε = (2 l +1) κ ⊕ kε , where by ν L is denoted the l + 1 –dimensional normal bundle over L .By the Smale-Hirsch construction an immersion ϕ : L l +1 R l +2 and a stablyskew-framing Ψ L of the normal bundle of this immersion are defined. Define δ st ([( f, Ξ f )]) to be the element of group Imm stsf (2 l + 1 , l + 1) correspondingto the pair ( ϕ, Ψ L ) .Consider the homomorphism Imm stsf (2 l +1 , l +1) h −→ Imm D (0 , l +2) = Z / , defined as the parity of number of double points of stably skew-framedimmersions (this invariant is called the stably Hopf invariant). The Browder-Eccles invariant ¯Θ st is defined as the composition Imm stfr (4 l + 2 , l + 1) h ◦ δ st −→ Imm D (0 , l + 2) = Z / . 10 ubgroup Imm stfr (4 l + 2 , l + 1) ∗ ⊂ Imm stfr (4 l + 2 , l + 1) . Define anauxiliary subgroup
Imm stfr (4 l + 2 , l + 1) ∗ ⊂ Imm stfr (4 l + 2 , l + 1) as thecomplete preimage δ − ( Imm sf (2 l + 1 , l + 1) ⊂ Imm stsf (2 l + 1 , l + 1)) of thehomomorphism ( ) .It is convenient to introduce the following equivalent geometrical definitionof the subgroup Imm stfr (4 l + 2 , l + 1) ∗ . This definition is valid for all thenatural l ≥ but not for l = 0 , , .The pair ( f : N l +2 R l +3 , Ξ N ) represents an element x ∈ Imm stfr (4 l +2 , l + 1) ∗ if the following is true.Take the pair ( ϕ : L l +1 R l +2 , Ψ L ) representing the element δ st ( x ) ;here L l +1 is the double point manifold of the immersion f . Consider thecanonical covering ¯ L l +1 → L l +1 of the double point manifold L l +1 (detailscan be found in [A]). Let ¯ g : ¯ L l +1 N l +2 be the parametrizing immersion, [ ¯ L ] ∈ H = H l +1 ( N l +2 ; Z / be the cycle obtained as the image of fundamentalclass of ¯ L l +1 by the immersion ¯ g . Consider the value π ([ ¯ L ]) , where π : H → Z / was defined in 4.Without losing generality we may assume that the stably framed immersion ( f, Ξ) is chosen from the regular cobordism class so that the manifold ¯ L l +1 is connected. This goal can be achieved through the 1-handles surgery on thedouble point manifold of immersion f . The techniques of such a surgery wasinvented in [H]. Then the condition x ∈ Imm stsf (4 l + 2 , l + 1) ∗ is equivalent to π ([ ¯ L l +1 ]) = 0 ,. The last condition is equivalent to the fact that the pull-back ¯ g ∗ ν f over ¯ L l +1 of the normal bundle ν f is not only stably trivial but it is alsotrivial (in the case l = 0 , , the normal bundle ν f is always trivial). It meansthat the immersion ϕ : L l +1 R l +2 is not only stably skew-framed but it isalso skew-framed. This is the equivalent geometrical definition of the subgroup Imm stfr (4 l + 2 , l + 1) ∗ .For convenience, all homomorphisms which have been constructed, includeinto the common commutative diagram: Imm fr (4 l + 2 , A,B ←→ Imm stfr (4 l + 2 , l + 1) ∗ ⊂ Imm stsf (4 l + 2 , l + 1) ↓ δ ↓ δ st ↓ δ st Imm sf (2 l + 1 , l + 1) = Imm sf (2 l + 1 , l + 1) ⊂ Imm stsf (2 l + 1 , l + 1) ↓ h ↓ hZ/ Imm D (0 , l + 2) = Imm D (0 , l + 2) The following lemma is necessary for our proof of Theorem 1.2
Lemma 4.1.
The twisted Arf-Kervaire homomorphism Θ st : Imm stfr (4 l +2 , l + 1) → Z / coincides with the twisted Browder-Eccles homomorphism ¯Θ st : Imm stfr (4 l + 2 , l + 1) → Z / , on the subgroup Imm stfr (4 l + 2 , l + 1) ∗ ⊂ Imm stfr (4 l + 2 , l + 1) .
11e shall derive Lemma 4.1 from the following lemma.
Lemma 4.2.
The homomorphisms Θ st and ¯Θ st coincide on the subgroup ( ImA = KerB ) ∩ Imm stfr (4 l + 2 , l + 1) ∗ ⊂ Imm stfr (4 l + 2 , l + 1) . Proof of Lemma 4.2.
From the main result of [E1] reformulated in [A-E1],in the required form, it follows that the homomorphisms Θ st and ¯Θ st coincideon the subgroup ImA ⊂ Imm stfr (4 l + 2 , l + 1) ∗ . Proof of Lemma 4.1.
Let ( f : N l +2 R l +3 , Ψ N ) be a stably framedimmersion. The double point manifold of this immersion equipped with thestable skew-framing will be denoted by ( L l +1 , Ξ L ) . Let η : W l +3 → R l +3 × R be a generic mapping of a stably parallelized manifold ( W l +3 , Ψ W ) defining theboundary of stably framed manifold ( N l +2 , Ψ) but not the immersion f itself,generally. The dimension of the critical point manifold Σ of η equals to l + 1 ;this is less than half of dimension of the manifold–preimage W l +3 . The criticalpoints of map η are of the type Σ , .By the Moren theorem (see [A-V-G, Ch. 1, Paragr. 9, Sect. 6, the case k = 2 ]) the critical point manifold has the normal form called the extendedWhitney umbrella. The formulae describing the singularities of Whitney um-brella R s → R s − can be found in [P, Ch. 1, Paragr. 4]. The notion "extended"means the inclusion of the standard singularity of umbrella into the identicalpolyparametrical collection of maps.One may assume w. l. o. g., after a corresponding repair of the singularityof map η , that the critical point manifold Σ satisfies the following properties.(1) Σ is connected with connected canonical double covering ¯Σ .(2) η (Σ) belongs to the hyperspace R l +3 ×{ } and the double point manifold K l +2 of map η with the boundary ∂K l +2 = L l +1 ∪ Σ l +1 is regular in a smallneighborhood of the boundary with respect to the height function on R insuch way that the subspace R l +3 × { } is higher than the manifold (i.e. K l +2 immerses into the subspace [+ ε, − ε ] × R l +3 outside its regular neighborhood.Let η − ε : N l +21 − ε R l +3 ×{ − ε } be the immersion defined as the restrictionof η on the complete preimage of the hyperspace R l +3 × { − ε } . Let L l +11 − ε bethe component of the double point manifold η − ε ( N l +21 − ε ) in the neighborhood ofthe critical point boundary Σ l +1 of K l +2 . From the assumption π ( ξ ) = 0 wemay deduce that the normal bundle ν L − ε of the manifold L l +11 − ε is decomposedinto the direct sum of the trivial bundle ν ε = (2 l + 1) ε and a nontrivial bundle ν κ = ν ε ⊗ κ , where κ is the orientation line bundle over L l +11 − ε . Since thecanonical covering is connected, κ is nontrivial. Construction of the stably framed immersion.
Let us construct the stably framed immersion ( ξ : N l +20 R l +3 , Ψ ) (6)such that the double point manifold L l +10 (equipped with a skew-framing Ξ ofthe normal bundle) coincides with an arbitrary given skew-framed immersion.Let us start the construction by the description of standard immersion g : S l +1 R l +2 with the self-intersection points at the origin of the co-ordinate system R l +11 ⊕ R l +12 = R l +2 . Let R l +1 diag , R l +1 antidiag be two coordinate12ubspaces defined by means of the sum and the difference of the base vectors inthe standard coordinate spaces R l +11 , R l +12 .Consider two standard unit disks D l +11 ⊂ R l +11 , D l +12 ⊂ R l +12 . Take amanifold C diffeomorphic to the cylinder S l × I defined as the collection of allthe segments such that each connects a pair of points in ∂D l +11 and ∂D l +12 with equal coordinates. The union of two disks D l +11 ∪ D l +12 with C (afterthe identification of corresponding components of the boundary) is the imageof the standard sphere S l +1 by a PL-immersion g into R l +2 , with one self-intersection point at the origin. After the smoothing of corners along ∂C weobtain the smoothly immersion of sphere under construction.Let us describe the manifold N l +20 , the stable framing Ξ N over this mani-fold and the immersion f : N l +20 R l +3 , [( f , Ξ N )] ∈ Imm stsf (4 l +2 , l +1) .Take the embedding η : L l +10 ⊂ R l +3 with the normal bundle ν L = ν ⊕ ν ⊗ κ ,where ν is a trivial (2 l + 1) –dimensional bundle (with the prescribed trivial-ization) and κ is the orientation line bundle over L l +10 . Consider the (2 l + 2) –dimensional bundle ν ⊗ κ ⊕ ε over L l +10 and define the manifold N l +20 as theboundary S ( ν ⊗ κ ⊕ ε ) of the disk bundle of this vector bundle.The locally trivial fibration p : N l +20 → L l +10 is well-defined. Because ν ⊗ κ ⊕ ε is the normal bundle of L l +10 , the manifold N l +20 admits an embeddingin codimension 1. This embedding determines the framing Ψ N over N l +20 suchthat the constructed framed manifold ( N l +20 , Ψ N ) is bounding.Let us define the immersion f : N l +20 R l +3 . Take the normal bundle ν ⊗ κ ⊕ ν of the immersion η and consider the collection of the standardimmersed spheres g ( S l +1 ) constructed above in each fiber of η . The pair ( f , Ψ ) is the stably framed immersion under construction. Calculation of invariants of the constructed stably framed immersion.
This section is devoted to the calculation of the twisted Arf-Kervaire and thetwisted Browder-Eccles invariants for the stably framed immersion (6). Thegroup H = H l +1 ( N l +20 ; Z / is generated by two elements. The first generator x ∈ H is represented by a spherical fiber of the fibration p : N l +20 → L l +10 .The fibration p has a standard section p − constructed by the trivial directsummand in the bundle ν ⊗ κ ⊕ ε . The image of the fundamental class of thebase L l +10 induced by p − , represents the second generator y ∈ H . Let us proveunder the assumption l + 1 = 1 , , that the homomorphism π : H → Z / isdefined by π ( x ) = 1 , π ( y ) = 0 .The condition π ( x ) = 1 holds since for an arbitrary immersion of a sphere f : S l +1 R l +2 with one self-intersection point the corresponding normal l + 1 –dimensional bundle is nontrivial.Let us prove π ( y ) = 0 . (7)The cycle y ∈ H is represented by the image of the fundamental class inducedby the map of section p − ( L l +10 ) → N l +20 of the fibration p . The collection ofthe bases in the fibers of the subbundle ν ⊂ ν L defines the trivialization of thenormal bundle of the immersion f over the submanifold p − ( L l +10 ) ⊂ N l +20 .This proves ( ) . 13he twisted Arf-Kervaire invariant of a stably framed immersion ( ξ , Ψ ) is equal to q ( y ) i.e. coincides with the twisted Browder-Eccles invariant. Thisgives the required computations.Let us finish the proof of Lemma 4.1. Consider a stably framed immersion ( f − ε , Ψ N − ε ) with the skew-framed double point manifold ( L l +11 − ε , Ξ L − ε ) . Be-cause of the condition (2) the stably framed immersion ( η − ε , Ψ N − ε ) is regularlycobordant to the immersion ( f, Ψ N ) from the beginning of the proof of Lemma4.1. Therefore [( f − ε , Ψ − ε )] ∈ Imm stfr (4 l +2 , l +1) ∗ ⊂ Imm stfr (4 l +2 , l +1) and the stably framed immersion ( f − ε , Ψ N − ε ) in fact is framed and the stablyskew-framed immersion ( L l +11 − ε , Ξ L − ε ) in fact is skew-framed.Let us apply the construction ( ) , where the standard stably framed immer-sion ( f , Ψ N ) is obtained such that its self-intersection points are opposite tothe points of the skew-framed immersion ( L l +11 − ε , Ξ L − ε ) .Note that the disjoint union ( η − ε , Ψ N − ε ) ∪ ( f , Ψ N ) is a stably framedboundary. Obviously, for the stably framed immersion ( ξ − ε , Ψ − ε ) ∪ ( ξ , Ψ ) both invariants are trivial. On other hand, by the calculations the twistedArf-Kervaire invariant and the twisted Browder-Eccles invariant of ( f , Ψ N ) coincide. Hence for the stably framed immersion ( η − ε , Ξ N − ε ) both invariantscoincide and moreover, for the stably framed immersion ( f, Ξ N ) ∈ Imm stfr (4 l +2 , l + 1) ∗ both invariants coincide. Lemma 4.1 is proved Proof of Theorem 1.2.
Let l + 2 = 30 or l + 2 = 62 . Consider a closedframed l –connected manifold ( N l +2 , Ψ) with the Arf-Kervaire invariant 1.Let us assume that for l = 7 ( l = 15 ) there exists an embedding J : N l +2 ⊂ R l +3 and that the normal (2 k + 2) –bundle ν ¯ I of this embedding is equippedwith (resp. ) linearly independent sections.The restriction of the normal bundle ν J on an arbitrary embedded sphere i S l +1 : S l +1 → N l +2 is the trivial l + 1 –dimensional, i.e. –dimensional(resp. –dimensional) bundle (the bundle i ∗ S l +1 ( ν J ) is stably dimensional andtrivial), equipped with (resp. ) linearly independent sections.There is only one stably trivial but nontrivial –dimensional (resp. –dimensional) bundle over S (resp. S ); this bundle is the tangent bundle T ( S l +1 ) (see [K-M] p. 534). The bundle i ∗ S l +1 ( ν J ) over S ( S ) is trivialif and only if when c ( S , i ∗ S ( ν J )) = 0 (resp. c ( S , i ∗ S ( ν J ) = 0 ). By theJ. F. Adams theorem (see Novikov’s survey [N, Ch. 3, Paragr. 8, p. 106 withthe reference on the Adams’ result]) the tangent bundle T ( S ) (resp. T ( S ) )admits no more than (resp. ) linearly independent sections. By our assump-tion the bundle i ∗ S l +1 ( ν J ) admits (resp. ) linearly independent sections.Therefore i ∗ S l +1 ( ν J ) is a trivial bundle. Hence the auxiliary homomorphism π : H → Z / is trivial and the Arf-Kervaire invariant of the stably framed man-ifold ( N l +2 , Ξ) is equal the twisted Arf-Kervaire invariant of the pair ( J, Ξ) , soboth are equal to 1.On the other hand, provided that J is an embedding, by Lemma 4.1 thetwisted Arf-Kervaire invariant coincides with the twisted Browder-Eccles in-variant. First assertion of Theorem 1.2 is proved.Let us assume that for l = 7 ( l = 15 ) there exists an embedding ¯ J : N l +2 × I ⊂ R l +4 and that the normal (2 k +2) –bundle ν ¯ I of this embedding is equippedwith (resp. ) linearly independent sections over the complement to the basesegment pt × I ∈ N l +2 × I .The restriction of the normal bundle ¯ ν ( ¯ J ) over an arbitrary embedded sphere14 S l +1 : S l +1 ⊂ N l +2 is the trivial l + 1 –bundle since it is equipped with (resp. ) linearly independent sections.By the Rourke-Sanderson compression theorem [R-S] we may assume, af-ter an appropriate isotopy of the framed embedding ¯ J , that the collection ofsegments I is vertically up with respect to the axis of projection. After theprojection we obtain an immersion J : N l +2 R l +3 , which is framed at leastoutside a neighborhood of a point. The double point manifold L l +1 of theimmersion J is stably skew-framed, hence in fact it is framed. Let us denotethis framing by Ξ .The framed manifold ( L l +1 , Ξ) determines an element of the group Imm stsf (2 l + 1 , l + 1) lying in the image of the homomorphism Π l +1 = Imm fr (2 l + 1 , l + 1) → Imm stsf (2 l + 1 , l + 1) . Therefore the twisted Browder-Eccles invariant of the stably framed immersion ( J, Ψ N ) is equal to the stableHopf invariant of the framed manifold ( L l +1 , Ξ L ) . By the Toda theorem for l + 1 = 15 and by the Adams theorem for l + 1 = 31 the Hopf invariant isequal to 0 (see [Mo-T, Sect. 18]).On the other hand, by Lemma 4.1 the twisted Browder-Eccles invariant of ( J, Ψ N ) is equal to the twisted Arf-Kervaire invariant of ( J, Ψ N ) . The last isequal to the Arf-Kervaire invariant of the framed manifold ( N, Ψ N ) becausethe auxiliary homomorphism π : H → Z / for the stably framed immersion ( J, Ψ N ) is trivial. Therefore the twisted Arf-Kervaire invariant is equal to 1.This contradiction shows that if the manifold N l +2 × I embeds in R l +4 thenthe collection of linearly independent sections of the normal bundle does notexist. Theorem 1.2 is proved. Acknowledgements
The first author was supported by the grant RFFI-08-01-00663a. The secondand the third author were supported by the Slovenian Research Agency grantsP1-0292-0101, J1-9643-0101, and J1-2057-0101.
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Email address : [email protected] of Mathematics and Physics, and Faculty of Education, University ofLjubljana, P. O. Box 2964, Ljubljana 1001, Slovenia