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A Remark on the Hopf invariant for Spherical 4-braids

Akhmet’ev PM*

Professor in IZMIRAN, Troitsk, Moscow region, Russia

*Corresponding Author:
Dr. Akhmet’ev PM
IZMIRAN, Troitsk, Moscow region,Russia
Tel: +7(4967) 510912
Fax: +7(495)3340124
E-mail: [email protected]

Received Date: October 31, 2013; Accepted Date: January 28, 2014; Published Date: March 10, 2014

Citation: Akhmet’ev PM (2014) A Remark on the Hopf invariant for Spherical 4-braids. J Phys Math 5:124 doi: 10.4172/2090-0902.1000124

Copyright: © 2014 Akhmet’ev PM. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

An approach by J.Wu describes homotopy groups πn(S2) of the standard 2-sphere as isotopy classes of spherical n+1--strand Brunnian braids. The case n=3 is investigated for applications.

Introduction

An approach by Wu describes homotopy groupsequation of the standard 2-sphere as isotopy classes of spherical equation-strand Brunnian braids, for more details, Theorem 1.2. This straightforward approach is not possible for n=3, i.e. for 4-strand braids the connection with π3(S2) was unknown.

The homotopy group π3(S2) in an infinite cyclic group, detected by the Hopf invariant

equation(1)

An element of π3(S2) is represented by a mapping equation , which is considered up to homotopy. The Hopf invariant H(h) is welldefined as the integer linking number of two oriented curves h-1(a), h-1(b), where a,b∈S2 be a pair of regular points of h. The Hopf invariant is very important for applications.

Proposition 7.1.1, sequence (17) gets an exact sequence, which algebraically describes the group Brunn4 of 4-straight Brunnian braids [1]. The key point of our elementary geometrical construction is to construct an alternative epimorphism onto the group equation, see Definition. The kernel of this epimorphism is a well-defined subgroup Brun4Br4of Brunnian braids in a new sense (let us remark that Brunn4 is not a subgroup of Brunn4. Define the Hopf invariant as a function of isotopy classes of spherical braids in Brunn4. An idea of the construction was coming from Graham and Roman [2]. However, the results by Ellis and Mikhailov are not adopted for physical applications.

The Hamiltonian provides an elegant method for generating simple geometrical examples of complicated braids and links, as is presented in Mitchell A Berger [3].

The paper is motivated by the following problems:

• Derive applications of higher-order winding numbers to generate turbulent motions of vortices in two dimensions. For a special Hamiltonian motion of 3 vortices on the plane this is done in Mitchell A Berger [3]. (Problem 1).

• To unify the approach Ch.3 to π*(S2) with the Wu’s approach (Problem 2) [4].

Let us clarify Problem 2. Let F be the space of functions equation with “right” boundary conditions at the infinity. The derivative of the order 1,2, and 3 of a function f ∈ F can nowhere be vanished simultaneously. Define the mapping,equation) , by

the formulaequation

V.I.Arnold (1996) conjectured that the induced homomorphism equation is an isomorphism forequationThis theorem was proved by V.A. Vassiliev in the special case n+2 and by Eliashberg and Mishachev in the general case.

The paper is organized as following. In Section 2 we recall required definitions concerning first-order stage of the construction and determine the linking numbers of spherical 4-component braids. In Section 3 the Hopf invariant for 4-component spherical braids is defined. This is a second-order particular defined invariant: to define this invariant we should assume that the all linking numbers (there are two) of components of a spherical braid are equal to zero. Results are formulated in Theorems 4, 6. The main result is the Corollary 8. In Section 4 we give proofs.

A possible application for turbulences (Problem 1)

Assume a motion of a large collection of n vortexes (or, particles) in a bounded domain U on the plane is investigated. The trajectories of vortexes (or, of particles) in the configuration space, i.e. in the Cartesian product, equation of the domain and the time, are represented by a braid F, components of the braid F correspond to vortexes in the collection. Assume that the windings numbers of components of the braid F are distributed as in the statement of Corollary. This means that the length of the segment (a,b), which is assumed sufficiently large, is bounded from below; the upper bound depends of the number n of vortexes in the collection. We may replace F by a colored braid, if b-a is sufficiently large, using the Arnol’d collection of the short paths, we have no loss of a generality.

Otherwise, assume that the bound k of the distribution of full angles of windings numbers is much less then the number n of partials. Consider the normalized sum of squares of Hopf invariants

equation(2)

this sum is taken over all collection of admissible quadruples of components of F, the number l of admissible quadruples could be sufficiently large by Corollary 8. The following statements will be proved, or disproved, elsewhere:

Υ is the universal constant of the motion, which depends no of the time scale and of the time interval [a, b] itself;

• The constant Υ is large (correspondingly, is small), if the motion of the system of vortexes (or, of partials) is turbulent (correspondingly, the system is closed to an integrable system);

• Assume the sum (2) is taken over all admissible quadruples, between which the distance is smaller then L. Then Υ (L) correlates with the spacial turbulent spectra of the motion up to the scale L.

Linking Numbers For Spherical Braids

By a spherical (ordered) n-braid we mean a collection of embeddings of the standard circles

equation

where the composition of this embedding with the standard projection S2 × S1 → S1 on the second factor in the target space, restricted to an arbitrary component 1 equation , i=1,…,n is the identity mapping.

The set of all ordered spherical n-braids up to isotopy is denoted by Brn. It is well-known that Brn is a group.

For a fixed value t ∈ S1, a braid f ∈ Brn intersects the level S2 × t by an (ordered) collection of n points {Z1(t),… Zn(t)}. Let assume that n=4. Denote by

equation

the 3-component braid, obtained from f by eliminating of the last componentequation.

component equation Let us identify the sphere S2 with the Riemann sphere, or with the complex projective line C. For a braid f let us consider the collection of Mobius transformations, which transforms the points z1, z2, z3 into 0, 1, ∞ correspondingly:

equation

The image F(f) is a 4-strand braid with the constant components {z1(t), z2(t), z3(t)}={0, 1, ∞}. Denote this braid by

equation(3)

The 3-strand braid g, constructed from fnorm is the constant in the points {0, 1, ∞}. The last component fnorm(S1) of F(f) is represented by a closed pathequation Note that, generally speaking, braids f, fnorm are not isotopic. Moreover, if f is a Brunnian in the sense [2], fnorm is, generally speaking, not a Brunnian.

For a given (ordered) 4-component braid f let us define the linking number Lk(f),

equation(4)

Consider the following 1-form

equation(5)

By definition we get

equation(5)

where log (z) is given by the formula:

equation(6)

assuming that log (1)=0, as a multivalued complex function.

Define Lk(f) by the formula:

equation(6)

where equation is the real part of the integral. By construction, Lk(f) is the winding number, i.e. the integer number of rotations of the path z4(t) of fnormwith respect to the origin and the infinity in C.

The permutation group Σ(4) of the order 24 acts on the space of ordered spherical braids:

equation(7)

The image of an ordered braid f by a transposition equation is well-defined by the corresponding re-ordering of components of f. Let us investigate the orbit of the linking numbers Lk(f) with respect to (7). Simply say, we investigate how many independent linking numbers of components of braids are well-defined?

Let us consider the following exact sequences of groups:

equation(8)

equation(9)

The subgroup A4 ⊂ Σ4 in the sequence (8) is represented by permutations, which preserve signs (equivalently, which is decomposed into an even number of elementary transpositions). The subgroup equationin the sequence (9) is generated by the permutations{(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)}.

Let us consider 2-primary subgroup K ⊂ Σ4 (the dihedral group of the order 8), which is defined as the extension of the subgroup equation from the sequence (9), which is included in the sequence (8). An epimorphism

equation(10)

is defined as follows: θ1(σ)=1 (the group Z/2 is in the multiplicative form), if σ preserves a (non-ordered) partition (1,3)(2,4), and θ1(σ), and θ1(σ)=-1, otherwise. Therefore θ1 is an epimorphism with the kernel equation from the left subgroup of the sequence (9). The epimorpism θ2(σ) is determined by the sign of a permutation σ, this is the restriction of the right epimorphism in the sequence (8) to the subgroup K ⊂ Σ4 . The kernel equation is the center of the dihedral group K.

Lemma 1

1. The function (4) is invariant with respect to the action (7) by an arbitrary permutation, which in the kernel of in (10).

2. The function (4) is skew-invariant with respect to the action by a permutation, which is in the kernel of θ1 (the composition of θ with the projection on the first factor, but not in the kernel of θ2 (the composition of θ with the projection on the second factor).

3. Denote by equation the ordered braid, which is obtained from f ∈ Br4 by the action (7) by the element (1,2) is the product of the generators of the factors). There exists an ordered braids f ∈ Br4, for which the linking numbers Lk(f), equation are arbitrary integers.

From Lemma one may deduce the following corollary.

Corollary 2

1. For an arbitrary braid f ⊂Br4the linking number Lk(f), is welldefined as the differences of the winding number of the component 2 between the components 1 and 3 with the winding number of the component 4 between the components 1 and 3.

2. For a braid , where f ∈Br4is an arbitrary, equation is defined in Lemma 1, the linking number equation is well-defined as the difference of the winding number of the component 2 between the components 1 and 3 with the winding number of the component 4 between the components 2 and 3.

Corollary (2) motivates the following definition.

Definition 3

Let f ∈ Br4be a (ordered) spherical braid. Define the total linking number by the following formula:

equation

The total linking number is a well-defined homomorphism

equation

Hopf Invariant of Braids

Let f ∈ Br4 be a (ordered) spherical braid with the trivial total linking number: Lk(f)=0. Such braids generate the subgroup in the group Br4, denote this subgroup by Brunn4 ⊂ Br4. Let us remark that this subgroup does not coincide with the subgroup of Brunnian braids Brun4, defined in Berrick et al. [1], Theorem 1.2.

Theorem 4

There exists a well-defined homomorphism

equation(11)

called the Hopf invariant. The homomorphism (11) is invariant with respect to the action (7) by an arbitrary permutation, which in the kernel of θ2 in (10) (this homomorphism is defined as the sign of a permutation of straights), and is skew-invariant with respect to the action by a permutation, which is not in the kernel of θ2.

Definition of the hopf invariant

In this section we present the construction, which is closed to Theorem 3 of Mitchell A Berger [3], using differential topology instead of homology algebra. Let f ∈ Brun4 be an arbitrary. Consider the braid fnorm, given by Mitchell A Berger [3]. Recall, the braid g ∈ Br3, which consists of the straits (1-3) of fnorm, is the constant braid at the points 0, 1, ∞ in correspondingly. Consider the strait (4) of the braid fnorm. This strait is represented by an oriented closed path 1 equation. This path determines a cycle, which is an oriented boundary, because of the condition LK(fnorm=0.

Let us prove that Lk(f)=0. Denote the group of Mobius transformations by M. The standard inclusion SO(3) ⊂ M is welldefined. This inclusion is a homotopy equivalence, therefore we getequation This proves that LK(2fnorm)=LK(2f). Because LK(2fnorm)=2LK(2fnorm), LK(2f)=2Lk(f), we get LK(fnorm)=Lk(f). The equality Lk(f)=0 is proved.

Consider the inclusions

equation

equation

equation

Because 1 1 equation the condition LK(fnorm)=0 implies I0,#([i])=0, for the homomorphism

equation

Analogously I∞,#([i])=0, I1,#([i])=0.

There exist the following 3 maps of the standard 2-disk

equation

equation

equation

Consider a 2-sphere, which is represented by a gluing 2 2 equation of the disks equation along the common boundary, which is identified with the circle equation . Denote this sphere byequation Analogously define spheresequation,equationBecause the target spaces of the mappings e0, e, e1 are aspherical, the corresponding mapping is well-defined up to homotopy.

Consider the following commutative diagram of inclusions:

equation(12)

Consider the mappings equationequation to the left bottom and to the right upper spaces of the diagram (12) correspondingly. The mapping 2 equation is well defined by gluing of the two mappings e0, e1 along the common mapping i of the boundaries. Consider the standard 3-ballequation (with corners along the curve equation with the boundaryequation.The mapping e0 e1 can be extended to the mapping

equation(13)

The target space of this mapping is the right bottom space of the diagram (12). Because the target space of the mapping d is contractible, the mapping d∞ is well-defined up to homotopy. By the analogous constructions the following mappings

equation(14)

equation(15)

are well-defined.

The mappings (13), (14), (15) determine the mapping

equation(16)

as follows. Take a 3-sphere S3, which is catted into 3 balls equation along the common circleequation The sphere S3 is represented as the join equationof the two standard circle. On the circleequation take 3 points equation The subsetsequation,equation,equation are 3 copies of 3D disks, which are glued along corresponding subdomains in its boundaries.

Let us identifyequation,equation,equation The boundary equationis identified with the balls equation equation which are glued along the common boundary equation . The boundary equation is identified with the balls equation,equation , which are identify along the common boundary equation . The boundary equation is identified with the ballsequation,equation , which are identified along the same boundary equation . The mappings d0,d1, don the corresponded balls are well-defined by the formulas (13-15) correspondingly. This mappings define the mapping (16) on the 3-sphere.

Definition 5: The Hopf invarian H(f) for a braid f ∈ Brunn4 in the formula (11) is defined as the Hopf invariant of the mapping h by the formula (1). The mapping h=h(f) is explicitly defined from the braid f by the formula (16).

A formula to calculate the Hopf invariant

Let us introduces an explicit formula to calculate the Hopf invariant for a braid f ∈ Brunn4 .Consider the complex plane C. The 4-th strain of the braid fnorm determines a curve on the plane without two points {0.1}, which is denoted by

equation(16)

Let us consider the closed 1-form (4). Define a complex 1-form

equation(17)

Define a real (multivalued) function λ0 by integration along the path γ (t), t ∈[0,t] ⊂ S1 of the real part of the form (18) as following:

equation(18)

Define a real (multivalued) function λ1 by integration along the path of the real part of the form (17) as following:

equation(19)

To take the multivalued functions (18), (19) well-defined, assume that the path γ starts at the point equation : λ0(0)=2, λ1(0)=2.

Define a closed 1-form Ψ (t) along a curve equation by the following formula:

equation(20)

Let us consider a function, which is well-defined as the real part of the integral

equation(21)

Theorem 6

The Hopf invariant of a braid f ∈ Brunn4 in the formula (11), which is defined by Definition 5, is calculated by the formula:

equation(22)

where γ is the closed path, determined by the 4-th straight of the braid fnorm by the formula (17).

From Theorem 6 we get a corollary.

Corollary 7

1. The Hopf invariant (11) is an epimorphism

2. Assume there is a braid f ∈ Brunn4 for which the braid fnorm is represented by a commutator of the straight (4) with straights (1) and (2) (such a braid is called the Borromean rings). Then H(f)=±1, where the sign in the formula depends on the sign of the commutator.

Proof of corollary

It is sufficient to prove --2. The right-hand side of the formula (21) coincides with the formula (28) [1], which is simplified for the considered example. The Berger’s formula is applied for the 3-uple configuration space, this gives the opposite sign for the last term in the formula (21) with respect to the origin formula. For the Borromean ring the formula (22) is non-trivial. The right side of the formula gives H(f)=1 for the right Borromean rings. Corollary is proved.

The following Corollary is the main result of the paper. The author hope that this result is the initial step toward the solution of the first problem, mentioned in Introduction.

Corollary 8

Assume we have a classical -conponent colored non-ordered braid F, equation, for which all pairwise winding (integer) numbers of components are distributed to the segment:equation, equation Let G is the spherical braid, which is defined as the image of F by the stereography projection equation. Then there exist at least

equation( equation whenequation) 4-component

subbraids fi ⊂ F, for which LK(gi)=0, gi ⊂ G,. In particular, the squares equation are well-defined.

Proof of corollary

Proof is evident: the number K of subbraids gi ⊂ F with trivial total linking number LK(gi)=0 is explicitly estimated from below using integers k, n.

Proofs

Proof of lemma

Proof of Statement 1. Take an oriented 3--manifold M3. Take two disjoin oriented cycles CI ⊂ M3, CII ⊂ M3, which represent the trivial homology class

equation(23)

The linking number link (LI, LII) is a well-defined integer the algebraic intersection coefficient of the boundaryequation.. The linking number link (CI, CII) is well defined, because of the condition.

Obviously, link (CI, CII)= link (CII,CI,), because the collections of signed points =equationandequation represent the same cycle [AI]=[AII] ∈ H0(M3; Z). The boundary of -[AI] ∪[AII] is given by the oriented curveequation .

Take M3=S2 ×S1. Take an arbitrary braid fnorm. The cycle CI is represented by the images of the following two closed paths [z1(t)]=0 ×- S1, [z3(t)]=∞ ×S1, t ∈[0,2π], where the path z1(t) is taken with the opposite orientation along S1. The cycle CII is represented by the two closed paths [z2(t)]=0 × -S1, [z4 (t)] ⊂ S2 × S1, where the path z2(t) is taken with the opposite orientation along S1.

Take σα=(1,2)(3,4), σα is the generator of Ker(θ). It is easy to see that Lk(fnorm)=link(CI, CII), Lk(σα × fnorm)=link(CII, CI). Statement 1 is proved.

Proof of Statement 2. Assume σb=(1,3), the case σb=(2,4) is analogous. Then Lk(fnorm)=link(CI, CII), Lk(σb × fnorm)=link(-CI, CII). Therefore we get Lk(fnorm)=-Lk(σb × fnorm). Statement 2 is proved.

Lemma 1 is proved.

Proof of Corollary 2

Statements 1,2 are obvious

Proof of Statement 3. The straights {z1(t), z2(t), z3(t), z4(t)} determines 6 pairs of -cycles in S2 × S1. A function of winding numbers of component is given by a linear combinations of linking numbers between the corresponding pairs of cycles. To prove that such a function is well-defined, we have to assume that the each cycle is a boundary. Denote the cycle, generated by the pair of paths –zi (t), zj (t) by Ci,j. We have the following identity: link (C1,2, C3,4)+link (C2,3, C1,4)+link (C3,1, C2,4x)=0, and the analogous 3 identities, which are obtained by the permutation of the indexes. Therefore we get a collection of 2 independent well-defined linking numbers. Statement is proved. Corollary 2 is proved.

Proof of theorem

Let us prove that the homomorphism (11) is skew-invariant with respect to the action (7) by an odd permutation. Assume that the permutation σ is given by an elementary transposition of straights with number (1-3) say by the transposition σ=(1,2). Then by the formula (16), the mappings h(f) is related with the mapping h(σ×f) by the composition with the reflectionS3→S3, which translates the curve equation ⊂ S3 to itself, and permutes the points x0, x1 on the circle equation . The reflection changes the homotopy class of h to the opposite. This proves Theorem in this case.

Assume that σ=(1, 4) (the cases σ=(2, 4), or (3,4) are analogous). Then we may calculate the Hopf invariants of the mappings h(f) and h(σ×f), using the formula (22) (Theorem 6 is proved below). The Mobius group is locally contractible. Therefore, the ordered braid (σ×f) is isotope to the braid f in which the components (1,4) are renumbered. The restriction of the considered isotopy on the common straight at ∞∈ C is the identity.

By Statement 2 of Corollary 7, the Hopf invariant h(σ×fnorm) is defined as the length of commutators of the straight (1) with straights (4) and (2).

The Hopf invariant for h(f) coincides with the commutator of the straight (4) with the straights (1,4). Therefore the Hopf invariant for h(σ×f) is opposite to the Hopf invariant for h(f), because the sign of the commutator is changed by a permutation of components.

Theorem 4 is proved.

Proof of Theorem 6

Consider the mapping equation, which is defined by the formula (16). Take two normalized volume forms 2 2 equation):

equation

The forms Ω0, Ω1 are defined as the standard ill-supported forms at the points 0, 1, correspondingly. The Hopf invariant (11) is calculated by the formula:

equation(24)

where x ∈ S3,equation is the pull-back of equation by h: S3→S2, β0 ∈ Λ1 (S3) is an arbitrary 1-form, such that d(β0)=h *(Ω0), the 1-form β1 ∈ Λ1(S3) is defined analogously to β0.

Evidently, the 1-forms β0 in the integral (24) is represented in its cohomology class by a cocycle, which satisfies the condition h*(Ω0)=dβ0.=0 inside the ball 0 D . This follows from the fact that the curve h-1(0) is outside the ball equation . In the formula (24) the first term is well-defined up to gauge transformation equation. We may put β0=0 in equation , and keep β0 on equation

Analogously, dβ1=0 in the ballequation . In the second term in the integral (24), using equation , we get β1=0 in equation , and keepβ1 on equation . Then we get the following simplification of (24):

equation

In the ball equation the 3-form equation is exact, we get equation

equation . Moreover, we may put equation over equation equation, overequation

In theequation ballthe 3-form equation is exact, we get equation, equation . We may put equation over equation , and α0=0 overequation .

Apply the 3D Gauss-Ostrogradsky formula, we get

equation

equation

The 2-form equation is exact. Because in the disk equation the 0-form equation is well defined, and equation, we get: equation

Analogously, the 2-form equation is exact. Because in the disk equation the 0-form equation is well defined, and equation, we get: equation.

Apply the 2D Green formula (singular points of β0, β1 give no contribution to the integral over the boundary) we get:

equation

equation

whereequation

The integral (24) is simplified as

equation

The integral (24) is simplified as

This formula coincides with the formula (22). Theorem 6 is proved.

Acknowledgements

The author is grateful to V.P.Leksin for the explication of the results of the paper [1], and to S.A.Melikhov for discussions. The results was presented at International Conference “Nonlinear Equations and Complex Analysis” in Russia (Bashkortostan, Bannoe Lake) during the period since March 18 (arrival day) till March 22 (departure day), 2013. The author was supported in part by Russian Foundation of Basic Research Grant No. 11-01-00822.

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