Medical, Pharma, Engineering, Science, Technology and Business

**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

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

An approach by Wu describes homotopy groups of the standard 2-sphere as isotopy classes of spherical -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 π_{3}(S^{2}) was unknown.

The homotopy group π_{3}(S^{2}) in an infinite cyclic group, detected by the Hopf invariant

(1)

An element of π_{3}(S^{2}) is represented by a mapping , 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),*

Proposition 7.1.1, sequence (17) gets an exact sequence, which algebraically describes the group *Brunn*_{4} of 4-straight Brunnian braids [1]. The key point of our elementary geometrical construction is to construct an alternative epimorphism onto the group , see Definition. The kernel of this epimorphism is a well-defined subgroup *Brun*_{4}⊂ *Br _{4}*of Brunnian braids in a new sense (let us remark that Brunn4 is not a subgroup of

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*(S^{2}) with the Wu’s approach (Problem 2) [4].

Let us clarify Problem 2. Let F be the space of functions 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,) , by

the formula

V.I.Arnold (1996) conjectured that the induced homomorphism is an isomorphism forThis 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, 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

(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*.

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

where the composition of this embedding with the standard projection S^{2} × S^{1} → S^{1} on the second factor in the target space, restricted to an arbitrary component 1 , i=1,…,n is the identity mapping.

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

For a fixed value t ∈ S^{1}, a braid f ∈ Br_{n} intersects the level S^{2} × t by an (ordered) collection of n points {Z_{1}(t),… Z_{n}(t)}. Let assume that *n*=4. Denote by

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

component Let us identify the sphere S^{2} 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 z_{1}, z_{2}, z_{3} into 0, 1, ∞ correspondingly:

The image *F(f)* is a 4-strand braid with the constant components {z_{1}(t), z_{2}(t), z_{3}(t)}={0, 1, ∞}. Denote this braid by

(3)

The 3-strand braid g, constructed from fnorm is the constant in the points {0, 1, ∞}. The last component f^{norm}(S^{1}) of *F(f)* is represented by a closed path Note that, generally speaking, braids *f, f ^{norm}* are not isotopic. Moreover, if

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

(4)

Consider the following 1-form

(5)

By definition we get

(5)

where log (z) is given by the formula:

(6)

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

Define *Lk(f) *by the formula:

(6)

where 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 z_{4}(t) of* f ^{norm}*with respect to the origin and the infinity in

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

(7)

The image of an ordered braid f by a transposition 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:

(8)

(9)

The subgroup A_{4} ⊂ Σ_{4} in the sequence (8) is represented by permutations, which preserve signs (equivalently, which is decomposed into an even number of elementary transpositions). The subgroup in 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 ⊂ Σ_{4} (the dihedral group of the order 8), which is defined as the extension of the subgroup from the sequence (9), which is included in the sequence (8). An epimorphism

(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 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 ⊂ Σ_{4} . The kernel 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 θ_{1} (the composition of θ with the projection on the first factor, but not in the kernel of θ_{2} (the composition of θ with the projection on the second factor).

3. Denote by the ordered braid, which is obtained from f ∈ Br_{4} by the action (7) by the element (1,2) is the product of the generators of the factors). There exists an ordered braids f ∈ Br_{4}, for which the linking numbers *Lk(f)*, are arbitrary integers.

From Lemma one may deduce the following corollary.

**Corollary 2**

1. For an arbitrary braid f ⊂Br_{4}the 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 ∈Br_{4}is an arbitrary, is defined in Lemma 1, the linking number 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 ∈ Br_{4}be a (ordered) spherical braid. Define the total linking number by the following formula:

The total linking number is a well-defined homomorphism

Let *f ∈ Br _{4}* be a (ordered) spherical braid with the trivial total linking number:

**Theorem 4**

*There exists a well-defined homomorphism*

(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 θ_{2} 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 θ_{2}.

**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 ∈ Brun _{4}* be an arbitrary. Consider the braid

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 get This proves that LK(2*f ^{norm}*)=LK(2f). Because LK(2

Consider the inclusions

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

Analogously I∞,#([*i*])=0, I_{1},#([*i*])=0.

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

Consider a 2-sphere, which is represented by a gluing 2 2 of the disks along the common boundary, which is identified with the circle . Denote this sphere by Analogously define spheres,Because the target spaces of the mappings e_{0}, e_{∞}, e_{1} are aspherical, the corresponding mapping is well-defined up to homotopy.

Consider the following commutative diagram of inclusions:

(12)

Consider the mappings to the left bottom and to the right upper spaces of the diagram (12) correspondingly. The mapping 2 is well defined by gluing of the two mappings e_{0}, e_{1} along the common mapping* i* of the boundaries. Consider the standard 3-ball (with corners along the curve with the boundary.The mapping e_{0} ∪_{∂} e_{1} can be extended to the mapping

(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

(14)

(15)

are well-defined.

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

(16)

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

Let us identify,, The boundary is identified with the balls which are glued along the common boundary . The boundary is identified with the balls , , which are identify along the common boundary . The boundary is identified with the balls, , which are identified along the same boundary . The mappings *d _{0},d_{1}, d_{∞}*on 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 ∈ Brunn_{4} 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 ∈ Brunn_{4} .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

(16)

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

(17)

Define a real (multivalued) function *λ0* by integration along the path γ (t), t ∈[0,t] ⊂ S^{1} of the real part of the form (18) as following:

(18)

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

(19)

To take the multivalued functions (18), (19) well-defined, assume that the path γ starts at the point : λ_{0}(0)=2, λ_{1}(0)=2.

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

(20)

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

(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:*

(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 ∈ Brunn_{4} 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, , for which all pairwise winding (integer) numbers of components are distributed to the segment:, Let G is the spherical braid, which is defined as the image of F by the stereography projection . Then there exist at least*

( when) 4-component

*subbraids f _{i} ⊂ F, for which LK(g_{i})=0, g_{i} ⊂ G,. In particular, the squares are well-defined.*

**Proof of corollary**

**Proof is evident: **the number *K* of subbraids *gi ⊂ F* with trivial total linking number* LK(g _{i})=0* is explicitly estimated from below using integers

**Proof of lemma**

Proof of Statement 1. Take an oriented 3--manifold M^{3}. Take two disjoin oriented cycles C_{I} ⊂ M^{3}, C_{II} ⊂ M^{3}, which represent the trivial homology class

(23)

The linking number *link (L _{I}, L_{II})* is a well-defined integer the algebraic intersection coefficient of the boundary.. The linking number

Obviously, *link (C _{I}, C_{II})*=

Take M^{3}=S^{2} ×S^{1}. Take an arbitrary braid f^{norm}. The cycle C_{I} is represented by the images of the following two closed paths* [z _{1}(t)]=0 ×- S^{1}, [z_{3}(t)]=∞ ×S1, t ∈[0,2π]*, where the path z

Take σα=(1,2)(3,4), σα is the generator of Ker(θ). It is easy to see that Lk(f^{norm})=link(C_{I}, C_{II}), Lk(σα × f^{norm})=link(C_{II}, C_{I}). Statement 1 is proved.

Proof of Statement 2. Assume σ_{b}=(1,3), the case σ^{b}=(2,4) is analogous. Then Lk(f^{norm})=link(C_{I}, C_{II}), Lk(σ_{b} × f^{norm})=link(-C_{I}, C_{II}). Therefore we get Lk(f^{norm})=-Lk(σ_{b} × f^{norm}). Statement 2 is proved.

Lemma 1 is proved.

**Proof of Corollary 2**

Statements 1,2 are obvious

Proof of Statement 3. The straights {z_{1}(t), z_{2}(t), z_{3}(t), z_{4}(t)} determines 6 pairs of -cycles in S_{2} × S_{1}. 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 –z_{i} (t), z_{j} (t) by C_{i,j}. We have the following identity: link (C_{1,2}, C_{3,4})+link (C_{2,3}, C_{1,4})+link (C_{3,1}, C_{2,4}x)=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 reflectionS^{3}→S^{3}, which translates the curve ⊂ S^{3} to itself, and permutes the points x0, x1 on the circle . 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×f ^{norm})* 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 , which is defined by the formula (16). Take two normalized volume forms 2 2 ):

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

(24)

where x ∈ S^{3}, is the pull-back of by h: S^{3}→S^{2}, β_{0} ∈ Λ^{1} (S_{3}) is an arbitrary 1-form, such that *d(β _{0})=h *(Ω_{0})*, the 1-form

Evidently, the 1-forms * β _{0}* in the integral (24) is represented in its cohomology class by a cocycle, which satisfies the condition h*(

Analogously, d*β _{1}*=0 in the ball . In the second term in the integral (24), using , we get

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

. Moreover, we may put over , over

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

Apply the 3D Gauss-Ostrogradsky formula, we get

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

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

Apply the 2D Green formula (singular points of *β _{0}, β_{1}* give no contribution to the integral over the boundary) we get:

where

The integral (24) is simplified as

The integral (24) is simplified as

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

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.

- JABerrick, FR Cohen, YL Wong, J Wu(2006) Configurations, braids, and homotopy groups. JAmer Math Soc19: 265-326.
- Graham Ellis, Roman Mikhailov(2010) A colimit of classifying spaces, Advances in Math, 223: 2097-2113; arXiv: 0804.3581.
- Mitchell A Berger(2001) Hamiltonian dynamics generated by Vassiliev invariants, J. Phys. A: Math. Gen. 34: 1363-1374.
- VAVassiliev(1994) Complements of discriminants of smooth maps: topology and applications, 2-d extended edition, Translations of Math. Monographs, 98, AMS, Providence, RI: 268.

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