A remark on the Hopf invariant for spherical 4-braids

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


Introduction
An approach by J.Wu describes homotopy groups π n (S 2 ) of the standard 2sphere as isotopy classes of spherical n + 1-strand Brunnian braids, for more details, see f.ex. [B-M-V-W], Sequence (1.1). This approach is not possible for n = 3 in the case of 4-strand braids.
The homotopy group π 3 (S 2 ) in an infinite cyclic group, detected by the Hopf invariant H : π 3 (S 2 ) → Z. (1) An element of π 3 (S 2 ) is represented by a mapping l : S 3 → S 2 , which is considered up to homotopy. The Hopf invariant H(l) is well-defined as the integer linking number of two oriented curves l −1 (a), l −1 (b), where a, b ∈ S 2 be a pair of regular points of l. The Hopf invariant is very important for applications. The goal of the paper is to modify the definition of Brunnian spherical 4-component braids, and to define the Hopf invariant as a function of isotopy classes of spherical braids, which are Brunnian in a new strong sense. The Hamiltonian provides an elegant method for generating simple geometrical examples of complicated braids and links, as is presented in [B].
Let us formulate the following problems: • Derive applications of higher-order winding numbers to generate Hamiltonian motion of 4 vortex in two dimensions on the sphere. For 3 vortex on the plane this is done in [B].
• To investigate integer lifts of the generator of π 4 (S 2 ) (the Arf invariant) by the Wu's approach.
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 well-defined. This is a secondorder 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. Main results are formulated in Theorems 4, 7. In Section 4 we give proofs of the main results.
In a private letter (August 2013) prof. Viktor Ginzbugr (about a draft of the paper): "'The subject is certainly interesting..."'. I am grateful to him for the interest.
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.

Linking numbers for spherical braids
By a spherical (ordered) n-braid we mean a collection of embeddings of the standard circles f : 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 S 1 i , i = 1, . . . , n is the identity mapping S 1 i → S 1 . The space 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 g = g(f ) : S 1 1 ∪ S 1 2 ∪ S 1 3 ⊂ S 2 × S 1 , the 3-component braid, obtained from f by eliminating of the last component S 1 4 . Let us identify the sphere S 2 with the Riemann sphere, or with the complex projective lineĈ. For a braid f let us consider the collection of Möbius 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 The 3-strand braid g, constructed from f norm is the constant braid at the points {0, 1, ∞}. The last component f norm For a given (ordered) 4-component braid f let us define the linking number Lk(f ), (3) Consider the following 1-form By definition we get where log(z) is given by the formula: assuming that log(1) = 0, as a multivalued complex function.
Define Lk(f ) by the formula: where ℜ 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) with respect to the origin and the infinity inĈ. The permutation group Σ(4) of the order 24 acts on the space of ordered spherical braids: The image of an ordered braid f by a transposition σ : (1, 2, 3, 4) → (σ 1 , σ 2 , σ 3 , σ 4 ) 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 (5). Simply say, we investigate how many independent linking numbers of components of braids are well-defined?
Lemma 1. -1. The function (3) is invariant with respect to the action (5) (the re-numbering of components) by an arbitrary permutation, which in the kernel of θ in (8), and is skew-invariant for 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).
From Lemma one may deduce the following corollary.
Corollary 2. -1. For an arbitrary braid f ∈ Br 4 the linking number Lk(f ) is well-defined 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 braidf ∈ Br 4 , where f ∈ Br 4 is an arbitrary,f is defined in Lemma 1, the linking number Lk(f ) is well-defined as 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.
-3. An arbitrary well-defined homomorphism Br 4 → Z, which is a function of the windings numbers between components is a linear combination of Lk(f ) and Lk(f ).
Definition 3. Let f ∈ Br 4 be a (ordered) spherical braid. Define the total linking number LK(f ) ∈ Z ⊕ Z by the following formula: The total linking number is a well-defined homomorphism LK : Br 4 → Z ⊕ Z.

Hopf invariant of braids
called the Hopf invariant. The homomorphism (9) is invariant with respect to the action (5) by an arbitrary permutation, which in the kernel of θ in (8), and 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).
Remark 5. Higher homotopy groups are described from the spherical braids groups with non-ordered components in the [B-M-V-W], Sequence (1.1).

Definition of the Hopf invariant
Let f ∈ Brun 4 be an arbitrary. Consider the braid f norm , given by (2). Recal, for the braid f norm the braid g ∈ Br 3 , which consists of the straits (1),(2),(3) of f norm , is the constant braid at the points 0, 1, ∞ inĈ correspondingly. Consider the strait (4) of the braid f norm . This strait is represented by an oriented closed path i : S 1 →Ĉ \ {0 ∪ 1 ∪ ∞}. This path determines a cycle, which is an oriented boundary, because of the condition LK(f norm ) = 0. (Evidently, LK(f norm ) = LK(f ), because the group of Möbius transformations is connected.) Consider the inclusions There exist the following 3 maps of copies of the standard 2-disk e 0 : D 2 0 →Ĉ \ {1 ∪ ∞}, e 0 | ∂D 2 = i, e ∞ : D 2 ∞ →Ĉ \ {0 ∪ 1}, e ∞ | ∂D 2 = i, e 1 : D 2 1 →Ĉ \ {0 ∪ ∞}, e 1 | ∂D 2 = i. Consider a 2-sphere, which is represented by a gluing D 2 0 ∪ ∂ D 2 ∞ of the disks D 2 0 , D 2 ∞ along the common boundary, which is identified with the circle S 1 4 . Denote this sphere by S 2 1 . Analogously define spheres S 2 0 = D 2 ∞ ∪ ∂ D 2 1 , S 2 ∞ = D 2 1 ∪ ∂ D 2 0 . Consider the following commutative diagram of inclusions: Consider the mappings e 0 : D 2 0 →Ĉ\{1∪∞}, e 1 : D 2 1 →Ĉ\{0∪∞} to the left bottom and to the right upper spaces of the diagram (10) correspondingly. The mapping e 0 ∪ ∂ e 1 : S 2 ∞ →Ĉ \ {∞} 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 D 3 ∞ (with corners along the curve S 1 4 ) with the boundary ∂D 3 ∞ = S 2 ∞ . The mapping e 0 ∪ ∂ e 1 can be extended to the mapping The target space of this mapping is the right bottom space of the diagram (10). 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 are well-defined. The mappings (11), (12), (13) determine the mapping braid f norm determines a curve on the plane without two points {0, 1}, which will be denoted by Let us consider the complex 1-form (4). Define a complex 1-form Define a real (multivalued) function λ 0 by integration along the path γ(t), t ∈ [0, t] ⊂ S 1 of the real part of the form (4) as following: Define a real (multivalued) function λ 1 by integration along the path of the real part of the form (16) as following: To take the multivalued functions (17), (18) well-defined, assume that the path γ starts at the point 2 ∈ C: λ 0 (0) = 2, λ 1 (0) = 2. Define a closed 1-form ψ(t) along a curve γ(t) ∈ C\{0∪1} by the following formula: Let us consider a function, which is well-defined as the real part of the integral Theorem 7. The Hopf invariant of a braid f ∈ Brun 4 in the formula (9), which is defined by Definition 6, is calculated by the formula: where γ is the closed path, determined by the 4-th straight S 1 4 of the braid f norm by the formula (15). From Theorem 7 we get a corollary.
-2. Assume that a braid f ∈ Brunn 4 is such that the braid f norm 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.