On a continuation of quaternionic and octonionic logarithm along curves and the winding number

This paper focuses on the problem of finding a continuous extension of the hypercomplex logarithm along a path. While a branch of the complex logarithm can be defined in a small open neighbourhood of a strictly negative real point, no continuous branch of the hypercomplex logarithm can be defined in any open set $A\subset \mathbb K\setminus \{0\}$ which contains a strictly negative real point $x_0$ (here $\mathbb K$ represents the algebra of quaternions or octonions). To overcome these difficulties, we introduced the logarithmic manifold $\mathscr E_\mathbb K^+$ and then showed that if $q\in\mathbb K,\ q=x+Iy$ then $E(x+Iy) %= (\exp (x + Iy), Iy) = (\exp x \cos y + I\exp x \sin y, Iy)$ is an immersion and a diffeomorphism between $\mathbb K$ and $\mathscr E_\mathbb K^+$. In this paper, we consider lifts of paths in $\mathbb K\setminus\{0\}$ to the logarithmic manifold $\mathscr{E}^+_\mathbb K$; even though $\mathbb K \setminus \{0\}$ is simply connected, in general, given a path in $\mathbb K \setminus \{0\}$, the existence of a lift of this path to $\mathscr{E}^+_\mathbb K$ is not guaranteed. There is an obvious equivalence between the problem of lifting a path in $\mathbb K \setminus \{0\}$ and the one of finding a continuation of the hypercomplex logarithm $\log_{\mathbb K}$ along this path.


INTRODUCTION
This paper focuses on the problem of finding a continuous extension of the hypercomplex logarithm along a path.As pointed out in [GPV], while a branch of the complex logarithm can be defined in a small open neighborhood of a strictly negative real point, no continuous branch of the hypercomplex logarithm can be defined in any open set A ⊂ K \ {0} which contains a strictly negative real point x 0 (here K represents the algebra of quaternions or of octonions).
To overcome these difficulties, in [GPV] we introduced the logarithmic manifold E + K and then showed that, if q ∈ K, q = x + Iy then E(x + Iy) = (exp x cos y + I exp x sin y, Iy) is an immersion and a diffeomorphism between K and E + K .In this paper, we consider lifts of paths in K \ {0} to the logarithmic manifold E + K ; even though K \ {0} is simply connected, in general, given a path in K \ {0}, the existence of a lift of this path to E + K is not guaranteed.There is an obvious equivalence between the problem of lifting a path in K\{0} and the one of finding a continuation of the hypercomplex logarithm log K along this path.
We want to recall that the slice regular logarithm log * (f ) of a slice regular function f (see [AdF1,GPV1]) over the quaternions or octonions, introduced as the slice regular inverse of the slice regular exponential exp * (f ) of a slice regular function f (see [AdF]), is not defined in general via the lift of f to E + K .In particular it turns out that, in general, log * (f )(q) = log K (f (q)).
The first and third authors were partly supported by INdAM, through: GNSAGA; IN-dAM project "Hypercomplex function theory and applications".It was also partly supported by MIUR, through the projects: Finanziamento Premiale FOE 2014 "Splines for accUrate NumeRics: adaptIve models for Simulation Environments".The second author was partially supported by research program P1-0291 and by research project J1-3005 at Slovenian Research Agency.
The paper is organized as follows: in Sections 2 and 3, after recalling the basic notions on slice regular exponential and logarithmic functions, we provide explicit examples of paths intersecting the real axis and show how a branch of the hypercomplex logarithm can be defined along certain curves even when they encounter the real axis at negative points, providing a so called continuation of the logarithm along a continuous curve.
Furthermore, we introduce the notion of path and of loop with a companion (see Subsection 4.1) and then give a definition of winding number with respect to 0 that has a full meaning for a class of loops in K \ {0} ≃ R 2 s \ {0} (s = 2, 3) with companion; this fact is quite novel and original since it is well known that a definition of winding number for a loop (with respect to a point) is not in general possible in R n when n is greater than 2. Moreover this notion of winding number is invariant for the class of c−homotopic loops with companion.
Finally, in the last Section 5, we extend the continuation of the hypercomplex logarithm to the case curves with an infinite number of intersections with the real axis.These represent the set of obstructions for such an extension.When these obstructions are "mild" and "reasonable", then we also present an effective way to calculate the winding numbers using the so-called notion of signature.

PRELIMINARY RESULTS
We denote by K either the algebra of quaternions or octonions.Let S be the sphere of imaginary units, i.e. the set of I ∈ K such that I 2 = −1.Given any z ∈ K\R, there exist (and are uniquely determined) an imaginary unit I, and two real numbers x, y (with y > 0) such that z = x + Iy.With this notation, the conjugate of z will be z := x − Iy and |z| 2 = zz = zz = x 2 + y 2 .Each imaginary unit I generates (as a real algebra) a copy of a complex plane denoted by C I .We call such a complex plane a slice.The upper half-plane in C I , namely {x + yI : y > 0} will be denoted by C + I .Similarly, the lower half-plane in C I {x + yI : y < 0} will be denoted by C − I ; each of these two half-planes will be called a leaf of C I .On any leaf C + I we define the function arg I : C + I → (0, π) as z = x + Iy ∈ C + I → cot −1 (x/y) := arg I (z).The function arg I can be continuously extended as a function arg I : It is also useful to define the imaginary unit function on K \ R in the following way: if z ∈ C + I , i.e. if z = x + Iy, with x, y ∈ R and y > 0, then I(z) = I; if z ∈ C − I , i.e. if z = x − Iy, with x, y ∈ R and y > 0, then I(z) = −I.
Remark 2.1.It is worthwhile noticing that the function I cannot be extended as a continuous function to any single point of the real axis R of K.At the same time, if we set S(−π, π) = {Iy : I ∈ S, y ∈ (−π, π)}, then the function Arg : K \ (−∞, 0] → S(−π, π) defined as the product Arg(q) := I(q) arg I (q) can be extended (as the zero function) to the positive real axis R + of K.

THE HYPERCOMPLEX EXPONENTIAL AND LOGARITHM
Let us recall that the exponential map on K is a slice regular and slice preserving entire function on K ( [AdF, GSS]).Let E + K = f (K + ) denote the logarithm manifold, i.e., the image T (K + ) of K + = {q ∈ K : Re q > 0} of the map T : K → K × Im(K) defined by T (x + Iy) = (sinh x cos y + I sinh x sin y, Iy) is an immersion and a diffeomorphism between K and E + K (see [GPV]).In the case of quaternions, it endows E + H with a structure of slice quaternionic manifold (see, e.g., [GGS]), which is different from the structure of hypercomplex Riemann manifold defined in Propostion 4.3.[GPV]) The next definition and result appear in [GPV].
K , then q = r exp p for r = |q| and our definition can be rewritten as: L(r exp p, p) = log r + p The hypercomplex manifold E + K plays the role of an "adapted" blow-up of K at points of the form x + 2Ikπ, for k ∈ Z and k = 0.
Proposition 3.2.The map , and a diffeomorphism from the logarithm manifold E + K to K. Note that if pr 1 : K × Im(K) → K denotes the projection on the first factor, then by definition the following equality holds for all q ∈ K. Indeed, the map L is a slice regular map from E + to H, with respect to the structure of slice regular manifold induced by E on E + K (see, e.g., [GGS]).This map allows the definition of the hypercomplex logarithm (see [GPV,GPV1]): ) is called a branch or a determination of the hypercomplex logarithm on pr 1 (Ω).
As one can expect, it holds K , then the projection on the first factor is injective.Therefore, in this way, one defines the principal branch of the logarithm (see [GV]) in pr 1 (Ω) = K \ (−∞, π] (see [GPV1,AdF1]).The principal branch of the hypercomplex logarithm is well defined and, for all I ∈ S, coincides with the principal branch of the complex logarithm in the slice C I \ (−∞, 0].As a consequence, log 0 is a slice regular function in the symmetric slice domain K \ (−∞, 0].
As already observed in the Introduction, despite the analogy with the complex holomorphic case, in general no continuous branch of the hypercomplex logarithm can be defined in any open set A ⊂ K \ {0} which contains a strictly negative real point x 0 .Nevertheless, we will now see how a branch of the hypercomplex logarithm can be defined along certain curves even when they encounter the real axis at negative points, providing a so called continuation of the logarithm along a continuous curve.
Throughout the paper, a continuous curve will be called a path, and a closed path will be called a loop.
The point γ(a) ∈ K will be called the initial point of the continuation γ.
The k-th branch of the hypercomplex argument is defined by setting As a consequence, Arg 2l+1 (q) = −I(q)(2π − arg I (q) + 2lπ) Therefore, the only different branches of the hypercomplex argument of a quaternion q ∈ K \ R, q = x + Iy with y > 0, can be listed for k ∈ Z as Arg 2k (q) := I(q) arg 2k (q).

CONTINUATION OF HYPERCOMPLEX LOGARITHMS ALONG PATHS
The construction of a continuation of the logarithm along a path naturally involves the notion of a lift of a path.
, if the the following diagram commutes: For (q, p) ∈ E + K , a lift Γ of γ such that Γ(a) = (q, p) will be said to have initial point (q, p).
The existence of a lift of a path γ is equivalent to the existence of a continuation of the hypercomplex logarithm along it.
Proposition 4.2.Let γ : [a, b] ⊂ R → K \ {0} be a path.Then, there exists a lift of γ to E + K if, and only if, there exists a continuation of the logarithm along γ.
Proof.Suppose that there exists a continuation of the logarithm γ along γ.Then the path Γ defined by Γ(t) = ((exp • γ)(t), Im( γ(t)) is obviously a lift of γ to E + K .
Thanks to the result just stated, we are left to find conditions under which a path γ : [a, b] → K \ {0} can be lifted to E + K .Since the map pr 1 : E + K → K \ {0} is not a covering, we have to specifically study the existence of lifts of γ.
Let us first consider the easy cases: it is not difficult to see that if we restrict the map pr 1 : E + K → K \ {0} to the preimage of K \ R, then the restriction pr 1|pr 1 −1 (K\R) becomes a covering.Indeed it becomes a trivial covering, since pr 1 −1 (K \ R) is homeomorphic (namely diffeomorphic) through the diffeomorphism to the countable collection of open simply connected domains given by Let us now set, for any k ∈ Z, 2k+1) .With this in mind, we can now state the following proposition.Namely, for all t ∈ [a, b], we have Finally, for all k ∈ Z, the map defined on the interval [a, b] by is the unique continuation of the hypercomplex logarithm along γ with initial point log |γ(a)| + Arg 2k (γ(a)), and is called the k-th branch of the hypercomplex logarithm along γ with initial point log |γ(a)| + Arg 2k (γ(a)).
Proof.For each k ∈ Z, the proof of the existence and uniqueness of Γ k as in the statement is a straightforward consequence of what already established.
To prove the last part of the statement, let us consider the graph Ω k of the lift Under the hypotheses of the preceding proposition, loops lift to loops, hence: Among the initial cases, there is the one corresponding to what is stated in Remark 2.1.
As pointed out in the Introduction, even though K \ {0} is simply connected, in general, given a path in K \ {0}, the existence of a lift of this path to E + K is not guaranteed.Indeed, consider the following examples.Example 4.6.
(a) Let σ : [π/2, 3π/2] → K\{0} be the path (depicted in Figure 1) defined by σ(t) = cos(t) + I(t) sin(t) where I : [π/2, 3π/2] → S is defined as The curve σ is continuous, but the function is not continuous at π (the left and right limits are different).Therefore σ cannot be lifted to where i, j are the usual orthogonal imaginary units (see Figure 2).Notice that the imaginary part of γ is continuous at all points of the interval [0, 1] (including 0).Nevertheless, for t near to 0, we have that the function has no limit for t approaching 0 + .Therefore γ cannot be lifted to E + K .
(c) Notice that in both the preceding cases, the paths σ := −σ and γ := −γ can be lifted to E + K , since their images are included in K \ (−∞, 0] (see Proposition 4.5).Complex slices {C J } J∈S are naturally parameterized by the elements of S/{± Id}, the real projective space is the classical 2 : 1 universal covering map and, as customary, for J ∈ S, the symbol [J] denotes the equivalence class whose representatives are the opposite (conjugate) imaginary units J, −J ∈ S. Each element [J] ∈ S/{± Id} uniquely defines the complex slice C If a companion I γ of the path γ exists, then γ is called a path with a companion and the pair (γ, I γ ) is called a path with companion I γ .
If the path γ has a unique companion I γ , then both γ and the pair (γ, I γ ) are called a tame path.
These last expressions are called canonical forms of (γ, I γ ).
Proof.Since γ(t) and I γ (t) are both continuous, then Re(γ(t)) = x(t) and It is easy to see that all paths lying entirely in a complex slice have a companion.Notice as well that a path γ : [a, b] → K \ {0} may have more than one companion: this happens for example when the path γ is such that γ([a, b]) ⊂ (0, ∞); in this case, for an arbitrary path [a, b] to a real number has more than one companion, and hence is not tame.
Remark 4.9.There exist paths in K \ {0} which can be lifted to E + K , but have no companion.Indeed, set where σ is the path defined in Example 4.6 (a).The path σ is the symmetric image of the path σ with respect to the plane of purely imaginary quaternions (see figure 1) and, as pointed out in Example 4.6 (c), it can be lifted to E + K .Obviously σ has no companion: the continuity of a companion cannot hold at t = π.
The following definition will play a central role in the sequel.
be the canonical forms of (γ, I γ ).The paths γ are called the (two conjugated) shadows associated with the pair (γ, I γ ).If the path γ is tame, then the paths γ I γ and γ −I γ are simply called the (two) shadows associated with the path γ.
Remark 4.11.The two shadows associated with the pair (γ, I γ ) are conjugate paths.
Paths with a companion are of interest because they can all be lifted to Then, the map defined on the interval [a, b] by Proof.The proof is a straightforward consequence of Proposition 4.12 and Proposition 4.2.
We will now turn our attention to the case of loops of K \ {0}.
If both γ and I γ are closed, then the path γ is called a loop with companion I γ , and the pair (γ, I γ ) is called a loop with companion.
The loop with companion (γ, In the most relevant case in which γ is tame, we can specialize the definition as follows. If both γ and I γ are closed, then γ is called a tame loop (with companion I γ ), and the pair (γ, I γ ) is called a tame loop.
The tame loop (γ, I γ ) is called untwisted if I γ is homotopic to a constant in S/{± Id}; if instead I γ is not homotopic to a constant, then (γ, I γ ) is said to be twisted.Winding number for untwisted loops with companion in K\{0}.It is well known that the definition of winding number for a loop (with respect to a point) is not natural in R n when n is greater than 2. Nevertheless, in our setting, we can start by giving a definition of winding number that has full meaning for loops with companion that are untwisted and lie in K \ {0}.
The following result opens a way to this definition of winding number.On the other hand, suppose the associated shadow γ is a loop by assumption, we obtain I γ (a) = I γ (b) and so the lift I γ of I γ is a loop.In conclusion, γ is untwisted.
We are now ready to use the well established definition of winding number for complex loops in C \ {0} to define the winding number in the case of untwisted loops with companion in K \ {0}.
Definition 4.19.Let the loop γ : [a, b] → K \ {0} with companion I γ be untwisted.The winding number (with respect to zero) of the loop (γ, I γ ), denoted wind(γ, I γ ), is defined as the absolute value of the winding number (with respect to zero), wind(γ I γ ), of a shadow γ I γ associated with I γ : In the case in which the loop (γ, I γ ) is tame, there is one and only one companion of γ, and hence we can simply denote the winding number of γ by wind(γ).
Of course, we need to show that the given definition of winding number of an untwisted loop with companion (γ, I γ ) does not depend on the choice of the shadow associated with I γ .Indeed, the two shadows associated with I γ are conjugate loops: as a consequence, their winding numbers are opposite.Therefore, Definition 4.19 is consistent.
One of the important features of the classical winding number (with respect to zero) of loops of C \ {0} is its invariance with respect to homotopy between such loops.The winding number of an untwisted loop with companion (in K \ {0}) just defined cannot be invariant with respect to standard homotopy in K \ {0}: all such loops are homotopic to a constant loop since K \ {0} is simply connected, and a constant loop has vanishing winding number.
A special notion of homotopy comes into the scenery in our setting.The next definition is useful to define such a notion.A continuous map If a companion I F of the map F exists, then F is called a continuous map with companion I F , and (F, I F ) is called a continuous map with companion.
If the map F has a unique companion I F , then it is called a tame map.
These last expressions are called canonical forms of (F, I F ).
The following simple result will be helpful in the sequel.(i) the map I F is a homotopy between I γ 1 and I γ 2 ; (ii) the homotopy I F can be lifted to a homotopy I F between a lift I γ 1 of I γ 1 and a lift I γ 2 of I γ 2 in such a way that the canonical form of is a homotopy between the canonical forms of γ 1 and γ 2 , respectively; (iii) the shadows of (γ 1 , I γ 1 ) and (γ 2 , I γ 2 ), respectively, are homotopic in C \ {0}.
Proof.The proofs of (i) and (ii) are a straightforward consequence of Definition 4.22 and of what is stated in Propositions 4.8 and 4.21.Let us prove (iii).To this aim, consider the canonical form of F that appears in (4.3) and the following continuous maps, for We will prove that homotopy between the two given shadows of γ 1 and γ 2 .Indeed, using directly formula (4.3) for the canonical form of F , it is easy to check that on The proof is now complete.
Example 4.24.To better illustrate the major difference between complex and quaternionic cases, consider the complex curve, defined by As a complex curve, i.e. with the constant companion i, the curve γ has winding number 0 and coincides with its own shadow.As a quaternionic curve, γ has a large family of companions I γ ; for example one can consider- and I γ (t) = [I γ (t)], where J : [π, 3π] → S is an arbitrary continuous curve with J (π) = i and J (3π) = −i.Correspondingly, the shadow of (γ, I γ ) is and so the winding number of (γ, I γ ) is 1.The pairs (γ, i) and (γ, I γ ) are not c-homotopic.
The notion of c-homotopy is particularly useful in this setting, because of the following result.
Proof.Suppose first that (2) holds.Then there exist: shadow of γ 1 and a shadow of γ 2 (its "conjugate" being a homotopy between the corresponding conjugate shadows).In this situation, the map is a homotopy between γ 1 and γ 2 .Indeed, F is obviously continuous, and such that, for all t ∈ [a, b] and all s ∈ [0, 1], Moreover, the continuous map G : [a, b] × [0, 1] → S defines, by construction, a companion of F given by Let us now suppose that (1) holds, i.e. that (γ 1 , I γ 1 ) and (γ 2 , I γ 2 ) are c-homotopic.In this case I γ 1 and I γ 2 are homotopic by definition, and the rest of the assertion follows from Proposition 4.23.Proof.If (γ, I γ ) is untwisted, then any lift I γ of the companion I γ with initial point I γ (a) is homotopic in S to the constant loop I γ (a), and therefore the loop γ is c-homotopic to its (closed) shadow in C I γ (a) (see Proposition 4.18).On the other hand, if the loop with companion (γ, I γ ) is c-homotopic to its shadow, then the lift of its companion I γ with initial point I γ (a) has to be homotopic in S to the constant loop I γ (a).As a consequence the loop (γ, I γ ) is untwisted by definition.
The notion of c-homotopy is suitable to comply with the meaning of the winding number of loops in the setting of K \ {0}.In this panorama, all untwisted tame loops play a special role: any such a loop has an "intrinsically defined" winding number that depends only on its geometric properties.Indeed, we can state the following result.Proof.By Proposition 4.25, γ and δ are c-homotopic if, and only if, the unique companions I γ and I δ are homotopic and a shadow of (γ, I γ ) is homotopic to a shadow of (δ, I δ ), in C \ {0}.According to Definition 4.19, the winding number of (γ, I γ ) (or (δ, I δ )) is defined as the absolute value of the winding number of one of the two (closed) shadows of (γ, I γ ) (or (δ, I δ )).Therefore the proof is a straightforward consequence of the properties of the fundamental group Π 1 (C \ {0}) ≡ Z, where the class of each loop is determined by its winding number (with respect to zero).
The given definition of winding number, which has particularly transparent geometrical meanings, cannot be adopted as it is in the twisted case, due to the two following results.Proof.The proof follows immediately from Proposition 4.28.
We might be encouraged to think that, in the case of a loop with companion which is twisted, we should first parameterise the loop in such a way that it has real endpoints (see Proposition 4.17), and then use Definition 4.19.Indeed, this approach gives a weird result, if tested, for instance, in the case of the twisted, tame loop λ presented in the next example.
Example 4.30.Consider the loop λ in the hyperplane of H generated by the orthogonal units {1, i, j}.The path consists of several arcs: the arc of parabola t+1+t 2 (i+j), t ∈ [−1, 1], the segments from (2, 1, 0) to (2, 1, 1), from (2, 1, 0) to (0, 1, 0) and from (0, 0, 1) to (0, 1, 1), the halfcircle cos t + i sin t, t ∈ [π/2, 3π/2] and the quarter of circle i cos t + j sin t, t ∈ [π/2, π].Let the orientation be such that it coincides with the positive orientation of the halfcircle part in the plane containing 1, i.The path intersects the real axis at points z = 1 and z = −1.In the previous example, the proposed winding number of the twisted, tame loop λ would be 1 if the loop is parameterised with real endpoints equal to 1 ∈ R (see Figure 3 (b)).On the other hand, the same loop λ parameterised with endpoints equal to −1 ∈ R would have winding number 0 (see Figure 3 (c)).What we just illustrated clarifies that a notion of winding number for twisted, tame loops in K \ {0} (if it exists) has to be given by following a different approach.
In the spirit of the above example and Proposition 4.28 the definition of the winding number for a closed tame twisted loop γ : [a, b] → K \ {0} cannot be given by considering the change of the argument since this depends on the choice of the initial point.
Assume that γ is a twisted loop in K \ {0} which intersects both the positive and the negative real axis; let γ and is not an integer multiple of 2π unless arg γ (a) = 0, π.Even if we set the initial point to be real, so that the change of argument is 2nπ, the number n can have more than one value as shown in the following example.
Example 4.31.Let γ 1 be the positively oriented unit circle with initial point −1 and companion i and define γ 2 (t the loop composed first of m copies of γ 1 followed by a copy of γ 2 .If the initial point is assumed to be the point −1 on the first copy of γ 1 , then the change of the argument is 2πm.If the initial point is the point −1 on the second copy of γ 1 , then the m − 1 copies of γ 1 before γ 2 give the winding number m − 1, but then the curve γ 2 reverses the orientation so the last copy of γ 1 has negative orientation with respect to the unit −i, hence the winding number is m − 2. Starting at −1 on the third copy of γ 1 , would therefore result in the winding number m − 4 and so forth.
A few words seem now appropriate, to present a suggestive geometrical explanation of the reason why the notion of winding number as given in the case of untwisted loops does not work for the case of twisted loops.Indeed, consider an untwisted loop γ : If we regard all points {x(t)} t∈ [a,b] as distinct points except the endpoints, such a γ has values in the surface a loop, and hence it is homotopic to the constant loop I γ (0) = I γ (1).As a consequence, the surface S γ is homeomorphic to a twodimensional cylinder.Therefore there is a notion of γ(t) being a point of this surface lying on one side or the other of the "real axis" formed by the points {x(t)} t∈ [a,b] , and hence a notion of winding number with respect to the origin becomes possible: the situation reduces, naively speaking, to a planar one.If instead γ : [a, b] → K \ {0} is twisted, then the path I γ : [a, b] → S has antipodal endpoints, and the surface S γ turns out to be homeomorphic to a Moebius strip.In this last situation, the lack of orientability seems to exclude the possibility of defining coherently a winding number for the loop γ.

OBSTRUCTIONS TO THE EXISTENCE OF LIFTS OF A PATH
In this section we present sufficient conditions for a path to have a lift, a companion and to be tame.
As already mentioned, if the path γ : It is clear that the necessary assumption for a lift of γ to E + K to exist is the requirement that γ has a lift on a neighbourhood of every parameter t, in particular, for each t ∈ T. It turns out that the existence of local lifts does not necessarily imply the existence of a global lift; recall that complex curves avoiding 0 always have local and global lifts.
In what follows, we start establishing the conditions on the behaviour of γ locally near its obstruction parameters in order to guarantee the existence first of local lifts and local companions and then of a global lift and a global companion.
As these conditions depend on the structure of the obstruction set, we start by considering paths with a finite obstruction set.
Let t s ∈ (a, b).Then 1) γ is tame at t s if both limits are either equal or opposite.In particular, if these limits are opposite, then the parameter t s is called a flip, whereas if they are the same it is called a bounce; 2) γ is semi-tame at t s if it is not tame at t s but both limits in (5.4) exist; 3) γ is not tame at t s if at least one of the limits in (5.4) does not exist.
If t s = a (resp.t s = b) then the path is tame at t s from the right (left) if the right (left) limit in (5.4) exists and not tame in all other cases.
If, in addition, the path γ is closed, we adapt the definition of tameness at the endpoints in the natural way.In particular, γ is semi-tame at a ≃ b if it is tame at a from the right and at b from the left.If the limits are the same, then a ≃ b is called a bounce and if they are opposite it is called a flip.In all other cases γ is not tame at a ≃ b.
Remark 5.3.A path γ in the Example 4.6 (b) does not have the limit (5.4) at t = 0.
The proposition below gives a motivation for the previous definitions.
If γ is a loop, then it is a tame loop if and only if it is tame at each t s , s = 1, . . ., p.If γ is a tame loop, the lift to E + K exists by Proposition 4.12.However, we want to present also a constructive proof, because we will use the same techniques to obtain lifts of non tame paths and to explain the definition of winding number through local data on the obstruction set.
Proof.Assume that the loop does not have flips.Then I γ equals I k 0 • γ for some k ∈ Z and so it is obviously a loop.
For the case of a loop with flips, we assume, without loss of generality, that k 0 = 0, so we have started with the principal branch and moreover, we also assume that the parameterization γ : As in the previous proof, all the functions arg 0 (γ s )(t), I 0 (γ s )(t) have continuous extensions to the endpoints of I s .
If a is a bounce, then the imaginary unit at γ(b) is the same as the one at γ(a), i.e.I 0 (γ 0 )(a) = I 0 (γ p )(b).The even number of flips ensures that the sign of the imaginary unit at the endpoint remains the same with respect to the one at the principal branch.
If the initial point is a flip, then the imaginary unit function at endpoint has the opposite sign with respect to the one at the initial point, I 0 (γ 0 )(a) = −I 0 (γ p )(b), and to end up with the same sign there must be an odd number of additional flips following the first one to ensure that the sign of the unit at the endpoint remains the same with respect to the one at the principal branch.
The proofs of Propositions 5.5 and 5.6 show that once the lift near the initial point is chosen, only the flips are relevant for the determination of the lift near the endpoint; bounces can be discarded.This enables us to calculate the change of argument and the winding number out of local data at the intersections of γ with the real axis.To determine the change of the argument we introduce a notion of signature.
The connection between the signature and the change of argument is described in the following Proof of Proposition 5.8.Assume that k 0 = 0, so arg(γ(a)) ∈ (0, π) and let the sequence of signs of flips be alternating starting with −1, i.e. −1, 1, −1, . . . .Then arg γ (γ(t)) increases when the path γ crosses the real axis, so k increases by 1 at each flip (because the sign of the flip changes); altogether, this occurs m l=1 (−1) l (−1) l = m l=1 sign(γ(ξ l ))(−1) l = σ(γ) times.This coincides with winding around the origin of the shadow in the positive direction.If the sequence of signs starts with 1, then arg γ (γ(t)) decreases when the path γ crosses the positive real axis and this results in the translation of the interval [0, π] by π m l=1 (−1) l−1 (−1) l = πσ(γ).To prove the general assertion it suffices to show what happens if the sequence is not alternating at one position.
Assume that we insert in the alternating sequence −1, 1 − 1 . . . the integer 1 in the second position, so the sequence is no longer alternating: −1, 1, 1, −1, . . . .This means that we have started from the upper half-plane, crossed the negative real axis, then the positive real axis with the arguments in [2π, 3π].Then we have crossed the positive real axis again, hence the choice of argument at this intersection must be 2π.Because the point is a flip, this means that the argument decreases and keeps decreasing till the end.This is faithfully reflected in the sequence s l = (−1) l sign(γ(ξ l )), because it equals 1, 1, −1, −1, ... and so the sum m l=1 sign(γ(ξ l ))(−1) l = σ(γ) multiplied by π corresponds with the total translation of the initial interval for the Arg.
Similarly, if we insert −1 on the second position, this means that we have crossed the negative real axis and we have the argument in [π, 2π] but then we have returned to the negative real axis and in order to have the argument continuous, at the second crossing the argument π must be chosen and because we have a flip, the argument arg γ (γ(t)) decreases and keeps decreasing till the end.The corresponding sequence s 1 , s 2 , . . .now equals 1, −1, −1, −1, . . .and m l=1 s l = σ(γ).The proof for k 0 even is the same.If k 0 is odd, this coincides with considering the conjugate shadow and hence reversed orientation compared, to k 0 even, so the signature has to be multiplied by −π to get the total translation of the initial interval for the Arg.
In practice this means that once the sequence of ±1-s is given, we start by cancelling the pairs of the same numbers until we end up with an alternating sequence.The number of elements multiplied by minus the first element is the signature.
If the path γ : [a, b] → K \ {0} is closed, i.e. γ(a) = γ(b), then we identify points a and b of [a, b] and consider the parameterization as γ : S 1 → K \ {0}, so there is no distinguished initial point.Therefore, in this case we require that for each s ∈ S 1 there exists a neighbourhood of U s of s in S 1 such that the lift of γ exists on U s .Proof.Because γ is not tame at ξ l we can only choose either k 0 = 0 or k 0 = −1 and lift the curve in a neighborhood of the point γ(ξ l ) to E + K using the principal branch of the logarithm.Assume that we have chosen k 0 = 0. Then on [ξ l+1 − δ, ξ l+1 ) for some small δ > 0 we may only have k = 0, −1, hence the signature can be either 0 or −1 in order to be able to extend the lift to ξ l+1 .If we have k 0 = −1, then, since we have to end up with k = 0, −1 near ξ i+1 , the condition is (−1) k 0 σ(γ| [ξ l ,ξ l+1 ] ) ∈ {0, 1} hence σ(γ| [ξ l ,ξ l+1 ] ) ∈ {0, −1}.
When we do not have additional information about the set γ −1 (R), we must assume that the continuous lift of γ exists on a neighbourhood U of γ −1 (R) ∩ (−∞, 0).This means that the path γ| U has a companion, since the restriction of the function arg γ to U is not vanishing.Recall that, on a neighbourhood of γ −1 (R) ∩ (0, ∞), the principal branch of the logarithm is well-defined and hence a lift of γ always exists.This does not imply that a global lift exists.
We now proceed with the detailed description of the possible situations when γ has a companion on a neighborhood of real points and omit the (trivial) case γ([a, b]) ⊂ R. Proposition 5.14.Let γ : [a, b] → K \ {0} be a path.Then γ has a companion if and only if it has a companion on a neighbourhood of the obstruction set.The same holds for a loop γ with γ(a) ∈ R.
In the sequel we explain how to extend the notion of signature to paths with infinite obstruction set.Since γ ([a, b]) is compact, there are only finitely many connected components of γ([a, b]) \ R with endpoints of opposite sign.Definition 5.15.Let L 1 , . . ., L m be all the connected components of γ([a, b]) \ R, L l (t) = γ(t), t ∈ (s l , e l ) ⊂ [a, b] satisfying γ(s l )γ(e l ) < 0 and a ≤ s l < e l ≤ s l+1 < e m ≤ b, l = 1, . . ., m.We call the components the big arcs and the subdivision a ≤ s l < e l ≤ s l+1 < e m ≤ b, l = 1, . . ., m the induced subdivision.The intervals [e l , s l+1 ] are called obstruction intervals.If γ is closed, then we identify a and b, e 0 := e m , s m+1 := s 1 and define also [e 0 , s 1 ] as the obstruction interval.
Because γ([e l , s l+1 ]) misses either the positive or the negative real axis, we define the sign of the obstruction interval as follows.
To define the winding number for a closed curve we have to take into account also the last interval I m and hence consider the closed signature.Let 1 ≤ j 1 < . . .< j k ≤ m be the indices for which γ, restricted to the neighbourhoods of the intervals J j l := [e j l , s j l+1 ] ⊂ (0, ∞), does not have a companion and assume γ has a companion on a neighbourhood of the closure of [a, b] \ ∪ k l=1 J j l .Then a lift of γ in E + K exists if and only if σ(γ| [s j l +1 ,e j l+1 ] ) ∈ {0, −1} for each l = 1, . . ., m − 1.If it exists, the lift is a loop.

FIGURE 2 .
FIGURE 2. The path γ (negative rocket) of the Example 4.6 (b) is drawn on the left: it cannot be lifted to E + K .Its reflection on the right (positive rocket) can be lifted to E + K .It is useful to point out that the existence of a lift Γ of a path γ to E + K is equivalent to the existence of a continuous function Arg γ : [a, b] → Im(H), such that Γ(t) = (γ(t), Arg γ (t)) ∈ E + K .As noticed in Remark 2.1, the function Arg γ will be decomposed, where possible, with obvious notation, as Arg γ := I γ arg γ where I γ : [a, b] → S and arg γ : [a, b] → R. The existence of I γ : [a, b] → S implies that we can assign to each t ∈ [a, b] a complex plane C I γ which contains the point γ(t) and hence determines the argument up to a multiple of 2π.Complex slices {C J } J∈S are naturally parameterized by the elements of S/{± Id}, the real projective spaceRP dim R K−2 of dimension dim K − 2. The projection [ ] : S → S/{± Id} = RP dim R K−2is the classical 2 : 1 universal covering map and, as customary, for J ∈ S, the symbol [J] denotes the equivalence class whose representatives are the opposite (conjugate) imaginary units J, −J ∈ S. Each element [J] ∈ S/{± Id} uniquely defines the complex slice C [J] = C J = C −J .A continuous imaginary unit function I γ : [a, b] → S naturally defines a continuous function I γ : [a, b] → S/{± Id} when we set I γ (t) = [I γ (t)].
a) ∈ [2kπ, (2k + 1)π].Proof.There exist exactly two continuous lifts I γ , −I γ of I γ to the universal covering S of S/{± Id}.Correspondingly, there exist two shadows γ I γ , γ −I γ : [a, b] → C \ {0} associated with I γ .Exchange I γ and −I γ if necessary, so that I γ is such that arg(γ I γ (a)) ∈ [0, π].As a complex path, γ I γ has a well defined argument arg γ I γ : [a, b] → R such that arg γ I γ (a) ∈ [0, π].Set arg γ := arg γ I γ .Then the chosen paths I γ and arg γ have the properties required in the statement.The rest of the proof is straightforward.The lifts Γ and Γ k (for k ∈ Z) appearing in the last Proposition are not unique, when Γ(a) and Γ k (a) are real.At this point, Proposition 4.2 implies directly the existence of all branches of the logarithm, along all paths in K \ {0} having a companion.Corollary 4.13.Let γ : [a, b] → K \ {0} be a path with companion I γ : [a, b] → S/{± Id}.For every k ∈ Z, let is a continuation of the hypercomplex logarithm along γ with initial point log |γ(a)| + Arg 2k (γ(a)).This map is called a k-th branch of the hypercomplex logarithm along γ with initial point log |γ(a)| + Arg 2k (γ(a)).
Remark 4.16.For any fixed I ∈ S, let γ : [a, b] → K \ {0} be a path lying in the complex slice C I .The path γ has always a particularly simple companion, namely I γ : [a, b] → S/{± Id} constantly equal to [I].Moreover, the two different lifts of I γ to S are both constantly equal to I or −I, respectively.As a consequence, if the given path γ is closed and tame, it is a tame loop and is untwisted.A twisted loop necessarily intersects the real axis.Indeed the following result holds.Proposition 4.17.Let γ : [a, b] → K \ {0} be a loop which misses the real axis.Then γ is a tame loop and is untwisted.Proof.By Proposition 4.5, the loop γ can be lifted to a path Γ : [a, b] → E + K with Γ = (γ, Arg γ ).Let us consider the map Arg γ = I γ arg γ : [a, b] → Im(K).By the hypothesis, there exists k ∈ Z such that the map arg γ = arg γ 2k : [a, b] → (2kπ, (2k + 1)π) is never vanishing and hence has constant sign.Now, since γ is closed, we have that I γ (a) arg γ 2k (a) = I γ (b) arg γ 2k (b).Since arg γ 2k (a) and arg γ 2k (b) have the same sign and both belong to the interval (2kπ, (2k + 1)π), we obtain arg γ 2k (a) = arg γ 2k (b) and hence I γ (a) = I γ (b).Therefore the path I γ : [a, b] → S is a loop, and hence the unique companion [I γ ] : [a, b] → S/{± Id} is a loop, homotopic to a constant.As a consequence the path γ is an untwisted, tame loop.4.1.
Proposition 4.18.A loop γ : [a, b] → K \ {0}, γ([a, b]) ⊂ R, with companion I γ is untwisted if, and only if, for any chosen non real initial point of γ, both shadows associated with I γ are loops.Proof.Let γ I γ : [a, b] → C \ {0}, γ I γ (t) = x(t) + iy(t), be one of the shadows associated with I γ .If the loop γ is untwisted, then any lift I γ of the companion I γ of γ is a loop, and hence it has coinciding endpoints.Therefore, the pathγ(t) = x(t) + I γ (t)y(t)being a loop, the continuous function y : [a, b] → R is such that y(a) = y(b).Hence the associated shadow γ I γ (t) = x(t) + iy(t) is closed.
Corollary 4.29.Let γ : [a, b] → K\{0} be a loop with companion I γ .Then the two shadows associated with I γ are closed if, and only if, the endpoints of γ are real.

FIGURE 3 .
FIGURE 3. From left to right: (a) the path γ and two of its shadows (b), (c) be one of the shadows associated with γ.Let arg γ (t), t ∈ [a, b], be the corresponding argument and choose the initial argument so that arg γ (a) ∈ [0, π].The set ∆ = {arg γ (a) for all possible initial points} is an interval contained in [0, π].Because the loop γ is twisted, the argument at b is arg γ (b) = 2nπ − arg γ (a) and hence arg γ (b) − arg γ (a) = 2nπ − 2 arg γ (a) [a, b] → K \ {0} misses the real axis, then the lift to E + K always exists.On the other hand, if γ([a, b]) ⊂ R, then, necessarily either γ([a, b]) ⊂ R − and we have the lifts of the form Γ(t) = log |γ(t)| + I(2k + 1)π, or γ([a, b]) ⊂ R + and then we have the lifts of the form Γ(t) = log |γ(t)| + I2kπ for any I ∈ S. From now on assume that γ([a, b]) is not entirely contained in the real axis but it intersects it.Definition 5.1.For a path γ : [a, b] → K \ {0} we define the set T := γ −1 (R) to be the obstruction set (for the lift of γ) and its points as obstruction parameters.
then a lift of γ on [a, b) can be extended continuously to b if and only if b is tame from the left.Corollary 5.10.Let γ be as in Proposition 5.8 and let γ(a) = γ(b) ∈ R. Then σ(γ) is even if and only if γ is tame and untwisted.If this is the case, then ω(γ) = |σ(γ)|/2.

FIGURE 4 .
FIGURE 4. A loop without a lift in the sense of Definition 5.11.