In Mathlib.

In Mathlib.

In Mathlib.

In Mathlib.

This is immediate from Definition 4 and Lemma 2.

By Lemma 2 for \(x\in T\) \(Re(cos(x))\not=-1\) and hence by Lemma 5 \(PBlog(x)\in S\).

By Lemma 2 \(CSexp(z)\in \mathbb {R}^-\) if and only if \(CSexp({\rm Im}(z))\in \mathbb {R}^-\) if and only if \({\rm Im}(z)\in {\pi +(2\pi )\mathbb {Z}⦄}\). Since, by Definition 8 for \(z∈ S\) \(-\pi {\lt} Im(z) {\lt} \pi \). It follows that \(CSexp(S)\subset T\). Conversely, by Lemma 6 if \(z\in T\) then \(PBlog(z)\in S\).

By Lemma 1 \(CSexp\) is continuous and, by Lemma 6, \(PBlog\) is continuous on \(T\). Suppose that \(z,w\in S\) and \(CSexp(z)=CSexp(w)\). By Lemma 4 there is an integer \(n\) such that \(z-w =2\pi *n*I\) and \(-2\pi {\lt} Im(z)-Im(w){\lt}2\pi \). It follows that \(n=0\) and hence that \(z=w\). This shows that \(CSexp|_S\) is one-to-one. Since \(CSexp|_S\) is one-to-one and \(CSexp({\rm PBlog}(z))=z\) for all \(z\in T\), it follows that \(CSexp\colon S\to T\) and \({PBlog}\colon T\to S\) are inverse functions. Since each is continuous, they are inverse homeomorphisms.

According to Definition 8 image of \(S\) under the translation action of \((2\pi )\mathbb {Z}\) on \(\mathbb {C}\) is the union of all strips \(S(2n-1)\pi ,(2n+1)\pi )\). By Definition 9 this union is \(\tilde S\). Thus we have a map \(S\times \mathbb {Z}\to \tilde S\) defined by \((z,n)\mapsto z+2\pi *n *I\). Since translation is a homeomorphism of \(\mathbb {C}\to \mathbb {C}\), this map is a local homeomorphism onto its image \(\tilde S\). If \(n ,n'\in \mathbb {Z}\) with \(n\not=n'\) then \(S((2n-1)\pi ,(2n+1)\pi )\cap S((2n'-1)\pi ,(2n'+1)\pi )=\emptyset \). Also \(\tilde S=\coprod _{n\in \mathbb {Z}}S((2n-1)\pi ,(2n+1)\pi )\). It follows that \(\varphi \) is a bijective map and hence a homeomorphism.

By Definition 10 \(\widetilde PBlog\) is the product of \(PBlog\colon T\to S\) and \({\rm Id}_\mathbb {Z}\colon \mathbb {Z}\to \mathbb {Z}\). By Lemma 1 the first of these factors is a homeomorphism. Since \({\rm Id}_\mathbb {Z}\) is a homeomorphism. it follows from basic properties of homeomorphisms that the product \(\widetilde{PBlog}\) is a homeomorphism.

\(\varphi \) is a homeomorphism by Lemma 7. By Lemma 8 \(\widetilde{PBlog}\colon T\times \mathbb {Z}\to S\times \mathbb {Z}\) is a homemorphism. Thus, the composition \(\varphi \circ \widetilde{PBlog}\colon T\times \mathbb {Z}\to \tilde S\) is a homeomorphism. For \((z,n)\in T\times \mathbb {Z}\),

By Lemma 4, \(CSexp(PBlog(z)+2\pi * n * I)=CSexp(PBlog(z))\), which by Lemma 8 equals \(z\). This establishes that \(\psi \) satisfies all the conditions of the Definition 1 on \(T⊆ \).

Since \(f\circ \tilde\varphi =\varphi \circ f\), we have \(\tilde\varphi \colon f^{-1}(U)\to f^{-1}(\varphi (U))\). Since \(\varphi \) and \(\tilde\varphi \) are homeomorphisms the induced map \(\tilde\varphi \colon f^{-1}(U) \to f^{-1}(\varphi (U))\) is a homeomorphism. Let \(\psi \colon f^{-1}(U)\to U\times I\) be a homeomorphism with \(p_1\circ \psi \) being the map \(f\colon f^{-1}U\to U\). Such a map is equivalent to a trivialization for \(f\) with base \(U\). Then

is a homeomorphism. Furthrmore, projection to the first factor is

This composition is \(f\colon f^{-1}(\varphi (U))\to \varphi (U)\), so that this homeomorphism determines a trivialization for \(f\) with base \(\varphi (U)\).

We have homeomorphism \(\mu \colon \mathbb {C}\to \mathbb {C}\) that sends \(z \to CSexp(\pi *I)z)\) and the homeomorphism \(\tilde\mu \colon \mathbb {C}\to \mathbb {C}\) defined by \(\tilde\mu (z)=z+\pi *I\) Clearly by Lemma 3 and Lemma 2 \(CSexp(\tilde\mu (z))= \mu (CSexp(z))\). By Definition 5 and Definition 11 \(\mu (T)=T'\). The result now follows from Lemma 9 and Proposition 2.

Suppose that \(x\in \mathbb {C}\) and \(x\not= 0\). Then either \(Re(x){\gt} 0\) or \(Re(x)\le 0\). If \(Re(x){\gt}0\), then by Definition 5 \(x\in T\). if \(Re(x){\lt} 0\) then by Definition 11 \(x\in T'\). Finally, if \(Re(z)=0\) and \(z\not=0\), then \(Im(z)\not= 0\) and \(z\in T\).

Immediate from Lemma 10 and Definition 6.

By Corollary 2 \(T\cup T'= Cstar\). By Proposition 2 and Corollary 1 \(CSexp\) is a trivialization on \(T\) and on \(T'\). Hence, every point of \(Cstar\) lies in the base of a trivialization for \(CSexp\). By definition, this shows that \(CSexp\colon \mathbb {C}\to \mathbb {C}\) is a covering on \(Cstar\). Since \(CSexp(z)\not=0\) for all \(z\in \mathbb {C}\), it follows that \(CSexp^{-1}(Cstar)=\mathbb {C}\). Lastly, by Lemma 5 if \(z\in \mathbb {C}\) and \(z\not= 0\) then \(CSexp(PBlog)(z)=z\). This proves that \(CSexp\) is onto \(\{ z\in \mathbb {C}| z\not=0\} \), which by Lemma 6, is equal to \(Cstar\).

By Corollary 3 and the basic result about covering projections.

This is immediate from Corollary 3 and the theorem that covering projections have the homotopy lifting property.

By Corollary 3 \(CSexp^{-1}(\omega (a))\not=\emptyset \). Fix a point \(x\in CSexp^{-1}(\omega (a))\) and let \(\tilde\omega _x\colon [a, b]\to \mathbb {C}\) be lift of \(\omega \) through the \(CSexp\) with initial point \(x\) as guaranteed by Corollary 4.

By the Definition 7 we have \(CSexp(\tilde\omega (b))=\omega (b)\) and \(CSexp(\tilde\omega (a)=\omega (a)\). By Definition 12 \(\omega (b)=\omega (a)\). Thus, \(CSexp(\tilde\omega (b))=CSexp(\tilde\omega (a))\). By Lemma 4, there is \(k\in \mathbb {Z}\), such that \(\tilde\omega (b)-\tilde\omega (b)=2\pi *k* I\). By Definition 14, the winding number of \(\tilde\omega \) is \(k\)

Let \(\tilde\omega '\) be another lift of \(\omega \). Since \(CSexp(\tilde\omega '(t))=CSexp(\tilde\omega (t))\) for every \(t\in [ a, b]\), there is an integer \(k(t)\in \mathbb {Z}\) with \(\tilde\omega '(t)-\tilde\omega _x(t)=2\pi k(t)*I\). Since \(\tilde\omega '\) and \(\tilde\omega \) are continuous functions of \(t\) so is \(k(t)\). Since the \([ a, b]\) is connected and \(\mathbb {Z}\) is discrete, \(k(t)\) is a constant function; i.e., there is an integer \(k_0\) such that for all \(t\in [ a, b]\), we have \(\tilde\omega '(t)=\tilde\omega (t)+2\pi * k_0*I\). Thus, \(\tildeω'(b) -\tildeω'(b)=\tildeω'(a)-\tildeω(a)\). It follows from Definition 14 \(w(\tildeω')=w(\tildeω).\)

This is immediate from Lemmas 12 and Lemma 11.

By Definition 13 for all \(\{ t∈ ℝ : 0≤t≤1\} \) \(H(a,t)=H(b,t)\). Let \(\mu \colon \{ t∈ ℝ : 0≤t≤1\} \to \mathbb {C}\) be the path \(μ(t)=H(a,t)\). By Corollary 5 since the image of \(H\) is contained in \(Cstar\), there is a lift \(\tilde H\colon [ a, b]\times I\) of \(H\) through \(CSexp\). Then \(\tilde H|_{\{ a\} \times I}\) and \(\tilde H|_{\{ b\} \times I}\) are two liftings of \(\mu \). So by Lemma 12 \(\tilde H(b,1)-\tilde H(b,0)=\tilde H(a,1)-\tilde H(a,0)\). Rewriting we have \(⁀ H(b,1)-⁀ H(a,1)= \tilde H(b,0)-\tilde H(a,0)\). Since \(\tilde H(t,0)\) is a lift of \(\omega \) through \(CSexp\) and \(\tilde H(t,1)\) is a lift of \(\omega '\) through \(CSexp\), by Definition 15 \(w(\omega ')=w(ω)\).

By Lemma 13 the winding number of the loop \(\omega \) is equal to the winding number of a constant loop. By Lemma 4 the lift of a constant loop through \(CSexp\) is a constant path. Thus, the endpoints of the lift of the constant loop are equal and hence by Definition 15 the winding number of a constant loop is zero.

Let \(ω \colon [ 0, 2\pi ] \to \mathbb {C}\) be the loop associated to \(ρ\). Then by Definition 16 for all \(t∈ [ 0 ,1 ]\) \(ω(t)=ρ(2\pi * t *I)∈ Cstar\).

By Definition 16 the loop associated with the constant map \(f\colon S^1\to Cstar\) is a constant loop at a point of \(Cstar\). By Lemma 17 the winding number of \(f\) is equal to the winding number of the constant loop at \(f(S^1)\in Cstar\). By Lemma 7 this winding number is zero.

Let \(H\colon S^1\times I\to \mathbb {C}\) be a homotopy from \(\psi \) to \(\psi '\) whose image lies in \(Cstar\). Let \(ω\) and \(ω'\) be the loops associated to \(ψ\) and \(ψ'\) respectively Define \(\hat H\colon [ 0, 2\pi ]\times [ 0, 1 ]\to X\) by \(\hat H(t,s)=H(CSexp(it),s)\). Then by Definition 16 \(\hat H\) is a homotopy from the loop \(\omega \) to the loop \(\omega '\). The images of \(H\) and \(\hat H\) are the same so that the image of \(\hat H\) lies in \(Cstar\). By Lemma 13 the winding numbers of \(\omega \) and \(\omega '\) are equal. By Definition 17 this means that the winding numbers of \(\psi \) that the winding numbers of \(ψ\) and \(ψ'\) are equal.

Define a continuous map \(J\colon S^1\times [ 0,1 ]\to D^2\) by \((z,t)\mapsto (1-t)z\). Then \(\hat f\circ J(z,0)= \rho (z)\) and \(\hat f\circ J(z,1)=\hat f(0)\) for all \(z\in S^1\). This is a homotopy in \(Cstar\) from \(\rho \) to a constant map of \(S^1\to Cstar\). By Lemma 16 the winding number of \(\rho \) is equal to the winding number of a constant map \(S^1\to C\star \). By Lemma 15, the winding number of a constant map \(S^1\to \hat f(0)\in Cstar\) is zero.

Since there is a homotopy \(H\) from \(\rho \) to a constant map with image in \(Cstar\), it follows from Lemma 16 that the winding number of \(\rho \) is zero.

By Definition‘16 and by Lemma 3 the loop \(\omega \colon [ 0, 2\pi ]\to \mathbb {C}\) associated to \(\psi _{\alpha _0,t}\) restricted to the circle of radius \(R\) is given by \(\omega (t)= \alpha _0 R^kCSexp(kt *I)\).

By Lemma 3 there is an \(\tilde\alpha _0\in \mathbb {C}\) with \(CSexp(\tilde\alpha _0)=\alpha _0 R^k\). Define \(\tilde\omega (t)=\tilde\alpha _0+kt *I\) for \(0\le t\le 2\pi \). Then by Lemma 3

By Definition 7 this means that \(\tilde\omega \) is a lift of \(\omega \) through \(CSexp\). By Definition 15 \(w(\omega )=(2\pi k*I-0)/2\pi * I = k\). By Definition 17, this means that the winding number of \(\psi _{\alpha _0,k}\) is \(k\).

Since for all \(z\in S^1\), \(|\psi (z)-|\psi '(z)|{\lt}|\psi (z)|\), it follows that \(|\psi (z)|{\gt}0\) and \(|\psi '(z)|{\gt}0\) for all \(z\in S^1\). Define a homotopy \(H\colon S^1× [ 0, 2\pi ]\to \mathbb {C}\) by \(H(z,t)=t\psi '(z)+(1-t)\psi (z)\). \(H(z,0)=\psi (z)\) and \(H(z,1)=\psi '(z)\), so \(H\) is a homotopy from \(\psi \) to \(\psi '\).

We establish that \(H(z,t)\not= 0\). For all \(z\in S^1\) and \(t\in [ 0, 1 ]\) \(|\psi (z)-(t\psi (z)-(1-t)\psi '(z)|=|(1-t)(\psi -\psi ')|\). Since \(0\le t\le 1\), \(0\le (1-t)\le 1\). Then, \(|\psi (z)-H(z,t)|=|\psi (z)-(t\psi (z)-(1-t)\psi '(z)|=(1-t)|\psi (z)-\psi '(z)|{\lt}|\psi (z)|\). So \(H(z,t)\not=0\) for all \(z\in S^1\) and all \(t\in [ 0, 1 ]\).

Consequently, \(H\) is a homotopy \(S^1\times [ 0 , 1 ]\to \mathbb {C}\) from \(\psi \) to \(\psi '\) whose image lies in \(Cstar\).

By Lemma 18, there is a homotopy \(H\) from \(\psi \) to \(\psi '\) whose image lies in \(Cstar\). Thus, by Lemma 16, \(\psi \) and \(\psi '\) have the same winding number.

For each \(1\le i\le k\) set \(\beta _i=\alpha _i/\alpha _0\) Choose \(R{\gt}\sum _{i=1}^k|\beta _j|\) and \(R{\gt}1\). For any \(z\in \mathbb {C}\) with \(|z|=R\), we have

By Lemma 19 for \(R{\gt}{\rm max}(1,\sum _{i=1}^k|\beta _j|)\), and for any \(z\in \mathbb {C}\) with \(|z|=1\) \(|\alpha _0(R*z)^k-f(z)| {\lt}|\alpha _0 R^k|\). By Lemma 8 the maps defined on \(S^1\) by \(z ↦\alpha _0*(R* z)^k\) and by \(f\) have the same winding number.

But according the Lemma 17 the winding number of the map \(S^1\mapsto \mathbb {C}\) given by \(z\mapsto \alpha _0(R*z)^k=(α_0R^k)*z^k\) is \(k\). Thus, the winding number of \(f\) is also \(k\).

The proof is by contradiction. Suppose that \(p(z)=\sum _{i=0}^k\alpha _iz^{k-i} \) with \(\alpha _0\not= 0\). Suppose that \(p(z)\not= 0\) for all \(z\in \mathbb {C}\). By Theorem  2 for \(R{\gt}0\) sufficiently large the winding number of the restriction of \(p(z)\) to the circle of radius \(R\) is \(k\). Fix such an \(R\)

Let \(D^2\to \mathbb {C}\) be the map \(z\mapsto Rz\). Define \(\rho \colon D^2\to \mathbb {C}\) by \(z\mapsto p(Rz)\). The restriction of this map to the boundary circle is the restriction of \(p(z)\) to the circle of radius \(R\). Since \(p(z)\not=0 \) for all \(z\in \mathbb {C}\), the image of \(\rho \) is contained in \(Cstar\). According to Lemma 1, this implies that the winding number of of \(p\) on the circle of radius \(R\) is zero.

Since \(k{\gt}0\), this is a contradiction.