Dependencies

Let \(\rho \colon S^1\to \mathbb {C}\) be a map with \(\rho (z)\in Cstar\) for all \(z\in S^1\). Suppose there is a map \(\hat f\colon D^2\to \mathbb {C}\) with \(\hat f|_{S^1}=f\) and with the image of \(\hat f\) contained in \(Cstar\). Then the winding \(w(\rho )=0\).

Suppose that \(\omega \colon [ a, b ]\to \mathbb {C}\) is a loop and \(\omega (t)\in Cstar\) for all \(t\in [ a, b ]\). Suppose that \(H\colon [ a, b ]\times [ 0, 1 ]\to \mathbb {C}\) is a homotopy of loops from \(\omega \) to a constant loop and \(H(t,s)\in Cstar\) for all \((t,s)\in [ a, b ]\times [ 0, 1 ]\). Then the winding number of \(\omega \) is zero

If \(f\colon S^1\to \mathbb {C}\) is a constant map to a point \(z\in Cstar\), then \(w(f)=0\).

Let \(\omega \colon [ a, b]\to \mathbb {C}\) be a loop with \(\omega (t)\in Cstar\) for all \(t\in [ a, b]\). There is a lift \(\tilde\omega \colon [ a, b]\to \mathbb {C}\) of \(\omega \) through \(CSexp\). There is a constant \(w(\omega )\in \mathbb {Z}\) such that for every lift \(\tilde\omega \colon [ a, b]\to \mathbb {C}\) the winding number of \(\tilde\omega \) is \(w(\omega )\).

\(CSexp\colon \mathbb {C}\to \mathbb {C}\) is continuous.

\(PBlog\) is continuous on \(T\) and if \(z\in T\) then \(PBlog(z)\in \{ z\in \mathbb {C}|-\pi {\lt} Im(z) {\lt} \pi \} \).

\(Cstar=\{ z\in \mathbb {C}| z\not= 0\} \)

\(CSexp\colon \mathbb {C}\to \mathbb {C}\) defined by the usual power series.

Let \(f\colon X\to Y\) be a continuous map and \(A\subset Y\). Then \(f\) is an even cover on \(A\subset X\) if every \(a\in A\) has a neighborhood which is contained in the target of a trivialization

Let \(f\colon X\to Y\) be a continuous map between topological spaces and \(\alpha \colon A\to Y\) a continuous map. A lift of \(\alpha \) through \(f\) is a continuous map \(\tilde\alpha \colon A\to X\) such that \(f\circ \tilde\alpha =f\).

There is a map \(PBlog\colon \mathbb {C}\to \mathbb {C}\).

Given a map of the circle \(\psi \colon S^1\to X\) the associated loop is \(\omega \colon [ 0, 2\pi ]\to X\) is defined by \(\omega (t)=\psi (CSexp(it))\).

For any \(a, b\in \mathbb {R}\) with \(a {\lt} b\) we define \(S(a,b)=\{ z\in \mathbb {C}| a {\lt} Im{z} {\lt} b⦄\). Define \(S\subset \mathbb {C}\) by \(S=S(-\pi ,\pi )\).

\(\tilde S\subset \mathbb {C}\) is the subset \(\{ r+\theta * I|r,\theta \in \mathbb {R}\text{\ and\ } \theta \not= (2k+1)\pi \text{ for any } k\in \mathbb {Z}\} \).

Let \(\widetilde{PBlog}\colon T\times \mathbb {Z}\to S\times \mathbb {Z}\) be defined by \(\widetilde{PBlog}(z,n)=(PBlog(z),n)\) for all \(z\in T\) and \(n\in \mathbb {Z}\).

Suppose given a loop \(\omega \colon a\colon [a, b]\to \mathbb {C}\) with \(\omega (t)\in Cstar\) for all \(t\in [a, b]\), and given a lift \(\tilde\omega \) of \(\omega \) through \(CSexp\) the winding number of the lift \(\tilde\omega \), denoted \(w(\tilde\omega )\), is \((\tilde\omega _x(b)-\tilde\omega _x(a))/2\pi *I\).

The winding number of a map \(\rho \colon S^1\to \mathbb {C}\) with \(\rho (z)\in Cstar\) for all \(z\in S^1\) is the winding number of the associated loop.

Let \(\omega \colon [ a, b]\to \mathbb {C}\) with \(\omega (t)\in Cstar\) for all \(t\in [ a ,b]\). Suppose that \(\tilde\omega \) and \(\tilde\omega '\) are lifts of \(\omega \) through \(CSexp\). Then \(w(\tilde\omega )\in \mathbb {Z}\) and \(w(\tilde\omega ')=w(\tilde\omega )\).

If \(\omega \colon [ a, b ]\to \mathbb {C}\) and \(\omega '\colon [ a, b ]\to \mathbb {C}\) are loops with \(\omega (t) , \omega '(t) \in Cstar\) for all \(t\in [ a, b ]\) and if \(H\colon [ a, b ]\times [ 0, 1 ]\to \mathbb {C}\) is a homotopy of loops from \(\omega \) to \(\omega '\) with \(H(t,s)\in Cstar\) for all \(t\in [ a, b ]\) and \(s\in [ 0, 1 ]\), then \(w(\omega )=w(\omega ')\)

for \(r,\theta \in \mathbb {R}\).

Let \(\omega \colon [a, b]\to \mathbb {C}\) be a loop. Assume that \(\omega (t)\in Cstar\) for all \(t\in [a, b]\). There is a lift of \(\omega \) through \(exp\).

Every complex polynomial of degree \(k{\gt}0\) has a complex root.

\(CSexp\colon \mathbb {C}\to \mathbb {C}\) is a covering projection over \(Cstar\) with source \(\mathbb {C}\). The image of \(CSexp\) is \(Cstar\).

\(CSexp\) satisfies the homotopy lifting property on \(Cstar\).

Given a path \(\omega \colon [ a , b]\to \mathbb {C}\) with \(\omega (t)\not=0\) for all \(t\in [ a, b]\), and \(\tilde a_0\in CSexp^{-1}(\omega (a))\), there is a unique map \(\tilde\omega \colon [ a, b ]\to \mathbb {C}\) with \(\tilde\omega (a)=\tilde a_0\) and \(exp(\tilde\omega )=\omega \).

Suppose \(f\colon E\to X\) is a map between topological spaces and \(U\subset X\) is an open subset and there is a trivialization for \(f\) on \(U\). Suppose also that there are homeomorphisms \(\varphi \colon X\to X\) and \(\tilde\varphi \colon E\to E\) with \(f\circ \tilde\varphi =\varphi \circ f\). The there is a trivialization for \(f\) on \(\varphi (U)\).

A homotopy of loops is a one parameter family \(\Omega \colon [a, b]\times [0, 1]\to X\) with \(\Omega |_{[a, b]\times \{ s\} }\) a loop for all \(s\in [0, 1]\). A homotopy of loops based at \(x_0\) is a one parameter family indexed by \([0, 1]\) of loops based at \(x_0\).

The image of \(PBlog\) is contained in \(\{ z\in \mathbb {C}|-\pi {\lt} Im(z)\le \pi \} \) and for all \(\{ z\in \mathbb {C}| z\not=0\} \) \(CSexp(PBlog(z))=z\).

Then \(CSexp\colon S\to T\) and \(PBlog\colon T\to S\) are inverse homeomorphisms.

Let \(X\) be a topological space and \(a, b ∈ ℝ\) with \(b {\gt} a\). A loop in \(X\) is a map \(\omega \colon [ a, b]\to X\) with \(\omega (b)=\omega (a)\). A loop is based at \(x_0\in X\) if \(\omega (a)=x_0\).

\(CSexp(z+w)=CSexp(z)* CSexp(w)\).

\(CSexp\colon \mathbb {C}\to \mathbb {C}\) is periodic of period \(2\pi i\) and with no smaller period.

Let \(\psi , \psi '\colon S^1\to \mathbb {C}\) be maps and \(H : S^1→ \mathbb {C}\) a homotopy between them whose image lies in \(Cstar\). Then the winding numbers of \(\psi \) and \(\psi '\) are equal.

Let \(ρ : S^1→ \mathbb {C}\) be a map with \(ρ(z)∈ Cstar\) for all \(z∈ S^1\). Let \(ω\) be the loop associated with \(ρ\). Then the image of \(ω\) is contained in \(Cstar\).

Suppose that \(\psi ,\psi '\colon S^1\to \mathbb {C}\) with \(|\psi (z)-\psi '(z)|{\lt}|\psi (z)|\) for all \(s\in [ 0, 2\pi ]\). Then \(\psi \) and \(\psi '\) have the same winding number.

\(T=\{ z\in \mathbb {C}|Re(z){\gt}0 \cup Im(z)\not= 0\} \)

\(T\cup T'=\{ z\in \mathbb {C}| z∈ Cstar\} \).

Define \(\varphi \colon S\times \mathbb {Z}\to \mathbb {C}\) by \(\varphi (z,k)=z+2k\pi *I\). Then \(\varphi \colon S\times \mathbb {Z}\to \tilde S\) is a homeomorphism.

Let \(T'=\{ z\in \mathbb {C}| Re(z){\lt}0\cup z\in \mathbb {C}| Im(z)\not= 0\} \).

\(f\colon X\to Y\) a local trivialization for \(f\) on \(U\) is:

• an open subset \(U\subset Y\)

• a discrete space set \(I\)

• a homeomorphism \(\varphi \colon f^{-1}(U)\to U\times I\)

such that letting \(p_1\colon U\times I\to U\) be the projection onto the first factor, we have \(p_1\circ \varphi (x)=f(x)\) for all \(x\in f^{-1}(U)\)

The composition \(\psi =\varphi \circ \widetilde{PBlog}\colon T\times \mathbb {Z}\to \tilde S\) defines a trivialization of \(CSexp\) on \(T\)

\(T'\) is the base of a trivialization for \(CSexp\colon \mathbb {C}\to \mathbb {C}\) with non-empty fiber.

Suppose that \(\psi \colon S^1\to \mathbb {C}\) and \(\psi '\colon S^1\to \mathbb {C}\) are maps and for each \(z\in S^1\), we have \(|\psi (z)-\psi '(z)|{\lt}|\psi (z)|\). Then there is a homotopy \(H\) from \(\psi \) to \(\psi '\) whose image lies in \(Cstar\).

\(\widetilde{PBlog}\colon T\times \mathbb {Z}\to S\times \mathbb {Z}\) is a homeomorphism.

Suppose that \(\omega \colon [ a, b ]\to \mathbb {C}\) is a loop with \(\omega (t)\in Cstar\) for all \(t\in [ a, b ]\). Then the constant \(w(\omega )\) given in Corollary 6 is the winding number of \(\omega \).

Let \(p(z)\) be a complex polynomial of degree \(k{\gt}1\) given by \(p(z)=\sum _{i=0}^k\alpha _iz^{k-i}\) with \(α_i∈ℂ\) for all \(i\) and \(α_0\not= 0\). Then for \(R\) sufficiently large, the map \(f : S^1\to \mathbb {C}\) given by \(f(z)= p(R* z)\) for \(z\in S^1\) has winding number \(k\).

For \(x\in \mathbb {C}\) with \(x\not= 0\), either \(x\in T\) or \(x\in T'\).

Let \(p(z)\) be a complex polynomial of degree \(k\); \(p(z)=\sum _{i=0}^k\alpha _iz^{k-i}\) with \(\alpha _i\in \mathbb {C}\) and \(\alpha _0\not= 0\). For all \(R\) sufficiently large \(|\alpha _0|R^k{\gt}|\alpha _0z^k - p(z)|\) for any \(z\) with \(|z|=R\).

For any \(\alpha _0\in \mathbb {C}\) and any \(k\in \mathbb {Z}\) \(k≥ 0\), define \(\psi _{\alpha _0,k}\colon \mathbb {C}\to \mathbb {C}\) by \(\psi _{\alpha _0,k}(z)=\alpha _0 z^k\). Then for any \(R{\gt}0\) if \(\alpha _0\not=0\) and \(k{\gt}0\) the winding number of the map of the restriction of \(\psi _{\alpha _0,k}\) to the circle of radius \(R\) is \(k\)