Commit d431d3ef by Felix Hilsky

### chapter 3.1 started

parent 228dec2f
 ... ... @@ -186,6 +186,48 @@ Let $Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞$ be non-orientable. Then there e - In chapter 3, orienting a line field is translated into lifting $Q$ from $ℝP^2$ to covering space $𝕊^2$. - Classical algebraic topology tools only consider continous maps, but we have $W^{1,p}(Ω)$ maps and want to preserve this regularity ### Continous line fields Chapter 3.1 deals with the lifting of continous line fields. Similar to "normal" algebraic topology. *Lemma 3*: - $Q : [t_1, t_2] → 𝒬$ continous path can be lifted in two ways (depending on choice of start). - If $Q$ stays close ($≤ ε s$ for $0 < ε < \sqrt{2}$) to some $\overline{Q} ∈ 𝒬$, then the lifted $n$ stays on the same half-circle as the start value ($|n(t) - \overline m| ≤ ε$) *Proof notes for Lemma 3*: First calculation: math | n ⊗ n - m ⊗ m |^2 = Σ_{i,j} | n_i n_j - m_i m_j |^2 = Σ_{i,j} ( n_i n_j - m_i m_j )^2  math = Σ_{i,j} (n_i n_j)^2 + (m_i m_j)^2 - 2 (n_i n_j m_i m_j) = (\underbrace{Σ_i n_i^2}_{=1} (\underbrace{Σ_j n_j^2}_{=1})) + (\underbrace{Σ_i m_i^2}_{=1} (\underbrace{Σ_j m_j^2}_{=1})) - 2 (\underbrace{Σ n_i m_i}_{=n · m} (\underbrace{Σ_j n_j m_j}_{= n·m}))  math = 1 + 1 - 2 (n·m)^2 = 2 (1 - (n·m)^2)  Define $n$ arbitrary (possibly non-continous) as $Q(t) = s(n(t) ⊗ n(t) - \frac13\operatorname{Id})$. Construct $n^+$ by starting at $n(t_1)$, then staying on the same side of the circle as $n^+(t_1)$ as long ("$δ$") as it is possible to prove. That makes it continous. Then start over at $n^+(t_1 + δ)$. This process is finite since a continous path into a bounded set on a compact domain is uniformly continous. There are only two continous liftings because otherwise we would have a jump by 180° where it changes from $n^+$ to $n^-$. Given that $Q$ stays close to some $\overline{Q}$, calculation shows that $n^+ · m$ can never be 0, so must be always $>0$. --- *Proposition 5*: $Ω$ simply connected, $Q$ continous. Then there exists a continous lifting $n: P(n) = Q$. *Proof*: See algebraic topology: q4-q23 Proposition 1.33, currently p. 70 --- *Theorem 1*: If $Q: G := Ω \setminus \bigcup_{i = 1}^N \overline{ω_i} → 𝒬_2$ is orientable at all $∂ω_i$, then $Q$ is orientable. *Proof sketch*: - For $f:[0,1] → \overline{Ω}$ define $f^*$ by replacing parts of $f$ in ω_i by (shorter) parts on $∂ω_i$. $f^*$ is continous (with proof in Lemma 4, but really?). - *Lemma 5*: 1) $\overline{Ω}$ is path-connected and simply connected. Proof: Schoenflies theorem: Jordan curve creates homeomorphism to unit disk. 2) $\overline{G}$ is path-connected. Proof: see previous bullet point. 3) $\overline{G}$ is locally path-connected: for every point $x ∈ \overline{G}$ and every radius $ε$, there is a ball $B_{δ}(x)$, such that all points in $B_{δ}(x) ∩ \overline{G}$ are path-connected to each other with paths within $B_{ε}(x) ∩ \overline{G}$ is path-connected. Proof: Use homeomorphism from Schoenflies theorem, build small balls and a straight line in unitdisk. - *Lemma 6*: ## Questions ... ...
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