Commit c5d131b0 by Jan-Bernhard Kordaß

### Used spell-checking program.

parent daf444bb
 ... @@ -13,12 +13,12 @@ Informally, corresponding primary invariants are Lefschetz numbers (Whitehead to ... @@ -13,12 +13,12 @@ Informally, corresponding primary invariants are Lefschetz numbers (Whitehead to \subsection{CW Complexes} \subsection{CW Complexes} \begin{dfn*} \begin{dfn*} A (finite) \CmMark{CW-complex} is a hausdorff space with a decomposition $E$ into (finitely many) cells (space hemeomorphic to some $\R^n$) such that for every $e \in E$ there is a continuous map $\phi_e \colon D^n \to X$ with $\phi_e \colon \mathring D^n \xrightarrow{\cong} e$ and $\Im(\phi_e|_{S^{n-1}}) \subset \bigcup_{f \in E, \dim f \leq n-1} f$. A (finite) \CmMark{CW-complex} is a hausdorff space with a decomposition $E$ into (finitely many) cells (space homeomorphic to some $\R^n$) such that for every $e \in E$ there is a continuous map $\phi_e \colon D^n \to X$ with $\phi_e \colon \mathring D^n \xrightarrow{\cong} e$ and $\Im(\phi_e|_{S^{n-1}}) \subset \bigcup_{f \in E, \dim f \leq n-1} f$. \end{dfn*} \end{dfn*} \begin{expl*} \begin{expl*} \begin{enumerate} \begin{enumerate} \item Simplicial complexes, e.g. triangles, pyramides, etc. \item Simplicial complexes, e.g. triangles, pyramids, etc. \item But CW-complexes are more general, the following graph is CW for example: \item But CW-complexes are more general, the following graph is CW for example: \begin{center} \begin{center} \begin{tikzpicture} \begin{tikzpicture} ... @@ -73,14 +73,14 @@ Then we can consider the \CmMark{n-skeleton} ... @@ -73,14 +73,14 @@ Then we can consider the \CmMark{n-skeleton} \begin{align*} \begin{align*} X^n := \sum_{e \in E, \dim e \leq n} e, X^n := \sum_{e \in E, \dim e \leq n} e, \end{align*} \end{align*} which yields a filtration $X^0 \subset X^1 \subset \cdots \subset X$ such there is a pushout diagram which yields a filtration $X^0 \subset X^1 \subset \cdots \subset X$ such there is a push-out diagram \begin{equation*} \begin{equation*} \begin{tikzcd} \begin{tikzcd} \coprod S^{n-1} \ar{r} \ar[hook]{d} & X^{n-1} \ar[hook]{d} \\ \coprod S^{n-1} \ar{r} \ar[hook]{d} & X^{n-1} \ar[hook]{d} \\ \coprod D^n \ar{r} & X^n \coprod D^n \ar{r} & X^n \end{tikzcd} \end{tikzcd} \end{equation*} \end{equation*} One could take this as an alternative definition of a CW-complex by a filtration with the pushout property. One could take this as an alternative definition of a CW-complex by a filtration with the push-out property. The cells can be recovered as connected components of $X^n\setminus X^{n-1}$. The cells can be recovered as connected components of $X^n\setminus X^{n-1}$. We have We have ... @@ -112,7 +112,7 @@ Recall that a map $f \colon X \to P$ between CW-complexes is \CmMark{cellular}, ... @@ -112,7 +112,7 @@ Recall that a map $f \colon X \to P$ between CW-complexes is \CmMark{cellular}, \end{thm*} \end{thm*} \begin{dfn*} \begin{dfn*} The \CmMark{Lefschetz number} of a self-map $f \colon X \to X$ of a finite CW-complex is defnined as The \CmMark{Lefschetz number} of a self-map $f \colon X \to X$ of a finite CW-complex is defined as \begin{align*} \begin{align*} \Lambda(f) = \sum_{i \geq 0} (-1)^i\tr C_i^{CW}(f) \in \Z. \Lambda(f) = \sum_{i \geq 0} (-1)^i\tr C_i^{CW}(f) \in \Z. \end{align*} \end{align*} ... @@ -168,14 +168,14 @@ Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-co ... @@ -168,14 +168,14 @@ Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-co \item homotopy invariance \item homotopy invariance \item\footnote{This is a deep theorem of Chapman.} If $f \colon X \to Y$ is a homeomorphism, then $\tau(f) = 0$. \item\footnote{This is a deep theorem of Chapman.} If $f \colon X \to Y$ is a homeomorphism, then $\tau(f) = 0$. \item additivity: \item additivity: A cellular pushout is a diagram A cellular push-out is a diagram \begin{equation*} \begin{equation*} \begin{tikzcd} \begin{tikzcd} X_0 \ar{r}{f} \ar[hook]{d}{i} & X_2 \ar{d}\\ X_0 \ar{r}{f} \ar[hook]{d}{i} & X_2 \ar{d}\\ X_1 \ar{r} & X X_1 \ar{r} & X \end{tikzcd} \end{tikzcd} \end{equation*} \end{equation*} with $X_i$ be CW-complexes, where $f$ is cellular and $i$ is an inclusion of a subcomplex. with $X_i$ be CW-complexes, where $f$ is cellular and $i$ is an inclusion of a sub-complex. If the diagram If the diagram \begin{equation*} \begin{equation*} \begin{tikzcd} \begin{tikzcd} ... @@ -185,7 +185,7 @@ Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-co ... @@ -185,7 +185,7 @@ Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-co & Y_1 \ar{rr}{\psi} & & Y & Y_1 \ar{rr}{\psi} & & Y \end{tikzcd} \end{tikzcd} \end{equation*} \end{equation*} is a map of cellular pushouts such that $f_i$ are homotopy equivalences. is a map of cellular push-outs such that $f_i$ are homotopy equivalences. Then $f$ is a homotopy equivalence and Then $f$ is a homotopy equivalence and \begin{align*} \begin{align*} \tau(f) & = "\tau(f_1) + \tau(f_2) - \tau(f_0)"\\ \tau(f) & = "\tau(f_1) + \tau(f_2) - \tau(f_0)"\\ ... @@ -462,7 +462,7 @@ The \CmMark{mapping cylinder} $\cyl(f_{*})$ is an $R$-chain complex with p-th di ... @@ -462,7 +462,7 @@ The \CmMark{mapping cylinder} $\cyl(f_{*})$ is an $R$-chain complex with p-th di \end{align*} \end{align*} \begin{rem*} \begin{rem*} For a continous map $f \colon X \to Y$ we have the topological mapping cylinder. For a continuous map $f \colon X \to Y$ we have the topological mapping cylinder. \begin{equation*} \begin{equation*} \begin{tikzcd} \begin{tikzcd} X \ar{r}{f} \ar[hook]{d}{i_0} & Y \ar[hook]{d}\\ X \ar{r}{f} \ar[hook]{d}{i_0} & Y \ar[hook]{d}\\ ... @@ -548,7 +548,7 @@ We have two exact sequences ... @@ -548,7 +548,7 @@ We have two exact sequences 0 \ar{r} & C_2 \ar{r}[swap]{c_2} & C_1 \ar{r}[swap]{c_1} \ar[bend right]{l}[swap]{\gamma_1} & C_0 \ar{r} \ar[bend right]{l}[swap]{\gamma_0} & 0 0 \ar{r} & C_2 \ar{r}[swap]{c_2} & C_1 \ar{r}[swap]{c_1} \ar[bend right]{l}[swap]{\gamma_1} & C_0 \ar{r} \ar[bend right]{l}[swap]{\gamma_0} & 0 \end{tikzcd} \end{tikzcd} \end{equation*} \end{equation*} is the full complex and thus contractibe'' means short exact''. is the full complex and thus contractible'' means short exact''. $C_1 \xrightarrow{\cong} C_0 \oplus C_2$ via $x \mapsto c_1 (x) + \gamma_1(x)$ with inverse $C_0 \oplus C_2 \xrightarrow{\cong} C_1, (x,y) \mapsto \gamma_0(x) + c_2(y)$. $C_1 \xrightarrow{\cong} C_0 \oplus C_2$ via $x \mapsto c_1 (x) + \gamma_1(x)$ with inverse $C_0 \oplus C_2 \xrightarrow{\cong} C_1, (x,y) \mapsto \gamma_0(x) + c_2(y)$. Let $\tilde\gamma$ be another chain contraction Let $\tilde\gamma$ be another chain contraction ... @@ -580,7 +580,7 @@ We have two exact sequences ... @@ -580,7 +580,7 @@ We have two exact sequences Define $\mu_n = (\gamma_{n+1} - \delta_{n+1}) \circ \delta_n$ and $\nu_n = (\delta_{n+1} - \gamma_{n+1}) \circ \gamma_n$. Define $\mu_n = (\gamma_{n+1} - \delta_{n+1}) \circ \delta_n$ and $\nu_n = (\delta_{n+1} - \gamma_{n+1}) \circ \gamma_n$. Then $(\id + \mu_{*})_{\text{odd}}$, $(\id + \nu_{*})_{\text{ev}}$ and $(c_{*} + \gamma_{*})_{\text{odd}} \circ (\id + \mu_{*})_{\text{odd}} \circ (c_{*} + \delta_{*})_{\text{ev}}$ are represented by upper triangular matrices with ones on the diagonal. Then $(\id + \mu_{*})_{\text{odd}}$, $(\id + \nu_{*})_{\text{ev}}$ and $(c_{*} + \gamma_{*})_{\text{odd}} \circ (\id + \mu_{*})_{\text{odd}} \circ (c_{*} + \delta_{*})_{\text{ev}}$ are represented by upper triangular matrices with ones on the diagonal. Thus $[A] = -[B]$ in $K_1(R)$ and $B$ is independent of the choice of $\gamma_{*}$, hence $A$ is independed of the choice of $\gamma_{*}$. Thus $[A] = -[B]$ in $K_1(R)$ and $B$ is independent of the choice of $\gamma_{*}$, hence $A$ is independent of the choice of $\gamma_{*}$. \end{proof} \end{proof} \begin{dfn*} \begin{dfn*} ... @@ -738,7 +738,7 @@ is a number. ... @@ -738,7 +738,7 @@ is a number. \end{dfn*} \end{dfn*} \begin{rem*}[Fact] \begin{rem*}[Fact] An integration by parts arguement shows that $\delta^{\nabla}$ exists and, when $\nabla$ is a metric connection, then An integration by parts argument shows that $\delta^{\nabla}$ exists and, when $\nabla$ is a metric connection, then \begin{align*} \begin{align*} \delta^{\nabla} \colon \Omega^1(M,E) \to \Omega^0(M,E), \delta^{\nabla} \colon \Omega^1(M,E) \to \Omega^0(M,E), \ \ ... @@ -846,7 +846,7 @@ Recall that above we considered $C^2$-maps $f \colon (M,g) \to (N,h)$ with tensi ... @@ -846,7 +846,7 @@ Recall that above we considered $C^2$-maps $f \colon (M,g) \to (N,h)$ with tensi \begin{align*} \begin{align*} \tau(f) := \tr_g(\nabla \dop f) = 0 \in \Omega^0(M,f^*\T N). \tau(f) := \tr_g(\nabla \dop f) = 0 \in \Omega^0(M,f^*\T N). \end{align*} \end{align*} In local co-ordinates $\{x^i\}$ on $M$ and $\{y^{\alpha}\}$ on $N$ this means \footnote{Use roman indices for the $M$ and greek ones for $N$.} In local co-ordinates $\{x^i\}$ on $M$ and $\{y^{\alpha}\}$ on $N$ this means \footnote{Use roman indices for the $M$ and Greek ones for $N$.} \begin{align*} \begin{align*} \tau(f)^{\gamma} \frac{\partial}{\partial y^{\gamma}} = (\Delta_g f^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})\partial_{\gamma} = 0, \tau(f)^{\gamma} \frac{\partial}{\partial y^{\gamma}} = (\Delta_g f^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})\partial_{\gamma} = 0, \end{align*} \end{align*} ... @@ -870,7 +870,7 @@ where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$. ... @@ -870,7 +870,7 @@ where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$. \begin{align*} \begin{align*} \tau(\eta)^{\gamma}\partial_{\gamma} = (\ddot \eta^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(\eta) \dot\eta^{\alpha}\dot\eta^{\beta})\partial_{\gamma} = 0, \tau(\eta)^{\gamma}\partial_{\gamma} = (\ddot \eta^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(\eta) \dot\eta^{\alpha}\dot\eta^{\beta})\partial_{\gamma} = 0, \end{align*} \end{align*} which is if and only if $\eta$ is a geodesic, i.e. the covariant derivate along $M$ of the curves speed vanishes: $\frac{\Dop}{\dop t} \dot \eta = 0$ and thus $E(\eta)|_a^b = \frac{1}{2}\int_a^b\|\dot\eta\|^2\dop t$. which is if and only if $\eta$ is a geodesic, i.e. the covariant derivative along $M$ of the curves speed vanishes: $\frac{\Dop}{\dop t} \dot \eta = 0$ and thus $E(\eta)|_a^b = \frac{1}{2}\int_a^b\|\dot\eta\|^2\dop t$. \item Now let $f \colon (M,g) \to \R$. \item Now let $f \colon (M,g) \to \R$. Here $\tau(f) = \Delta_gf = 0$. Here $\tau(f) = \Delta_gf = 0$. \begin{prop*} \begin{prop*} ... @@ -961,7 +961,7 @@ There are various useful applications of the synthetic view'' on harmonic func ... @@ -961,7 +961,7 @@ There are various useful applications of the synthetic view'' on harmonic func \subsection{Bochner formulas} \subsection{Bochner formulas} Let $(E,\nabla,a)$ be a riemannina vector bundle, i.e. $a$ is a metric on $E$, $\nabla$ is a connection on $E$ preserving $a$ ($\nabla a = 0$) and there is a vector bundle projection map $E \to (M,g)$. Let $(E,\nabla,a)$ be a riemannian vector bundle, i.e. $a$ is a metric on $E$, $\nabla$ is a connection on $E$ preserving $a$ ($\nabla a = 0$) and there is a vector bundle projection map $E \to (M,g)$. Let $\omega \in \Omega^p(M,E)$ and let $\nabla$ be a connection on $\Omega^p(M,E)$. Let $\omega \in \Omega^p(M,E)$ and let $\nabla$ be a connection on $\Omega^p(M,E)$. \begin{align*} \begin{align*} ... @@ -1037,7 +1037,7 @@ From now on, best'' means harmonic with respect to some riemannian metric. ... @@ -1037,7 +1037,7 @@ From now on, best'' means harmonic with respect to some riemannian metric. In this case it depends on the curvature of $(N,h)$. In this case it depends on the curvature of $(N,h)$. Consider the flat torus $\mathbb T^2$ and the round sphere $\mathbb S^2$. Consider the flat torus $\mathbb T^2$ and the round sphere $\mathbb S^2$. For a dregree 1 map $\mathbb T^2 \to \mathbb S^2$ there is no harmonic map in the homotopy class (see the book by Lin on geometry of harmonic maps). For a degree 1 map $\mathbb T^2 \to \mathbb S^2$ there is no harmonic map in the homotopy class (see the book by Lin on geometry of harmonic maps). \end{expl*} \end{expl*} \begin{thm*}[Eells-Sampson 1964] \begin{thm*}[Eells-Sampson 1964] ... @@ -1047,10 +1047,10 @@ From now on, best'' means harmonic with respect to some riemannian metric. ... @@ -1047,10 +1047,10 @@ From now on, best'' means harmonic with respect to some riemannian metric. Try to take $\tau(u) = 0$ for some $u \sim f$. Try to take $\tau(u) = 0$ for some $u \sim f$. In this approach $E \colon C^2(M,N) \to \R, f \mapsto \frac{1}{2} \int_M \|\dop f\|^2 dV_g$ such that $E(f_n) \to \inf_{f \in C^2}E(f)$, we would have to weaken to topology considering Sobolov spaces $W^{1,2}(M,N)$ In this approach $E \colon C^2(M,N) \to \R, f \mapsto \frac{1}{2} \int_M \|\dop f\|^2 dV_g$ such that $E(f_n) \to \inf_{f \in C^2}E(f)$, we would have to weaken to topology considering Sobolev spaces $W^{1,2}(M,N)$ The other approach using gradient flow goes as follows. The other approach using gradient flow goes as follows. Try to solve initival value problem (IVP). Try to solve initial value problem (IVP). Let $f \colon M \times (0,\infty) \to N$, such that $\frac{\partial f}{\partial t} = \tau(f_t)$ and $f(\blank, 0) = f$. Let $f \colon M \times (0,\infty) \to N$, such that $\frac{\partial f}{\partial t} = \tau(f_t)$ and $f(\blank, 0) = f$. Recall the first variation of every $f_t \colon M \to N, \frac{\dop}{\dop t}f_t|_{t = 0} = 0$. Recall the first variation of every $f_t \colon M \to N, \frac{\dop}{\dop t}f_t|_{t = 0} = 0$. ... ...
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