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 \subsection{CW Complexes} \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*} \begin{expl*} \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: \begin{center} \begin{tikzpicture} ... ... @@ -73,14 +73,14 @@ Then we can consider the \CmMark{n-skeleton} \begin{align*} X^n := \sum_{e \in E, \dim e \leq n} e, \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{tikzcd} \coprod S^{n-1} \ar{r} \ar[hook]{d} & X^{n-1} \ar[hook]{d} \\ \coprod D^n \ar{r} & X^n \end{tikzcd} \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 $X^n\setminus X^{n-1}$. We have ... ... @@ -112,7 +112,7 @@ Recall that a map $f \colon X \to P$ between CW-complexes is \CmMark{cellular}, \end{thm*} \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*} \Lambda(f) = \sum_{i \geq 0} (-1)^i\tr C_i^{CW}(f) \in \Z. \end{align*} ... ... @@ -168,14 +168,14 @@ Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-co \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 additivity: A cellular pushout is a diagram A cellular push-out is a diagram \begin{equation*} \begin{tikzcd} X_0 \ar{r}{f} \ar[hook]{d}{i} & X_2 \ar{d}\\ X_1 \ar{r} & X \end{tikzcd} \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 \begin{equation*} \begin{tikzcd} ... ... @@ -185,7 +185,7 @@ Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-co & Y_1 \ar{rr}{\psi} & & Y \end{tikzcd} \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 \begin{align*} \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 \end{align*} \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{tikzcd} X \ar{r}{f} \ar[hook]{d}{i_0} & Y \ar[hook]{d}\\ ... ... @@ -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 \end{tikzcd} \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 $C_0 \oplus C_2 \xrightarrow{\cong} C_1, (x,y) \mapsto \gamma_0(x) + c_2(y)$. Let $\tilde\gamma$ be another chain contraction ... ... @@ -580,7 +580,7 @@ We have two exact sequences Define $\mu_n = (\gamma_{n+1} - \delta_{n+1}) \circ \delta_n$ and $\nu_n = (\delta_{n+1} - \gamma_{n+1}) \circ \gamma_n$. 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} \begin{dfn*} ... ... @@ -738,7 +738,7 @@ is a number. \end{dfn*} \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*} \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 \begin{align*} \tau(f) := \tr_g(\nabla \dop f) = 0 \in \Omega^0(M,f^*\T N). \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*} \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*} ... ... @@ -870,7 +870,7 @@ where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$. \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, \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 $f \colon (M,g) \to \R$. Here $\tau(f) = \Delta_gf = 0$. \begin{prop*} ... ... @@ -961,7 +961,7 @@ There are various useful applications of the synthetic view'' on harmonic func \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 $\Omega^p(M,E)$. \begin{align*} ... ... @@ -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 $(N,h)$. 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*} \begin{thm*}[Eells-Sampson 1964] ... ... @@ -1047,10 +1047,10 @@ From now on, best'' means harmonic with respect to some riemannian metric. Try to take $\tau(u) = 0$ for some $u \sim f$. 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. 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 $f(\blank, 0) = f$. 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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