Used spell-checking program.

contents.tex

@@ -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 $f_t \colon M \to N, \frac{\dop}{\dop t}f_t|_{t = 0} = 0$.