@@ -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}

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@@ -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

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

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@@ -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*}

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@@ -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

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*}

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@@ -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*}

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@@ -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]

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