Commit a07bad41 by Jan-Bernhard Kordaß

### Added Romans fourth lecture.

parent 15e2fdd0
 ... ... @@ -442,6 +442,7 @@ The inverse homomorphism is given by $R^n \cong \{ P \oplus Q \xrightarrow{f \ot One can see that$1 - t - t^{-1}$is a unit in$\Z[\Z/5]$, since$(1 - t - t^{-1})( - t^2 - t^3) = 1$and thus$\tau([1 - t - t^{-1}]) \neq 1$\end{expl*} \subsection{Whitehead torsion for chain complexes} In the following let us repeat some preliminaries on chain complexes. ... ... @@ -659,14 +660,194 @@ Then$\tau(g_{*}) = \tau(f_{*}) + \tau(h_{*})$. \end{proof} Recall that last time we proved the following Lemma. \begin{lemma*} \begin{enumerate}[label=(\arabic*)] \item Additivity \begin{equation*} \begin{tikzcd} 0 \ar{r} & C_{*}' \ar{r} \ar{d}{f_{*}}[swap]{\simeq} & D_{*}' \ar{r} \ar{d}{g_{*}}[swap]{\simeq} & E_{*}' \ar{r} \ar{d}{h_{*}}[swap]{\simeq} & 0\\ 0 \ar{r} & C_{*} \ar{r} & D_{*} \ar{r} & E_{*} \ar{r} & 0 \end{tikzcd} \end{equation*} Then$\tau(g_{*}) = \tau(f_{*}) + \tau(h_{*})$. \item Homotopy invariance. If$f_{*} \simeq g_{*} \colon C_{*} \xrightarrow{\simeq} D_{*}$, then$\tau(f_{*}) = \tau(g_{*})$. \item Composition formula.$\tau(g_{*} \circ f_{*}) = \tau(g_{*}) + \tau(f_{*})$. \end{enumerate} \end{lemma*} \begin{proof} \begin{enumerate}[label=ad (\arabic*),leftmargin=1.5cm] \setcounter{enumi}{2} \item If$h_{*} \colon f_{*} \simeq g_{*}, then we have an isomorphism \begin{align*} \cone(f_{*}) \colon C_{*-1} \oplus D_{*} \xrightarrow{\begin{pmatrix}\id & 0 \\ h_{*-1} & \id\end{pmatrix}} C_{*-1} \oplus D_{*} = \cone(g_{*}). \end{align*} Thus \begin{align*} & \tau(f_{*}) - \tau(g_{*}) = \tau(\cone(f_{*})) - \tau(\cone(g_{*}))\\ & \qquad = \sum_{p \geq 0} (-1)^p \begin{pmatrix}\id & 0 \\ h_{*-1} & \id\end{pmatrix} = 0 \in K_1(R). \end{align*} \end{enumerate} \end{proof} \subsection{Whitehead torsion of maps between CW-complexes} LetX,Y$be connected finite CW-complexes and let$f \colon X \xrightarrow{\simeq} Y$be a homotopy equivalence. Further pick base points$x \in X, y = f(x) \in Y$and set$\pi := \pi_1(Y,y)$. We identify$\pi_1(X,x)$with$\pi$via$\pi_1(f)$and considering the universal covers we obtain the diagram \begin{equation*} \begin{tikzcd} \tilde X \ar{r}{\tilde f}[swap]{\simeq} \ar{d}{\pr_X} & \tilde Y \ar{d}{\pr_Y}\\ X \ar{r}{f}[swap]{\simeq} & Y \end{tikzcd} \end{equation*} There is a unique lift$\tilde f$with$\tilde f(\tilde x) = \tilde y$. Further,$\tilde f$is$\pi$-equivariant.$\tilde X$caries a CW structure$\tilde X^n = \pr_X^{-1}(X^n)$. Thus$C^{\text{CW}}_{*}(\tilde f) \colon C_{*}^{\text{CW}}(\tilde X) \xrightarrow{\simeq} C_{*}^{\text{CW}}(\tilde Y)$, where$C_{n}^{CW}(\tilde X) = H_n(\tilde X^n, \tilde X^{n-1})$, are chain homotopy equivalences of$\Z\pi$-chain complexes. We obtain a$\Z\pi$-basis of$C_n^{\text{CW}}(\tilde X)$by choosing ($\pi-)push-outs: \begin{equation*} \begin{tikzcd} \coprod_{I_n} \pi \times S^{n-1} \ar{r} \ar[hook]{d} & \tilde X^{n-1} \ar[hook]{d} \\ \coprod_{I_n} \pi \times D^n \ar{r} & \tilde X^n \end{tikzcd}. \end{equation*} This yields a cellular basis \begin{align*} C_n^{\text{CW}}(\tilde X) = H_n(\tilde X^n, \tilde X^{n-1}) \xleftarrow{\cong} \bigoplus_{I_n} H_n(\pi \times (D^n, S^{n-1})) = \bigoplus_{I_n}\Z\pi \end{align*} The matrix of base change when we take another push-out looks like \begin{align*} \bigoplus_{I_n} \Z\pi \xrightarrow{% \begin{pmatrix}\pm g_1 & & \\ & \ddots &\\ & & \pm g_n\end{pmatrix}P} \bigoplus_{I_n} \Z\pi, \end{align*} whereP$is a permutation matrix. Hence we obtain a well-defined element$\tau(C_{*}^{\text{CW}}(\tilde f)) \in \Wh(\pi)$independent of the choices of cellular bases of$\tilde X, \tilde Y$. It may still depend on the choice of basepoints. Let$y,y' \in Y$be connected by two paths$\alpha$and$\beta$.\marginpar{TODO} \begin{equation*} \text{figure: paths} \end{equation*} Then$\pi_1(Y,y) \xrightarrow{\alpha_{*}, \beta_{*}} \pi_1(Y,y')$differ by an inner automorphism of$\pi_1(Y,y')$, which induces the identity on$\Wh(\pi_1(Y,y')). So there is a canonical isomorphism \begin{align*} \phi_{y,y'} \colon \Wh(\pi_1(Y,y)) \xrightarrow{\cong} \Wh(\pi_1(Y,y')) \end{align*} induced by any choice of path fromy$to$y'$. This yields a basepoint free definition of the Whitehead group. \begin{dfn*}$\Wh(\pi_1(Y)) := \colim_{y \in Y} \Wh(\pi_1(Y,y)) = \coprod_{y \in Y}\Wh(\pi_1(Y,y))/\sim$for$z \sim \phi_{y,y'}$. \end{dfn*} One verifies that$\tau(C_{*}^{\text{CW}}(\tilde f)) \in \Wh(\pi(Y))$is independent of all choices of basepoints. \begin{dfn*} Let$X \xrightarrow{f} Ybe a homotopy equivalence of finite CW-complexes. Define \begin{align*} \Wh(\pi(Y)) := \bigoplus_{C \in \pi_0(Y)} \Wh(\pi(C)) \end{align*} and \begin{align*} \tau(f) := \left(\tau(C_{*}^{\text{CW}}(\tilde f \colon \tilde{f^{-1}(C)} \to \tilde C))\right)_{C \in \pi_0(Y)}, \end{align*} which is called the \CmMark{Whitehead torsion} off$. \end{dfn*} \begin{rem*} The elementary properties of$\tau(f)$stated in the beginning of chapter 2 are now direct consequences of the lemma in section 2.3. \end{rem*} In the following we will discuss the topological meaning of$\tau(f)$. For this let$S^{n-2} \subset S_+^{n-1} \subset S^{n-1} \subset D^n$, where$S_+^{n-1}$denotes the upper hemisphere.\marginpar{TODO: figure} \begin{equation*} \begin{tikzcd} S_+^{n-1} \ar{r}{q} \ar[hook]{d}{\simeq} & X \ar[hook]{d}{\simeq} \\ D^n \ar{r}{\bar q} & Y \end{tikzcd} \end{equation*} Such a homotopy equivalence$X \to Y$as in the diagram is called \CmMark{elementary expansion}. A map$r \colon Y \to X$such that$r \circ j = \id_X$is called \CmMark{elementary collapse} ($r$will be a homotopy inverse of$j$). \begin{rem*} If$X$is a CW-complex and$q(S^{n-2}) \subset X^{n-2}$with$q(S^{n-1}_+) \subset X^{n-1}$, then$Y$inherits a natural CW-structure. Then$Y$is the result of attaching an$(n-1)$-cell and then an$n$-cell to$X$. \end{rem*} \begin{dfn*} Let$f \colon X \to Y$be a map of finite CW-complexes. We call$f$a \CmMark{simple homotopy equivalence}, if it is homotopic to a zig-zag of elementary expansions and collapses, i.e. there exists maps$f(i)$such that \begin{equation*} \begin{tikzcd} X = X(0) \ar{r}{f(0)} \ar[bend right]{rrr}{f} & X(1) \ar{r} & \cdots \ar{r}{f(n-1)} & X(n) = Y \end{tikzcd}, \end{equation*} commutes up to homotopy, where each$f(i)$is an elementary expansion or collapse. \end{dfn*} \begin{thm*} \begin{enumerate}[label=(\arabic*)] \item A homotopy equivalence$f \colon X \to Y$is simple if and only if$\tau(f) = 0$. \item For every element$a \in \Wh(\pi(X))$there is a homotopy equivalence$X \xrightarrow{f} Y$such that$f_{*}^{-1}(\tau(f)) \in \Wh(\pi(X))$. \end{enumerate} \end{thm*} \begin{proof} \begin{enumerate}[label=ad (\arabic*),leftmargin=1.5cm] \item We will only prove the direction simple$\implies \tau(f) = 0$''. We may assume that$f$is an elementary expansion \begin{equation*} \begin{tikzcd} 0 \ar{r} & C_{*}(\tilde X) \ar{r}{f_{*}} & C_{*}(\tilde Y) \ar{r} & C_{*}(\tilde Y, \tilde X) \ar{r} & 0\\ 0 \ar{r} & C_{*}(\tilde X) \ar{r}{\id} \ar{u}{\id} & C_{*}(\tilde X) \ar{r} \ar{u}{f_{*}} & 0 \ar{r} \ar{u}{0} & 0 \end{tikzcd} \end{equation*} Thus, by additivity, we have$\tau(f) = \tau(C_{*}(\tilde Y, \tilde X))$. But this is a very simple CW-complex with differential$\id \colon \Z\pi \to \Z\pi$(for$\pi = \pi_1(Y)$) from degree$n$to degree$n-1$and thus$\tau(C_{*}(\tilde Y,\tilde X)) = 0$. The converse is more complicated. \item Let$A \in \GL_n(\Z\pi)$($n \geq 3$) and let$X' = X \vee \bigvee_{j = 1}^nS^{n-1}$, where we denote the$j$-th inclusion of$S^{n-1}$into the wedge by$b_j$. We attach$nn$-cells to$X'$via attaching maps$f_j \colon S^{n-1} \to X'$(relevant:$[f_j] \in \i_{n-1}(X')$which yields$Y. \begin{align*} \begin{pmatrix} f_1 \\ \vdots \\ f_n \end{pmatrix} = A \cdot \begin{pmatrix} b_1 \\ \vdots \\ b_n \end{pmatrix} \ni \bigoplus_{j = 1}^n \pi_{n-1}(X') \text{ for } b_j \in \pi_{n-1}(X') \curvearrowleft \pi_1(X') = \pi_1(X) = \pi. \end{align*} One sees by closer inspection that \begin{align*} \tau(\tilde X \hookrightarrow \tilde Y) = \tau(C_{*}^{\text{CW}}(\tilde Y, \tilde X)) = \tau\left(\bigoplus_{j=1}^n \Z\pi \xrightarrow{A} \bigoplus_{j=1}^n \Z\pi\right) \end{align*} for[A] \in \Wh(\pi)$. \end{enumerate} \end{proof} \chapter{Harmonic Maps [Andy Sanders]} Also consider the notes \url{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}. \section{Basics of harmonic maps} In the following let every manifold be oriented (for integration safety reasons). \subsection{Background differential geometry} Let$E \to M$be an$\R$-vector bundle over$M$(second countable, hausdorff manifold) of rank$r\$. ... ...
 ... ... @@ -87,6 +87,7 @@ \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\colim}{colim} % colimit \DeclareMathOperator{\pr}{pr} % projection \DeclareMathOperator{\Hom}{Hom} % homomorphisms \DeclareMathOperator{\Diff}{Diff} % diffeomorphisms ... ...
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