Commit f7e4dc1a by Jan-Bernhard Kordaß

### Aded draft of the first rtg lecture on the third rtg day.

parent 422d5e6f
 ... ... @@ -442,6 +442,222 @@ 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*} \section{Whitehead torsion for chain complexes} In the following let us repeat some preliminaries on chain complexes. Let$R$be a (not necessarily commutative) ring. Let$f_{*} \colon C_{*} \to D_{*}$be an$R$-chain map. The \CmMark{mapping cylinder}$\cyl(f_{*})$is an$R-chain complex with p-th differential \begin{align*} C_p \oplus C_{p-1} \oplus D_p \xrightarrow{% \begin{pmatrix} c_p & -\id & 0\\ 0 & -c_{p-1} & 0\\ 0 & f_{p-1} & f \end{pmatrix}} C_{p-1} \oplus C_{p-2} \oplus D_{p-1}, \quad \end{align*} \begin{rem*} For a continous mapf \colon X \to Y$we have the topological mapping cylinder. \begin{equation*} \begin{tikzcd} X \ar{r}{f} \ar[hook]{d}{i_0} & Y \ar[hook]{d}\\ X \times [0,1] \ar{r} & \cyl(f) \end{tikzcd} \end{equation*} If$X,Y$are CW-complexes and$fis cellular, then \begin{align*} \cyl(C_{*}(f) \colon C_{*}(X) \to C_{*}(Y)) = C_{*}(\cyl(f)). \end{align*} \end{rem*} The \CmMark{mapping cone}\cone(f_{*} \colon C_{*} \to D_{*})$is a quotient of$\cyl(f_{*})$by the obvious copy of$C_{*}, so its differential is \begin{align*} C_{p-1} \oplus D_+ \xrightarrow{% \begin{pmatrix} -c_{p-1} & 0 \\ f_{p-1} & d_p \end{pmatrix}} C_{p-2} \oplus D_{p-1} \end{align*} \begin{rem*} Again there is a topological analogue, the topological mapping cone\cone(f) = \cyl(f)/X \times \{1\}. These are related via \begin{align*} \cone_i(C_{*}(f)) = C_i(\cone(f)) \text{ for } i > 0. \end{align*} \end{rem*} The \CmMark{suspension}\Sigma C_{*}$of an$R$-chain complex$C_{*}$is a chain complex with$p-th differential \begin{align*} C_{p-1} \xrightarrow{-c_{p-1}} C_{p-2}, \end{align*} which is isomorphic to a quotient of\cone(\id_{C_{*}})$by$C_{*}. We have two exact sequences \begin{align*} & 0 \to C_{*} \to \cyl(f_{*}) \to \cone(f_{*}) \to 0 & 0 \to D_{*} \to \cone(f_{*}) \to \Sigma C_{*} \to 0 \end{align*} \begin{dfn*} AnR$-chain complex$C_{*}$is \CmMark{finite}, if$|C_p| = 0$for$p >> 0$and each$C_p$is finitely generated. It is called \CmMark{projective}, if each$C_p$is projective; \CmMark{free}, if each$C_p$is free, and \CmMark{based free}, if each$C_p$is based free with a preferred basis. \end{dfn*} \begin{rem*} Let$f_{*} \colon C_{*} \to D_{*}$be a chain map between projective chain complexes. Then the following statements are equivalent \begin{enumerate} \item$f_{*}$is a homology isomorphism (i.e.$H_i(f_{*})$is an isomorphism for all$i$) \item$f_{*}$is a chain homotopy equivalence \item$\cone(f_{*})$is contractible (i.e.$\cone(f_{*}) \simeq 0$). \end{enumerate} This can be seen from the following sequence (together with the fundamental lemma in homological algebra to show the equivalence of the first two statements). \begin{equation*} \begin{tikzcd} 0 \ar{r} & C_{*} \ar{r} & \cyl(f) \ar[<->]{d}{\simeq} \ar{r} & \cone(f_{*}) \ar{r} & 0\\ & & D_{*} & & \end{tikzcd} \end{equation*} A short exact sequence of chain complexes induces a long exact sequence in homology associated to$C_{*}$, which implies$\cone(f_{*}) = 0 \rightsquigarrow H_{*}(\cone(f_{*})) = 0\rightsquigarrow f_{*}$is homology isomorphism. \end{rem*} \begin{lemma*} Let$C_+$be a based free, finite$R$-chain complex that is contractible. Let$\gamma_p \colon C_p \to C_{p+1}$for$p \in \Zbe a chain contraction, i.e. \begin{align*} c_{p+1} \circ \gamma_p + \gamma_{p-1} \circ c_p = \id - 0. \end{align*} Then theR$-homomorphism$(c_{*} + \gamma_{*}) \colon C_{\text{odd}} \to C_{\text{ev}}$(where$C_{\text{odd}} = \bigoplus_p C_{2p+1}$and$C_{\text{ev}} = \bigoplus_p C_{2p}$) is an isomorphism. Let$A$be its representing matrix. Its class$[A] \in K_1(R)$is independent of the choice of$\gamma_{*}$. \end{lemma*} \begin{expl*} Let$C_p = 0$unless$i \in \{0,1,2\}$. Then \begin{equation*} \begin{tikzcd} 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''.$C_1 \xrightarrow{\cong} C_0 \oplus C_2$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 \begin{equation*} \begin{tikzcd} C_0 \oplus C_2 \ar{r}{\cong}[swap]{(\tilde \gamma_1, c_2)} & C_1 \ar{r}{\cong}[swap]{(c_1,\gamma_1)} & C_0 \oplus C_2 \end{tikzcd} \end{equation*} Let$x + y \in C_0 \oplus C_2. Then \begin{align*} \tilde \gamma_0(x) + c_2(y) \mapsto \ & c_1\tilde \gamma_0(x) + \gamma_1 \tilde\gamma_0(x) + \gamma_1 c_2(y)\\ & = (x - \underbrace{\tilde\gamma_2c_0(x))}_{=0} + \gamma_1\tilde\gamma_0(x) + (y - \underbrace{c_3\gamma_2(y)}_{=0})\\ & = x + y + \gamma_1 \tilde\gamma_0(x), \end{align*} which is represented by a matrix\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}$. \end{expl*} \begin{proof} Let$\gamma_{*}, \delta_{*}$be two chain contractions of$C_{*}. Then we consider \begin{align*} (c_{*} + \delta_{*})_{\text{odd}} \colon C_{\text{odd}} \xrightarrow{A} C_{\text{ev}} \text{ and } (c_{*} + \delta_{*})_{\text{ev}} \colon C_{\text{ev}} \xrightarrow{B} C_{\text{odd}} \end{align*} represented by matricesA$and$B$. 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$\gamma_{*}$. \end{proof} \begin{dfn*} \begin{enumerate}[label=(\roman*)] \item For a contractible, based free, finite$R$-chain complex$C_{*}define \begin{align*} \tau(C_{*}) := [(c_{*} + \gamma_{*})_{\text{odd}}] \in K_1(R) \end{align*} (for some or every) choice of\gamma_{*} \colon C_{*} \simeq 0$). \item Let$f_{*} \colon C_{*} \to D_{*}$be a chain homotopy equivalence of based free, finite$R$-chain complexes. The \CmMark{Whitehead torsion} of$f_{*}is \begin{align*} \tau(f_{*}) := \tau(\cone(f_{*})) \in K_1(R). \end{align*} \end{enumerate} \end{dfn*} We say that a short exact sequence of based free modules \begin{align*} 0 \to A \xrightarrow{j} B \xrightarrow{p} C \to 0 \end{align*} is \CmMark{based exact}, if\text{basis}_B = B_1 \coprod B_2$, such that$B_1 = j(\text{basis}_{A)}$and$p(B_2) = \text{basis}_C$. \begin{lemma*} Consider the following diagram with based exact rows. \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$\tau(g_{*}) = \tau(f_{*}) + \tau(h_{*}). \end{lemma*} \begin{proof} We have the following diagrams with exact rows and columns. \begin{equation*} \begin{tikzcd} & 0 \ar{d} & 0 \ar{d} & 0 \ar{d} & \\ 0 \ar{r} & C_{*}' \ar{r} \ar{d}{\simeq} & D_{*}' \ar{r} \ar{d}{\simeq} & E_{*}' \ar{r} \ar{d}{\simeq} & 0\\ 0 \ar{r} & \cyl(f_{*}) \ar{r} \ar{d} & \cyl(g_{*}) \ar{r} \ar{d} & \cyl(h_{*}) \ar{r} \ar{d} & 0\\ 0 \ar{r} & \cone(f_{*}) \ar{r} \ar{d} & \cone(g_{*}) \ar{r} \ar{d} & \cone(h_{*}) \ar{r} \ar{d} & 0\\ & 0 & 0 & 0 \end{tikzcd} \end{equation*} We may assume that the short exact sequence given by the columns \begin{align}\label{eq:ses-1} 0 \to C_{*} \xrightarrow{j_{*}} D_{*} \xrightarrow{p_{*}} E_{*} \to 0 \end{align} is a based exact sequence of contractible based exact sequence of contractible based free, finite chain complexes and then have to prove that\tau(D_{*}) = \tau(C_{*}) + \tau(E_{*})$. The sequence \eqref{eq:ses-1} splits as chain complexes.\footnote{This heavily depends on contractibility and is not true for arbitrary chain complexes.} Let$e_*$be a contraction of$E_{*}$and let$\sigma_i \colon E_i \to D_i$be a split of$p_i$for all$i$. Set$s_i \colon E_i \to D_i, s_i := d_{i+1} \circ \sigma_{i+1} \circ \varepsilon_i + \sigma_i \circ \varepsilon_{i-1} \circ e_i$. Claim:$s_{*}$is a chain map and$p_{*} \circ s_{*} = \id_{E_{*}}. Hence we obtain an isomorphism of chain complexes \begin{align*} (j_{*},s_{*}) \colon C_{*} \oplus E_{*} \xrightarrow{\cong} D_{*}, \end{align*} which has a corresponding matrix of the form\begin{pmatrix}\Id & * \\ 0 & \Id \end{pmatrix}$. We can finish the proof with the following remark. If$u_{*} \colon C_{*} \to D_{*}is a chain isomorphism of then \begin{align*} \tau(C_{*}) - \tau(D_{*}) = \sum_p (-1)^p [u_p] \in K_1(R). \end{align*} This can be shown by transporting a chain contraction\gamma_{*}$for$C_{*}$to one for$D_{*}$via$u_{*}\$, which yields a diagram: \begin{equation*} \begin{tikzcd} C_{\text{odd}} \ar{r} \ar{d}{u_{\text{odd}}}[swap]{\simeq} & C_{\text{ev}} \ar{d}{u_{\text{ev}}}[swap]{\simeq}\\ D_{\text{odd}} \ar{r} & D_{\text{ev}}. \end{tikzcd} \end{equation*} \end{proof} \chapter{Harmonic Maps [Andy Sanders]} ... ...
 ... ... @@ -55,6 +55,9 @@ \DeclareMathOperator{\Characteristic}{char} \DeclareMathOperator{\aff}{aff} % affine Hülle \DeclareMathOperator{\cone}{cone} % \DeclareMathOperator{\cyl}{cyl} % % Lie Theory \DeclareMathOperator{\Ad}{Ad} % Ad \DeclareMathOperator{\ad}{ad} % ad ... ... @@ -68,6 +71,7 @@ \DeclareMathOperator{\lk}{lk} % link \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\rg}{rank} \DeclareMathOperator{\rank}{rank} ... ...
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