diff --git a/thesis/2_ChainComplexes.tex b/thesis/2_ChainComplexes.tex index cf574b2..a1d377c 100644 --- a/thesis/2_ChainComplexes.tex +++ b/thesis/2_ChainComplexes.tex @@ -36,9 +36,9 @@ In order to see why we are interested in the construction of homology groups, we The topology on $\Delta^n$ is the subspace topology. \end{definition} -In particular $\Delta^0$ is simply a point, $\Delta^1$ a line and $\Delta^2$ a triangle. There are nice inclusions $\Delta^n \inject \Delta^{n+1}$ which we need later on. For any $n \in \N$ we define: +In particular $\Delta^0$ is simply a point, $\Delta^1$ a line and $\Delta^2$ a triangle. There are nice inclusions $\Delta^n \mono \Delta^{n+1}$ which we need later on. For any $n \in \N$ we define: \begin{definition} - For $i \in \{0, \ldots, n+1\}$ the $i$-th face map $\delta^i : \Delta^n \inject \Delta^{n+1}$ is defined as: + For $i \in \{0, \ldots, n+1\}$ the $i$-th face map $\delta^i : \Delta^n \mono \Delta^{n+1}$ is defined as: $$ \delta^i (x_0, \ldots, x_n) = (x_0, \ldots, x_{i-1}, 0, x_{i+1}, \ldots, x_n) \text{ for all } x \in \Delta^n.$$ \end{definition} diff --git a/thesis/3_SimplicialAbelianGroups.tex b/thesis/3_SimplicialAbelianGroups.tex index d384446..0fc9abf 100644 --- a/thesis/3_SimplicialAbelianGroups.tex +++ b/thesis/3_SimplicialAbelianGroups.tex @@ -115,20 +115,22 @@ Note that indeed $\Hom{\DELTA}{X}{[n]} \in \Set$, because the collection of morp \delta^1(\delta_0\sigma_0) &= \delta_0 \sigma_0 \delta_1 = \delta_0 \\ \delta^1(\delta_1\sigma_0) &= \delta_0 \sigma_0 \delta_1 = \delta_1. \end{align*} - \begin{tikzpicture} + + $$ \Delta[1] = + \begin{tikzpicture}[baseline=-0.5ex] \matrix (m) [matrix of math nodes] { - \Delta[1] = & \{x_0, x_1\} & \{\sigma^0 x_0, \id, \sigma^0 x_1\} & \{ \} & \cdots \\ + \{x_0, x_1\} & \{\sigma^0 x_0, \id, \sigma^0 x_1\} & \{ \} & \cdots \\ }; - \foreach \r in {-5, 5} \draw [raise line=\r, <-] (m-1-2) -> (m-1-3); - \foreach \r in {0} \draw [raise line=\r, ->] (m-1-2) -> (m-1-3); + \foreach \r in {-5, 5} \draw [raise line=\r, <-] (m-1-1) -> (m-1-2); + \foreach \r in {0} \draw [raise line=\r, ->] (m-1-1) -> (m-1-2); - \foreach \r in {-10, 0, 10} \draw [raise line=\r, <-] (m-1-3) -> (m-1-4); - \foreach \r in {-5, 5} \draw [raise line=\r, ->] (m-1-3) -> (m-1-4); + \foreach \r in {-10, 0, 10} \draw [raise line=\r, <-] (m-1-2) -> (m-1-3); + \foreach \r in {-5, 5} \draw [raise line=\r, ->] (m-1-2) -> (m-1-3); - \foreach \r in {-15, -5, 5, 15} \draw [raise line=\r, <-] (m-1-4) -> (m-1-5); - \foreach \r in {-10, 0, 10} \draw [raise line=\r, ->] (m-1-4) -> (m-1-5); - \end{tikzpicture} + \foreach \r in {-15, -5, 5, 15} \draw [raise line=\r, <-] (m-1-3) -> (m-1-4); + \foreach \r in {-10, 0, 10} \draw [raise line=\r, ->] (m-1-3) -> (m-1-4); + \end{tikzpicture}.$$ \end{example} As we are interested in simplicial abelian group, it would be nice to make these $n$-simplices into simplicial abelian groups. We have seen how to make an abelian group out of any set using the free abelian group. We can use this functor $\Z[-] : \Set \to \Ab$ to induce a functor $\Z^\ast[-] : \sSet \to \sAb$ as shown in the diagram~\ref{fig:diagram_Z}. diff --git a/thesis/4_Constructions.tex b/thesis/4_Constructions.tex index 9d9c46b..5f2a084 100644 --- a/thesis/4_Constructions.tex +++ b/thesis/4_Constructions.tex @@ -52,4 +52,41 @@ Now this is a functor, because it is a composition of functors. Furthermore it i $$ \Hom{A}{F \Z^{\ast} \Delta (-)}{-} : A \to \sAb $$ where we are supposed to fill in the second argument first, leaving us with a simplicial abelian group. -Now we know that $\Ch{\Ab}$ is an abelian group and we have actually two functors $C, N : \sAb \to \Ch{\Ab}$, so we now have functors from $\Ch{\Ab} \to \sAb$. +Now we know that $\Ch{\Ab}$ is an abelian group and we have actually two functors $C, N : \sAb \to \Ch{\Ab}$, so we now have functors from $\Ch{\Ab} \to \sAb$. Of course we will be interested in the one using $N$. So we define the functor: +$$ K(C) = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[-]}{C} \in \sAb. $$ +This definitions is very abstract, but luckily we can also give a more explicit definition. By writing it out for low dimensions we see: + +$$ K(C)_0 = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[0]}{C} = \Bigg\{ +\begin{tikzpicture}[baseline=-0.5ex] + \matrix (m) [matrix of math nodes, row sep=1em, column sep=1em] { + \cdots & 0 & 0 & \Z \\ + \cdots & C_2 & C_1 & C_0 \\ + }; + + \foreach \x in {1, 2} + \foreach \i/\j in {1/2, 2/3, 3/4} \draw[->] (m-\x-\i) -- (m-\x-\j); + + \foreach \i/\j in {2/2, 3/1, 4/0} \draw[->] (m-1-\i) -- node {$f_\j$} (m-2-\i); +\end{tikzpicture} +\Bigg\} \iso C_0, $$ +because for $f_1, f_2, \ldots$ there is now choice at all, and for $f_0 : \Z \to C_0$ we only have to choose an image for $1 \in \Z$. In the next dimension we see: + +$$ K(C)_1 = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[1]}{C} = \Bigg\{ +\begin{tikzpicture}[baseline=-0.5ex] + \matrix (m) [matrix of math nodes, row sep=1em, column sep=1em] { + \cdots & 0 & \Z & \Z^2 \\ + \cdots & C_2 & C_1 & C_0 \\ + }; + + \foreach \x in {1, 2} + \foreach \i/\j in {1/2, 2/3, 3/4} \draw[->] (m-\x-\i) -- (m-\x-\j); + + \foreach \i/\j in {2/2, 3/1, 4/0} \draw[->] (m-1-\i) -- node {$f_\j$} (m-2-\i); +\end{tikzpicture} +\Bigg\} \iso C_1 \oplus C_0, $$ + +because again we can choose $f_1$ anyway we want, which gives us $C_1$. But then we are forced to choose $f_0(x, x) = \del(f_1(x))$ for all $x \in \Z$, so we are left with choosing an element $c \in C_0$ for defining $f(1,-1) = c$. Adding this gives $C_1 \oplus C_0$. This pattern can be continued and gives the following result: + +\begin{proposition} + For any chain complex $C$ we have $K(C)_n \iso \bigoplus_{[n] \epi [p]} C_p$. +\end{proposition} diff --git a/thesis/preamble.tex b/thesis/preamble.tex index 6ce5af9..ccc9390 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -40,7 +40,8 @@ \newcommand{\eps}{\varepsilon} \newcommand{\I}{\,\mid\,} \newcommand{\then}{\Rightarrow} -\newcommand{\inject}{\hookrightarrow} +\newcommand{\mono}{\hookrightarrow} +\newcommand{\epi}{\twoheadrightarrow} \newcommand{\del}{\partial} \newcommand{\nsubgrp}{\trianglelefteq}