Commit 8e6ffa3c authored by Han-Miru Kim's avatar Han-Miru Kim
Browse files

Alg Topo le 1 - 11

parent 77b0f321
......@@ -170,8 +170,9 @@ Packages for sectioning and positioning.
\DeclareMathOperator{\Bil}{Bil}
\DeclareMathOperator{\ev}{ev}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator{\kernel}{ker}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\image}{im}
\DeclareMathOperator{\spn}{span}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\rang}{rang}
......@@ -191,10 +192,13 @@ Packages for sectioning and positioning.
% Cat
\newcommand*{\Top}{\text{\sffamily Top}}
\newcommand*{\Htpy}{\text{\sffamily Htpy}}
\newcommand*{\Ab}{\text{\sffamily Ab}}
\newcommand*{\hTop}{\text{\sffamily hTop}}
\newcommand*{\Grp}{\text{\sffamily Grp}}
\newcommand*{\Set}{\text{\sffamily Set}}
\newcommand*{\Ring}{\text{\sffamily Ring}}
\newcommand*{\Comp}{\text{\sffamily Comp}}
\newcommand*{\hComp}{\text{\sffamily hComp}}
%\newcommand{\coprod}{\amalg}
% Analysis
......@@ -312,9 +316,11 @@ Setup numbering system and define a function that creates environment.
#+BEGIN_SRC latex
\defboxenv{dfn}{Definition}{orange!50}
\defboxenv{edfn}{(Definition)}{orange!20}
\defboxenv{cor}{Corollary}{blue!20!green!20!white}
\defboxenv{thm}{Theorem}{blue!40}
\defboxenv{lem}{Lemma}{blue!20}
\defboxenv{xmp}{Example}{black!20}
\defboxenv{prop}{Proposition}{blue!30}
\defboxenv{exr}{Exercise}{black!50}
\defboxenv{rem}{Remark}{red!20}
#+END_SRC
\section{Notation}
$\Top^{2}$ is the category consisting of pairs of topological spaces $(X,X')$, where $X' \subseteq X$ is a subspace and a morphism $f:(X,X') \to (Y,Y')$ is a continuous map $f:X \to Y$ such that $f(X') \subseteq Y'$.
\section{Category Theory}
For alternative introduction, read my notes for Commutative Algebra.
\begin{rem}
In the earlier exams, some very basic definitions were exam-relevant, so it is important that you still know the definitions given in the lecture.
\end{rem}
\subsection{Categoric definitions}
\begin{dfn}[Diagram]
Let $I$ be a small category.
A \textbf{diagram of shape} $I$ in a category $\textsf{C}$ is a functor
\begin{align*}
\bm{\cdot}: I \to \textsf{C}
\quad
i \mapsto X_i
\end{align*}
\end{dfn}
Consider the diagram $(\delta)$:
\begin{center}
\begin{tikzcd}[ ]
A \arrow[]{r}{f_1} \arrow[]{d}{f_2}& B\\
C
\end{tikzcd}
\end{center}
\begin{dfn}[]
A \textbf{solution} to the diagram $(\delta)$.
\end{dfn}
\subsection{Notation}
For a path $\alpha: [0,1] \to X$ from $x_1$ to $x_2$, let $\overline{\alpha}$ denote the backwards path from $x_2$ to $x_1$.
\subsection{Free products with amalgamation}
\begin{dfn}[Coproduct]
The coproduct is the product in the opposite cateogry.
\end{dfn}
\begin{rem}[]
\begin{itemize}
\item In $\Set$, it's the disjoint union.
\item In $\Grp$, it's the free product.
\item If it exists, it's unique up to isomorphism.
\end{itemize}
\end{rem}
\begin{lem}[]
The pushout in $\Grp$ of the diagram
\begin{center}
\begin{tikzcd}[ ]
G
\arrow[]{r}{\phi_1}
\arrow[]{d}{\phi_2}
&
H_1\\
H_2
\end{tikzcd}
\end{center}
exists and is the quotient of the group $H_1 \ast H_2$ by the subgroup $N$ generated by elements of the form $\phi_1(g^{-1}) \cdot \phi_2(g)$ for $g \in G$
\end{lem}
\begin{proof}
Let $N$ be as above and $K = \faktor{H_1 \ast H_2}{N}$.
We first check that $K$ is a solution.
To do so, define the morphisms $\psi_1, \psi_2$
\begin{center}
\begin{tikzcd}[ ]
G \arrow[]{r}{\phi_1} \arrow[]{d}{\phi}
&
H_1 \arrow[]{d}{\psi_1}
\\
H_2 \arrow[]{r}{\psi_2} & K
\end{tikzcd}
\end{center}
given by $\psi_i(h) = h N \in K$.
The diagram commutes, as by choice of $N$:
\begin{align*}
(\psi_1 \circ \phi_1)(g) = \phi_1(g) N = \phi_2(g) N
\end{align*}
For the universal property, let
\begin{center}
\begin{tikzcd}[ ]
G \arrow[]{r}{\phi_1} \arrow[]{d}{\phi}
&
H_1 \arrow[]{d}{\theta_1}
\\
H_2 \arrow[]{r}{\theta_2} & L
\end{tikzcd}
\end{center}
be another solution.
To produce the homomorphism $\tau$
\begin{center}
\begin{tikzcd}[ ]
G \arrow[]{r}{\phi_1} \arrow[]{d}{\phi}
&
H_1 \arrow[]{d}{\psi_1} \arrow[]{ddr}{\theta_1}
\\
H_2 \arrow[]{r}{\psi_2} \arrow[]{drr}{\theta_2}
& K \arrow[dashed]{dr}{\tau}
\\
& & L
\end{tikzcd}
\end{center}
we use the universal property of the free product, which gives us a morphism $\mu$: such that TFDC
\begin{center}
\begin{tikzcd}[column sep=0.8em]
&
H_1 \ast H_1
\arrow[]{dd}{\mu}
\\
H_1
\arrow[]{ur}{F}
\arrow[]{dr}{\theta_1}
& &
H_2
\arrow[]{ul}{F}
\arrow[]{dl}{\theta_2}
\\
& L
\end{tikzcd}
\end{center}
\end{proof}
\begin{thm}[Seifert -- van Kampen]
Let $X = X_1 \cup X_2$ for $X_1,X_2 \subseteq X$ open.
Assume $X_1,X_2$ and $X_0 := X_1 \cap X_2$ are non-empty and path connected.
Fix $p \in X_0$ and let
\begin{center}
\begin{tikzcd}[column sep=0.8em]
&
X_0
\arrow[]{dl}{i_1}
\arrow[]{dr}{i_2}
\\
X_1
\arrow[]{dr}{j_1}
& &
X_2
\arrow[]{dl}{j_2}
\\
& X
\end{tikzcd}
\end{center}
Then, the fundamental group $\pi_1(X,p)$ is the pushout of the diagram
\begin{center}
\begin{tikzcd}[ ]
\pi_1(X_0,p)
\arrow[]{r}{i_1^{\ast}}
\arrow[]{d}{i_2^{\ast}}
&
\pi_1(X_1,p)
\\
\pi_2(X_2,p)
\end{tikzcd}
\end{center}
\end{thm}
\begin{center}
See Topology lecture notes for proof.
Alternatively, view the drawing.
\end{center}
\begin{proof}
First, check that $\pi_1(X,p)$ is a solution of the diagram.
This follows as $\pi_1$ is a functor.
Check that if a loop $u$ stays entirely in $X_1$ or $X_2$, the morhism $\Psi$ is already deterined by
\begin{align*}
\Psi([u]) = \phi_1([u]) \quad \text{or} \quad \Psi([u]) = \phi_2([u])
\end{align*}
Because we know what to do with loops that stay entirely in $X_1$ or in $X_2$,
we want to decompose $\gamma$ into loops that stay in $X_1$ or $B$, respectively.
We claim there there are timesteps $0 = t_0 < t_1 < \ldots < t_n = 1$ such that
\begin{align*}
\gamma_i := \gamma|_{[t_i,t_{i+1}]} \subseteq X_1
\quad \text{or} \quad
\gamma|_{[t_i,t_{i+1}]} \subseteq B
\end{align*}
the reason this is possible is because $X_1,X_2$, are open, so for every $t \in [0,1]$, there exists an open interval $I_t$ such that its closure has an image $\gamma(\overline{I_t})$ that is contained in $X_1$ or $X_2$.
Doing this for every $t \in [0,1]$, we get an open covering of $[0,1]$ and by compactness of $[0,1]$, there exist finitely many such $t_i$.
So let $\gamma_i: [0,1] \to X$ be the paths that stay in $X_1$ or $X_2$.
To make loops out of $\gamma_i$, we connect the basepoint $p$ once with $x_i$ and once with $x_{i+1}$.
These paths sadly aren't loops with basepoint $x_0$, but since $X$ is path connected, we can connect the start and endpoints of each $\gamma_i$ with $x_0$ by paths $\beta_i, \beta_{i+1}$.
So the actual loops can be constructed as
\begin{align*}
\tilde{\gamma}_i = \beta_{i+1}^{-1} \gamma_i \beta_i
\end{align*}
And so, because the $\beta_i$ cancel eachother out when composing all $\tilde{\gamma}_i$ we get that
\begin{align*}
[\gamma] = [\gamma_1][\gamma_2] \dots [\gamma_k] \in \pi_1(X)
\end{align*}
And since each $\gamma$ stays in $A$ and $B$, we finally get
\begin{align*}
[\tilde{\gamma}_i] \in \image(j_A) \cup \image(j_B) \implies [\tilde{\gamma}_i] \in \image(\phi) \implies [\gamma] \in \image(\phi)
\end{align*}
We will use the notation $\phi_{\ast}$ to mean ``either $\phi_1$ or $\phi_2$''.
\end{proof}
\begin{thm}[Seifert -- van Kampen, the full version]
Let $\textsf{Grpd}$ be the category of groupoids and let $\Pi: \Top \to \textsf{Grpd}$, the be fundamental groupoid.
Suppose $\mathcal{U}$ is an open cover of $X$ with the property that the intersection of finitely many elements of $\mathcal{U}$ again belongs to $\mathcal{U}$.
Then regard $\mathcal{U}$ as a category where the objects are the open sets $U \in \mathcal{U}$ and $\Hom(U,V)$ consit of the inclusion mapping $\iota: U \hookrightarrow V$, if $U \subseteq V$ and empty otherwise.
Then the fundamental groupoid $\Pi(X)$ is the colimit of the diagram $\Pi|_{\mathcal{U}}$.
In other words, if $\Pi(X)$ is uniquely characterised by the following universal property:
Suppose $\mathcal{G}$ is a grupoid and that $\tau: \Pi \to \mathcal{G}$ is a functor.
Then the following diagram commutes
\begin{center}
\begin{tikzcd}[column sep=0.8em]
\Pi(U)
\arrow[]{rr}{\Pi(i)}
\arrow[]{dr}{}
\arrow[]{ddr}{\tau(U)}
& &
\Pi(V)
\arrow[]{dl}{}
\arrow[]{ddl}{\tau(V)}
\\
&
\Pi(X)\\
&
G
\end{tikzcd}
\end{center}
\end{thm}
\begin{proof}[Proof sketch]
On points of $X$, define $\eta(x) = \tau(U)(x)$, for $U \in \mathcal{U}$ and $x \in U$.
On morphisms, a path $u$ form $x$ to $y$ entirely containted in $U \subseteq \mathcal{U}$, define
\begin{align*}
\eta(u): \eta(x) \to \eta(y)
\end{align*}
to be $\tau(U)(u)$.
\end{proof}
\section{Singular Homology}
A group homomorphism from a free abelian group is determined by its basis.
\begin{dfn}[$n$-simplex]
An ordered tuple $(z_0,\ldots,z_n)$ in $\R^{m}$ is said to be \textbf{affinely independent}, if $\{z_1-z_0,\ldots,z_n-z_0\}$ is linearly independent.
Given such an affinely independent tuple the $n$-simplex spanned by $(z_{0}, \ldots, z_{n})$ is the set
\begin{align*}
[z_0,z_1,\ldots,z_n] := \left\{
x \in \R^{m} \big\vert x = \sum_{i=0}^{n}s_i z_I, 0\leq s_i \leq 1, \sum_{i=0}^{n}s_i = 1
\right\}
\end{align*}
\end{dfn}
\begin{dfn}[]
The face opposite to $z_i$ is
\begin{align*}
[z_0,\ldots,\hat{z}_i,\ldots,z_n] := \left\{
x \in [z_{0}, \ldots, z_{n}] \big\vert s_i = 0
\right\}
\end{align*}
The boundary of $[z_{0}, \ldots, z_{n}]$ is the union of all its faces.
The standard $n$-simplex is $\Delta^{n}\subseteq \R^{n+1}$ spanned by the canonical basis vectors.
A singular $n$-simplex is a continuous map $\sigma: \Delta^{n} \to X$
The restriction of $\sigma$ to its $i$-th face induces a $n-1$ simplex $\sigma \circ \epsilon_i: \Delta^{n-1} \to X$.
The alternating sum of the restriction maps is the \textbf{boundary} of the simplex
\begin{align*}
\del \sigma := \sum_{i=0}^{n}(-1)^{i} \sigma \circ \epsilon_i
\end{align*}
\end{dfn}
As a motivation for the alternating sign, we can now (with some abuse of notation) write the fundamental theorem of calculus as
\begin{align*}
\int_{I} \del F dx = F_{|\del I} = F(b) - F(a)
\end{align*}
\begin{dfn}[Singular $n$-chain]
$C_n(X)$ is the free abelian group with basis the set of $n$-simplices in $X$.
Its elements are called singular $n$-chains.
We have the \textbf{singular boundary operators} which are induced by the map $\sigma \mapsto \del \sigma$.
\end{dfn}
\begin{prop}[]
$\del^{2} = 0$.
\end{prop}
This follows from the alternating definition of the boundary.
\begin{dfn}[$n$-th singular homology group]
Singular $n$-chains that are in the kernel of $\del_n$ are called \textbf{singular $n$-cycles}, $Z_n(X) := \kernel \del_n \subseteq C_n(X)$
Those who are in the image of $\del_{n+1}$ are called \textbf{singular $n$-boundaries}, $B_n(X) := \image \del_{n+1} \subset C_n(X)$
The \textbf{$n$-th singular homology group} is
\begin{align*}
H_n(x) := \faktor{Z_n(X)}{B_n(X)}
\end{align*}
Which is an abelian subgroup of $C_n(X)$
For some singular $n$-cycle $c$, denote by $\scal{c}$ the coset $c + \kernel \del_n$ called the \textbf{homology class} of $c$.
\end{dfn}
If $f: X \to Y$ is continuous, this defines a map
\begin{align*}
f_{\#}: C_n(X) \to C_n(Y), \quad \sigma \mapsto f \circ \sigma
\end{align*}
Moreover, $f_{\#}$ commutes with $\del$ and induces a map
\begin{align*}
H_n(f): H_n(X) \to H_n(Y), \quad \scal{c} \mapsto \scal{f_{\#}(c)}
\end{align*}
\begin{cor}[]
For each $n \geq 0$ $H_n: \Top \to \Ab$ is a functor.
\end{cor}
\begin{cor}[]
If $X,Y$ are homeomorphic, then $H_n(X) \iso H_n(Y)$ for all $n \geq 0$
\end{cor}
Follows directly from functor properties.
\section{The homotopy axiom}
Recall that the fundamental group $\pi_1. \Top_{\ast}\to \Grp$ had homotopy-preserving properties that let us interpret it as a functor $\pi_1: \hTop \to \Grp$.
We wish to show that $H_n:\Top \to \Ab$ has some homotopy-preserving properties aswell, which will let us interpret $H_n$ as a functor $\hTop \to \Ab$.
In particular we want to show the \textbf{homotopy axiom}, which states that for maps $f,g: X \to Y$
\begin{align*}
[f] = [g] \implies H_n(f) = H_n(g) \quad \forall n \geq 0
\end{align*}
The terminology will be explained when we cover the \emph{Eilenberg-Steenrod axioms}.
%In the case of the fundamental group, two maps are \emph{homotopic}, if there exists a map $h: I^{2} \to X$ such that
\begin{prop}[Dimension axiom]
Let $X$ be a one-point space $\{\ast\}$.
Then $H_n(X) = 0$ for all $n \geq 0$.
\end{prop}
\begin{proof}
There is only the constant $n$-simplex $\sigma_n: \Delta^{n} \to \{\ast\}$, meaining $C_n(X) = \scal{\sigma_n} \iso \Z$ for all $n \geq 0$.
Moreover, $\del_0(\sigma_0) = 0$ and $\del_1(\sigma_1) = 0$ but $\del_2(\sigma_2) = \sigma_1$.
More generally, for all $n > 0$, we have
\begin{align*}
\del_n(\sigma_n) = \left\{\begin{array}{ll}
0 & \text{ if $n$ is odd} \\
\sigma_{n-1} & \text{ if $n$ is even}
\end{array} \right.
\end{align*}
Therefore
\begin{align*}
H_n(X) = \faktor{\kernel \del_n}{\image \del_{n+1}} \iso
\left\{\begin{array}{ll}
\faktor{\Z}{\Z} = 0 & \text{ if $n$ is odd}\\
\faktor{0}{0} =0 & \text{ if $n$ is even}
\end{array} \right.
\end{align*}
\end{proof}
\begin{prop}[]
Let $X$ be a topological space.
Let $\{X_{\lambda} \big\vert \lambda \in \Lambda\}$ denote the path components of $X$.
Then for every $n \geq 0$ one has
\begin{align*}
H_n(X) \iso \bigoplus_{\lambda \in \Lambda} H_n(X_ljk )
\end{align*}
\end{prop}
It therefore suffices to compute $H_n(X)$ only on the path-connected spaces $X$.
However, computing $H_n(X)$ for $n > 0$ is generally difficult, but it is always possible to compute $H_0(X)$.
\begin{prop}[]
If $X$ is a non-empty path connected space, then $H_0(X) = \Z$.
A generator is given by $\scal{x}$ for any $x \in X$ and if $x,y \in X$, then $\scal{x} = \scal{y}$.
Moreover if $\scal{c}$ is any generator, then $\scal{c} = \scal{x}$ for some $x \in X$.
\end{prop}
\begin{proof}
As $\del_0: C_0(X) \to 0$, $Z_0(X) = C_0(X)$.
As $0$-cycles can be identified with their image, a general $0$-chain $c \in C_0(X)$ is of the form
\begin{align*}
c = \sum_{x \in X}m_x x, \quad m_x \in Z
\end{align*}
where all but finitely of $m_x$ are zero.
We claim that
\begin{align*}
B_0(X) = \left\{
\sum_{x \in X}m_x x \big\vert \sum_{x \in X}m_x = 0
\right\}
\end{align*}
For ``$\supseteq$'', assume $\sum_{i = 1}^{n}m_ix_i = 0$.
To pick a $1$-chain $a \in C_1(X)$ such that $\del(a) = c$, chose any point $p \in X$.
A $\Delta^{1}$ is isomorphic to the interval $I$, we can identify $1$-simplices with paths.
As $X$ is path connected, we consider a path $\sigma_ii$ (i.e. a $1$-simplex) starting at $p$ and ending at $x_i$.
Then $\del \sigma_i = x_i - p \in C_0(X)$.
Set $a := \sum_{i=1}^{n}m_i \sigma_i$. Then
\begin{align*}
\del a = \del \left(
\sum_{i=1}^{n}m_i \sigma_i
\right)
=
\sum_{i=1}^{n}m_i \del \sigma_i
=
\sum_{i=1}^{n}m_i x_i - \underbrace{\sum_{i=1}^{n}m_i}_{=0} p
= c
\end{align*}
Now for ``$\subseteq$'' in the claim, let $d \in B_0(X)$.
So there exists a $1$-chain $b \in C_1(X)$ such that $\del b = d$.
Since the general form of a $1$-chain is
\begin{align*}
b = \sum_{j=1}^{k}l_j \tau_j, \quad \text{where} \quad \tau_j: \Delta^{1} \to X, \quad l_i \in \Z
\end{align*}
we have
\begin{align*}
d = \del b = \sum_{j=1}^{k}l_j (\tau_j(e_1) - \tau_j(e_0))
\end{align*}
so every coefficient $l_j$ appears twice with opposite sign satisfying the condition in the RHS of the claim.
Therefore we have a short exact sequence
\begin{align*}
0 \to B_0(X) \hookrightarrow Z_0(X) \stackrel{\phi}{\to} \Z \to 0
\end{align*}
where $\phi$ is the map extracting the coefficients
\begin{align*}
\phi: Z_0(X) = C_0(X) \to \Z, \quad \sum_{x \in X}x \mapsto \sum_{x \in X}m_x
\end{align*}
thus $H_0(X) \iso \image \phi = \Z$.
The rest of the proof is trivial.
\end{proof}
\begin{cor}[]
Let $X,Y$ be path connected spaces and $f: X \to Y$ continuous.
Then $H_0(f) : H_0(X) \to H_0(Y)$ maps generators of $H_0(X)$ to generators of $H_0(Y)$.
\end{cor}
\begin{prop}[]\label{prop:homology-interval}
Let $X$ be a topological space and define includsions $\iota,j: X \to X \times I$ with
\begin{align*}
\iota(x) = (x,0), \quad j(x,) = (x,1)
\end{align*}
Then $H_n(i) = H_n(j)$.
\end{prop}
There is a very nice proof using the \emph{acyclic models theorem} which is presented at the end of the course.
We give a more elementary proof below.
\begin{lem}[]\label{lem:homology-p}
Let $f,g: X \to Y$ continuous. Assume for each $n \geq -1$ there is a homomorphism
\begin{align*}
P: C_n(X) \to C_{n+1}(Y)
\end{align*}
with
\begin{align*}
f_{\#} - g_{\#} = \del P + P \del
\end{align*}
Then $H_n(f) = H_n(g)$ for all $n \geq 0$
\end{lem}
The maps $P_n$ look like this
\begin{center}
\begin{tikzcd}[]
\ldots
\arrow[]{r}{}
&
C_{n+1}(X)
\arrow[]{r}{\del}
&
C_n(X)
\arrow[]{r}{\del}
\arrow[]{dl}{P}
\arrow[]{d}{f_{\#}-g_{\#}}
&
C_{n-1}(X)
\arrow[]{r}{}
\arrow[]{dl}{P}
&
\ldots
\\
\ldots
\arrow[]{r}{}
&
C_{n+1}(Y)
\arrow[]{r}{\del}
&
C_n(Y)
\arrow[]{r}{\del}
&
C_{n-1}(Y)
\arrow[]{r}{}
&
\ldots
\end{tikzcd}
\end{center}
Beware that this diagram does not necessarily commute!
\begin{proof}
Let $c \in \Z_n(X)$. Then
\begin{align*}
(f_{\#} - g_{\#})c = (\del P + P \del)c = \del P c \in B_n(Y)
\end{align*}
thus $H_n(f) \scal{c} = H_n(g) \scal{c}$.
\end{proof}
The first step in the proof of Proposition \ref{prop:homology-interval}is the following observation:
The square $\Delta^{1} \times I$ is the union of two triangles $\Delta^{2}$.
The prism $\Delta^{2} \times I$ is the union of three tetraeders $\Delta^{3}$.
More generally:
\begin{lem}[]