Alg Topo le 1 - 11

05/12_AlgTopo-I/README.org

@@ -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

05/12_AlgTopo-I/files/01.tex

+\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 $f(X') \subseteq Y'$.
+

05/12_AlgTopo-I/files/05.tex

+
+\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 $x_1$.
+
+

05/12_AlgTopo-I/files/06.tex

+
+\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}

05/12_AlgTopo-I/files/07-singular-homology.tex

+\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.
+

05/12_AlgTopo-I/files/08-homotopy-axiom.tex

+\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}[]