\section{Singular cohomology}

Cohomology forms a dual notion to homology. To every topological space we assign 
a graded group $H^*(X)$ equipped with a ring structure given by a product 
$\cup: H^i(X)\times H^j(X)\to H^{i+j}(X)$. In this
section we give basic definitions and properties of singular
cohomology groups which are very similar to those of homology groups.

\begin{cislo}\label{COchain}{\bf Cochain complexes.}
A \emph{cochain complex}\index{cochain complex}
$(C,\delta)$ is a  sequence of
Abelian groups (or modules over a ring) and their homomorphisms
indexed by integers
$$\dots\xrightarrow{\ \delta^{n-2}\ } C^{n-1}\xrightarrow{\ \delta^{n-1}\ }
C^n\xrightarrow{\ \delta^{n}\ }
C^{n+1}\xrightarrow{\ \delta^{n+1}\ }\dots$$
such that
$$\delta^{n}\delta^{n-1}=0.$$
$\delta^n$ is called a
coboundary operator. A \emph{cochain homomorphism}\index{cochain homomorphism}
of cochain complexes $(C,\delta_C)$ and $(D,\delta_D)$  is a
sequence of homomorphisms of Abelian groups (or modules over a
ring) $f^n:C^n\to D^n$ which commute with the coboundary operators
$$\delta_D^nf_{n}=f^{n+1}\delta_C^n.$$
\end{cislo}

\begin{cislo}\label{COhomology}{\bf Cohomology of cochain complexes.}
The $n$-th \emph{cohomology group}\index{cohomology group of a cochain
complex} of a cochain complex $(C,\delta)$ is the group
$$H^{n}(C)=\frac{\ker\delta^n}{\im\delta^{n-1}}.$$
The elements of $\ker\delta^n=Z^n$ are called
\emph{cocycles}\index{cocycle} of dimension $n$ and the elements
of $\im\delta^{n-1}=B^n$ are called
\emph{coboundaries}\index{coboundary} (of dimension $n$).
If a cochain complex is exact, then its cohomology groups are
trivial.

The component $f^n$ of the cochain homomorphism $f:(C,\delta_C)\to
(D,\delta_D)$ maps cocycles into cocycles and
coboundaries into coboundaries. It enables us to define
$$H^n(f):H^{n}(C)\to H^n(D)$$
by the prescription $H^n(f)[c]=[f^n(c)]$ where $[c]\in H^n(C)$
and $[f^n(c)]\in H^n(D)$ are classes  represented by the
elements $c\in Z^n(C)$ and $f^n(c)\in Z^n(D)$, respectively.
\end{cislo}

\begin{cislo}\label{COlesch}{\bf Long exact sequence in cohomology.}
A sequence of cochain homomorphisms
$$\dots\to
A\xrightarrow{\ f\ }B\xrightarrow{\ g\ }C\to\dots$$
is exact if for every $n\in\mathbb Z$
$$\dots\to
A^n\xrightarrow{\ f^n\ }B_n\xrightarrow{\ g^n\ }C^n\to\dots$$
is an exact sequence of Abelian groups. Similarly as for homology
groups we can prove


\begin{thm*} Let $0\to A\xrightarrow{\ i\ }B\xrightarrow{\ j\ }C\to 0$
be a short exact sequence of cochain complexes. Then there is a so
called \emph{connecting homomorphism}\index{connecting
homomorphism} $\delta^*:H^n(C)\to H^{n+1}(A)$ such that the
sequence
$$\dots\xrightarrow{\ \delta^*\ } H^{n}(A)\xrightarrow{H^{n}(i)} H^n(B)
\xrightarrow{H^n(j)}H^{n}(C)\xrightarrow{\ \delta^*\ }H^{n+1}(A)
\xrightarrow{H^{n+1}(i)}
\dots$$
is exact.
\end{thm*}
\end{cislo}

\begin{cislo}\label{COchhmtp}{\bf Cochain homotopy.} Let $f,g:C\to
D$ be two cochain homomorphisms. We say that they are \emph{cochain
homotopic}\index{cochain homotopic} if there are homomorphisms
$s^n:C^n\to D^{n-1}$ such that
$$\delta_D^{n-1}s^n+s^{n+1}\delta_C^{n}=f^n-g^n \quad
\text{for all }n.$$
The relation to be cochain homotopic is an equivalence.
The sequence of maps $s^n$ is called a \emph{cochain
homotopy}.\index{cochain homotopy} Similarly as for homology we
have


\begin{thm*} If two cochain homomorphism $f,g:C\to D$ are cochain
homotopic, then
$$H^n(f)=H^n(g).$$
\end{thm*}
\end{cislo}

\begin{cislo}\label{COsing}{\bf Singular cohomology groups of a
pair.} Consider a pair of topological spaces $(X,A)$, an
inclusion $i:A\hookrightarrow X$ and an Abelian group $G$.
Let
$$C(X,A)=(C_n(X)/C_n(A),\partial_n)$$
be the singular chain complex of the
pair $(X,A)$. The \emph{singular cochain complex}\index{singular
cochain complex} $(C(X,A;G),\delta)$
for the pair $(X,A)$ is defined as
\begin{align*}
C^n(X,A;G)=\hom\left(C_n(X,A),G\right)&\cong\{h\in\hom(C_n(X),G);
\ h\vert C_n(A)=0\}\\
&=\ker i^{*}:\hom(C_n(X),G)\longrightarrow \hom(C_n(A),G).
\end{align*}
and
$$\delta^n(h)=h\circ\partial_{n+1}\quad\text{for
}h\in\hom(C_n(X,A),G).$$
The $n$-th \emph{cohomology group}\index{cohomology group}
of the pair $(X,A)$ with
coefficients in the group $G$ is the $n$-th cohomology group of
this cochain complex
$$H^n(X,A;G)=H^n(C(X,A;G),\delta).$$
We write $H^n(X;G)$ for $H^n(X,\emptyset;G)$.
A map $f:(X,A)\to(Y,B)$ induces the cochain homomorphism
$C^n(f):C^n(Y;G)\to C^n(X;G)$ by
$$C^n(f)(h)=h\circ C_n(f)$$
which restricts to a cochain
homomorphism $C^n(Y,B;G)\to C^n(X,A;G)$ since $f(A)\subseteq B$.
In cohomology it induces the homomorphism
$$f^*=H^n(f):H^n(Y,B)\to H^n(X,A).$$

Moreover, $H^n(\id_{(X,A)})=\id_{H^n(X,A;G)}$ and
$H^n(fg)=H^n(g)H^n(f)$.
We can  conclude that $H^n$ is a contravariant functor (cofunctor)
from the category $\top^2$ into the category $\abAG$ of Abelian groups.
\end{cislo}

\begin{cislo}\label{COles}{\bf Long exact sequence for singular
cohomology.} Consider inclusions of spaces $i:A\hookrightarrow X$,
$i':B\hookrightarrow Y$
and maps $j:(X,\emptyset)\to (X,A)$,
$j':(Y,\emptyset)\to (Y,B)$ induced by
$\id_{X}$ and $\id_Y$, respectively.
Let $f:(X,A)\to (Y,B)$ be a map. Then there are connecting
homomorphisms $\delta_X^*$ and $\delta^*_Y$ such that the following
diagram
$$
\xymatrix{
\dots \ar[r]^-{\delta_X^*}  & H^n(X,A;G) \ar[r]^-{j^*} &
H^n(X;G) \ar[r]^-{i^*} &H^n(A;G)
\ar[r]^-{\delta_X^*}
& H^{n+1}(X,A;G) \ar[r]^-{j^*} & \dots   \\
\dots \ar[r]^-{\delta_Y^*}  & H^n(X,B;G) \ar[r]^-{j'^*}
\ar[u]_{f^*} &
H^n(Y;G) \ar[r]^-{i'^*} \ar[u]_{f^*}&H^n(B;G) \ar[r]^-{\delta_Y^*}
\ar[u]_{(f/B)^*}
&H^{n+1}(Y,B;G) \ar[r]^-{j'^*}\ar[u]_{f^*}& \dots
}
$$
commutes and its horizontal sequences are exact.

The proof follows from Theorem \ref{COlesch} using the fact that
$$0\to C^n(X,A;G)\xrightarrow{C^n(j)} C^n(X;G)\xrightarrow{C^n(i)}
C^n(A;G)\to 0$$
is a short exact sequence of cochain complexes as it follows directly
from the definition of $C^n(X,A;G)$.

\begin{remark} Consider the functor $I:\top^2\to\top^2$ which
assigns to every pair $(X,A)$ the pair $(A,\emptyset)$. The
commutativity of the last square in the diagram above means
that $\delta^*$ is a natural transformation of contravariant functors
$H^n\circ I$ and $H^{n+1}$ defined on $\top^2$.
\end{remark}

\begin{remark} It is useful to realize how
$\delta^*:H^n(A;G)\to H^{n+1}(X,A;G)$ looks like. Every element of
$H^{n}(A;G)$ is represented by a cochain $q\in \hom(C_n(A);G)$ with a
zero coboundary $\delta q\in \hom(C_{n+1}(A);G)$. Extend $q$ to
$Q\in \hom(C_n(X);G)$ in arbitrary way. Then
$\delta Q\in\hom(C_{n+1}(X),G)$ restricted to $C_{n+1}(A)$ is
equal to $\delta q=0$. Hence it lies in $\hom(C_{n+1}(X,A);G)$
and from the definition in \ref{COlesch} we have
$$\delta^*[q]=[\delta Q].$$
\end{remark}
\end{cislo}

\begin{cislo}\label{COhmtp}{\bf Homotopy invariance.}
If two maps $f,g:(X,A)\to (Y,B)$ are homotopic, then they
induce the same homomorphisms
$$f^*=g^*:H^n(Y,B;G)\to H_n(X,A;G).$$


\begin{proof} We already know that the homotopy between $f$ and $g$
induces a chain homotopy $s_*$ between $C_*(f)$ and $C_*(g)$.
Then we can define a cochain homotopy between $C^*(f)$  and
$C^*(g)$ as
$$s^n(h)=h\circ s_{n-1}\quad\text{for }h\in \hom(C_n(Y);G)$$
and use Theorem \ref{COchhmtp}.
\end{proof}

\begin{cor*} If $X$ and $Y$ are homotopy equivalent spaces, then
$$H^n(X)\cong H^n(Y).$$
\end{cor*}
\end{cislo}

\begin{cislo}\label{COet}{\bf Excision Theorem.} Similarly as for
singular homology groups there are two
equivalent versions of this theorem.

\begin{thma}[Excision Theorem, 1st version] Consider spaces
$C\subseteq A\subseteq X$ and suppose  that
$\bar C\subseteq \inte A$. Then the inclusion
$$i:(X-C,A-C)\hookrightarrow (X,A)$$
induces the isomorphism
$$i^*:H^n(X,A;G)\xrightarrow{\cong} H^n(X-C,A-C;G).$$
\end{thma}

\begin{thma}[Excision Theorem, 2nd version] Consider two
subspaces $A$ and $B$ of a space $X$. Suppose that
$X=\inte A\cup\inte B$. Then the inclusion
$$i:(B,A\cap B)\hookrightarrow (X,A)$$
induces the isomorphism
$$i^*:H^n(X,A;G)\xrightarrow{\cong} H^n(B,A\cap B;G).$$
\end{thma}


The proof of Excision Theorem for singular cohomology follows
from the proof of the homology version.
\end{cislo}


\begin{cislo}\label{COsum}{\bf Cohomology of finite disjoint union.}
Let $X=\coprod_{\alpha=1}^k X_{\alpha}$ be a disjoint union.
Then
$$H^n(X;G)=\bigoplus_{\alpha=1}^kH^n(X_{\alpha}).$$
The statement is not generally true for infinite unions.
\end{cislo}

\begin{cislo}\label{COred}{\bf Reduced cohomology groups.} For
every space $X\ne\emptyset$ we define the \emph{augmented cochain complex}
\index{augmented cochain complex}
$(\tilde C^*(X;G),\tilde\delta)$ as follows
$$
\tilde C^n(X;G)= \hom(\tilde C_n(X);G)
$$
with $\tilde\delta^nh=h\circ\tilde\partial_{n+1}$ for
$h\in\hom(\tilde C_n(X);G)$. See 3.14. %\ref{HOred}.
The \emph{reduced cohomology groups}\index{reduced cohomology groups}
$\tilde H_n(X;G)$ with coefficients in $G$ are the cohomology groups
of the augmented cochain
complex. From the definition it is clear that
$$\tilde H^n(X;G)=H^n(X;G)\quad \text{for }n\ne 0$$
and
$$\tilde H^n(*;G)=0\quad\text{for all }n.$$

For pairs of spaces we define $\tilde H^n(X,A;G)=H^n(X,A;G)$ for all
$n$. Then theorems on long exact sequence, homotopy invariance
and excision hold for reduced cohomology groups as well.

Considering a space $X$ with base point $*$ and applying
the long exact sequence for the pair $(X,*)$, we get that for all
$n$
$$\tilde H^n(X;G)=\tilde H^{n}(X,*;G)=H^{n}(X,*;G).$$
Using this equality and the long exact sequence for unreduced
cohomology we get that
$$H^0(X;G)\cong H^0(X,*;G)\oplus H^0(*;G)\cong \tilde H^0(X)\oplus
\mathbb G.$$

Analogously as for homology groups we have

\begin{lemma*} Let $(X,A)$ be a pair of CW-complexes. Then
$$\tilde H^n(X/A;G)=H^n(X,A;G)$$
and we have the long exact sequence
$$\dots\to \tilde H^n(X/A;G)\to \tilde H^n(X;G)\to \tilde H^n(A;G)\to \tilde
H^{n+1}(X/A;G)\to\dots$$
\end{lemma*}
\end{cislo}

\begin{cislo}\label{COtriple}{\bf The long exact sequence of a
triple.} Consider a triple $(X,B,A)$,
$A\subseteq B\subseteq X$. Denote
$i:(B,A)\hookrightarrow (X,A)$ and $j:(X,A)\to (X,B)$ maps induced by
the inclusion $B\hookrightarrow X$ and $\id_X$, respectively.
Analogously as for homology one can derive the long exact
sequence of the triple $(X,B,A)$
$$\dots\xrightarrow{\ \delta^*\ }
H^n(X,B;G)\xrightarrow{\ j^*\ } H^n(X,A;G)\xrightarrow{\ i^*\ }
H^{n}(B,A;G)\xrightarrow{\ \delta^*\ }H^{n+1}(X,B;G)\xrightarrow{\ j^*\ }\dots$$
\end{cislo}

\begin{cislo}\label{COsphere}{\bf Singular cohomology groups of
spheres.} Considering the long exact sequence of the triple
$(\Delta^n,\delta\Delta^n,V=\delta\Delta^n-\Delta^{n-1})$:
we get that
$$H^{i}(\Delta^n,\partial\Delta^n;G)=
\begin{cases}
G\quad& \text{for }i=n,\\
0\quad& \text{for }i\ne n.
\end{cases}$$

The pair $(D^n,S^{n-1})$ is homeomorphic to
$(\Delta^n,\partial\Delta^n)$. Hence it has the same cohomology
groups. Using the long exact sequence for this pair we obtain
$$\tilde H^{i}(S^{n};G)=H^{i+1}(D^{n+1},S^{n})=
\begin{cases} 0\quad & \text{for }i\ne n,\\
G\quad& \text{for }i=n.
\end{cases}$$
\end{cislo}

\begin{cislo}\label{COmves}{\bf Mayer-Vietoris exact sequence.}
\index{Mayer-Vietoris exact sequence}
Denote inclusions $A\cap B\hookrightarrow A$,
$A\cap B\hookrightarrow B$, $A\hookrightarrow X$, $B\hookrightarrow X$
by $i_A$, $i_B$, $j_A$, $j_B$, respectively. Let $C\hookrightarrow
A$, $D\hookrightarrow B$ and suppose that $X=\inte A\cup\inte B$,
$Y=\inte C\cup\inte D$. Then there is
the long exact sequence
\begin{multline*}
\dots \xrightarrow{\ \delta^*\ }
H^n(X,Y;G)\xrightarrow{\ (j_{A}^*,j_{B}^*)\ } H^n(A,C;G)\oplus
H^n(B,D;G)\\
\xrightarrow{\ i_{A}^*-i_{B}^*\ }
H_{n}(A\cap B,C\cap D;G)\xrightarrow{\ \delta^*\ }H^{n+1}(X,Y;G)
\xrightarrow{\ \ \ } \dots
\end{multline*}


\begin{proof} The coverings $\mathcal{U}=\{A,B\}$ and
$\mathcal{V}=\{C,D\}$ satisfy conditions of Lemma 3.12. %\ref{HOet}. 
The sequence of chain complexes
$$0\longrightarrow C_n(A\cap B,C\cap D)\overset{i}\longrightarrow
C_n(A,C)\oplus C_n(B;D)\overset{j}\longrightarrow
C_n^{\mathcal{U},\mathcal{V}}(X,Y)\longrightarrow 0$$
where $i(x)=(x,x)$ and $j(x,y)=x-y$ is exact. Applying $\hom(-,G)$
we get a new short exact sequence of cochain complexes
$$0\longrightarrow C^n_{\mathcal{U},\mathcal{V}}(X,Y;G)\overset{j^*}\longrightarrow
C^n(A,C;G)\oplus C^n(B,D;G)\overset{i^*}\longrightarrow C^n(A\cap B,C\cap
D;G)
\longrightarrow 0$$
and it
induces a long exact sequence. Using Lemma 3.12 %\ref{HOet} 
we get that $H^n(C_{\mathcal{U},\mathcal{V}}(X,Y;G))=H^n(X,Y;G)$,
which completes the proof.
\end{proof}
\end{cislo}

\begin{cislo}\label{COcomp}{\bf Computations of cohomology of
CW-complexes.} If we know a CW-structure of a space $X$, we can
compute its cohomology in the same way as homology. Consider the
chain complex from Section 4
\begin{equation*}
(H_n(X^n,X^{n-1}),d_n).
\end{equation*}

\begin{thm*} Let $X$ be a CW-complex. The $n$-th cohomology group
of the cochain complex
$$(\hom(H_n(X^n,X^{n-1};G),d^n)\quad d^n(h)=h\circ d_n$$
is isomorphic to the $n$-th singular cohomology group
$H^n(X;G)$.
\end{thm*}

\begin{ex} After reading the next section try to prove the theorem above 
using the results and proofs from Section 4. % \ref{HOCWcomp}.
\end{ex}

\begin{ex} Compute singular cohomology of real and complex
projective spaces with coefficients $\mathbb Z$ and $\mathbb
Z_2$.
\end{ex}
\end{cislo}