\section{Homotopy excision  and Hurewicz theorem}

One of the reasons why the computation of homotopy groups is so
difficult is the fact that we have no general excision
theorem at our disposal. Nevertheless, there is a restricted
version of such a theorem. It has many consequences, one of them
is the Freudenthal suspension theorem which enables us to compute
$\pi_n(S^n)$. At the end of this section we define the Hurewicz
homomomorphism which under certain conditions compares homotopy
and homology groups.

\begin{cislo}\label{EHex}{\bf Homotopy excision theorem.}
Excision theorem for homology groups has the following restricted analogue
for homotopy groups.
\begin{thm*} [Blakers-Massey theorem]  Let $A$ and $B$ be subcomplexes of
CW-complex $X=A\cup B$. Suppose that $C=A\cap B$ is connected,
$(A,C)$ is $m$-connected and $(B,C)$ is $n$-connected. Then the
inclusion
$$j:(A,C)\hookrightarrow (X,B)$$
is $(m+n)$-equivalence, i.~e. $j_*:\pi_i(A,C)\to\pi_i(X,B)$
is an isomorphism for $i<m+n$ and an epimorphism for $i=m+n$.
\end{thm*}

\begin{proof} We distinguish several cases.
\vskip 1mm
\noindent
{\bf 1.} Suppose that $A=C\cup \bigcup_{\alpha}e^{m+1}_{\alpha}$
and $B=C\cup e^{n+1}$. First we prove that
$j_*:\pi_i(A,C)\to\pi_i(X,B)$ is surjective for $i\le m+n$.

Consider $f:(I^i,\partial I^i,J^{i-1})\to (X,B,x_0)$. Using
simplicial approximation lemma 12.4 %\ref{HCWsal} 
we can suppose that
there are simplices $\Delta^{m+1}_{\alpha}\subset e^{m+1}_{\alpha}$ and
$\Delta^{n+1}\subset e^{n+1}$ such that their inverse images
$f^{-1}(\Delta^{m+1}_{\alpha})$, $f^{-1}(\Delta^{n+1})$ are unions
of convex polyhedra on each of which $f$ is a linear surjection
$\R^i$ onto $\R^{m+1}$ and $\R^{n+1}$, respectively.
We will  need the following statement.

\begin{lemma*} If $i\le m+n$ then there exist points
$p_{\alpha}\in\Delta^{m+1}_{\alpha}$, $q\in\Delta^{n+1}$ and a
continuous function $\p:I^{i-1}\to [0,1)$ such that
\begin{enumerate}
\item[(a)] $f^{-1}(p_{\alpha})$ lies above the graph of $\p$,
\item[(b)] $f^{-1}(q)$ lies below the graph of $\p$,
\item[(c)] $\p=0$ on $\partial I^{i-1}$.
\end{enumerate}
\end{lemma*}

%\begin{figure}[H]
%\centering
%\includegraphics[height=6cm,width=13cm]{fg7.eps}%
%\caption{}\label{fg7}
%\end{figure}

\begin{figure}[htb]
  \centering
  \def\svgwidth{13cm}
  \input{img/fig-13_1.pdf_tex}
  \caption{The graph of $\varphi$}
\end{figure}

Let us postpone the proof of the lemma for a moment. The subspace
$M=\{(s,t)\in I^{i-1}\times I;\ t\ge\p(s)\}$ is a deformation
retract of $I^i$ with deformation retraction $h:I^i\times I\to
I^i$, $h(x,0)=x$, $h(x,1)\in M$. Then
$$H=fh:I^i\times I\to X$$
provides a homotopy between $f$ and
$$g:(I^i,\partial I^i,J^{i-1})\to
(X-\{q\},X-\{q\}-\bigcup\{p_{\alpha}\},x_0).$$
Obviously, $g$ is homotopic to $\tilde g:(I^i,\partial I^i,J^{i-1})\to
(A,C,x_0)$. Hence $j_*[\tilde g]=[f]$.

The fact that $j_*:\pi_i(A,C)\to\pi_i(X,B)$ is monomorphism for
$i\le m+n-1$ can be proved by the same way as above replacing
$f$ by homotopy $h:I^i\times I\to (X,B)$. (Notice that
$i+1\le m+n$ now.)

\vskip 2mm
\noindent
{\it Proof of the lemma.} Choose arbitrary $q\in\Delta^{n+1}$. Then
$f^{-1}(q)$ is a union of convex simplices of dimension $\le
i-n-1$. Denote $\pi:I^i\to I^{i-1}$ the projection given by
omitting the last coordinate. $\pi^{-1}(\pi(f^{-1}(q)))$ is the
union of convex simplices of dimension $\le i-n$. On the set
$\pi^{-1}(\pi(f^{-1}(q)))\cap f^{-1}(\Delta^{m+1}_{\alpha})$
is $f$ linear, hence
$$f(\pi^{-1}(\pi(f^{-1}(q))))\cap \Delta^{m+1}_{\alpha}$$
is the union of simplices of dimension at most
$i-n<m+1$  for $i\le m+n$. Consequently, there is $p_{\alpha}\in
\Delta^{m+1}_{\alpha}$ such that
$$f^{-1}(p_{\alpha})\cap\pi^{-1}(\pi f^{-1}(q))=\emptyset.$$
Since $\im f$ meets only finite number of cells
$e^{m+1}_{\alpha}$, the set $\bigcup\pi(f^{-1}(p_{\alpha}))$
is compact and disjoint from $\pi(f^{-1}(q))$. Hence there is
continuous function $\p$, $\p=0$ on $\bigcup\pi(f^{-1}(p_{\alpha}))$
and $\p=1-\varepsilon$ on $\pi(f^{-1}(q))$ with required
properties.

\vskip 2mm
\noindent
{\bf 2.} Suppose that $A$ is obtained from $C$ by attaching cells
$e^{m+1}_{\alpha}$ and $B$ is obtained by attaching cells
$e^{n_{\beta}}_\beta$ of dimensions $\ge n+1$. Consider a map
$f:(I^i,\partial I^i,J^{i-1})\to (X,B,x_0)$. $f$ meets only
finite number of cells $e^{n_{\beta}}_{\beta}$. According to the
case 1 we can show that $f$ is homotopic to
\begin{align*}
f_1:&(I^i,\partial I^i)\to (X-e^{n_1}, B-e^{n_1}),\\
f_2:&(I^i,\partial I^i)\to (X-e^{n_1}-e^{n_2}, B-e^{n_1}-e^{n_2}),\\
    &\dots\\
f_r:&(I^i,\partial I^i)\to (A, C).
\end{align*}

\vskip 2mm
\noindent
{\bf 3.} Suppose that $A$ is obtained from $C$ by attaching cells
of dimensions $\ge m+1$ and $B$ is obtained by attaching cells
of dimensions $\ge n+1$. We may assume that the dimensions of
new cells in $A$ is $\le m+n+1$ since higher dimensional ones
have no effect on $\pi_i$ for $i\le m+n$ by cellular
approximation theorem 12.5. %\ref{HCWca}. 
Let $A_k$ be a CW-subcomplex
of $A$ obtained from $C$ by attaching cells of dimension $\le k$,
similarly let $X_k$ be a CW-subcomplex
of $X$ obtained from $B$ by attaching cells of dimension $\le
k$. Using the long exact sequences for triples $(A_k,A_{k-1},C)$
and $(X_k,X_{k-1},B)$, we get the diagram
$$
\xymatrix{
  \pi_{i+1}(A_k,A_{k-1})\ar[r]\ar[d]^{\cong}
& \pi_{i}(A_{k-1},C)\ar[r]\ar[d]
& \pi_{i}(A_k,C)\ar[r]\ar[d]
& \pi_{i}(A_k,A_{k-1})\ar[r]\ar[d]^{\cong}
& \pi_{i-1}(A_{k-1},C)\ar[d]\\
  \pi_{i+1}(X_k,X_{k-1})\ar[r]
& \pi_{i}(X_{k-1},B)\ar[r]
& \pi_{i}(X_k,B)\ar[r]
& \pi_{i}(X_k,X_{k-1})\ar[r]
& \pi_{i-1}(X_{k-1},B)
}
$$
Applying the previous step for $X_k=A_k\cup X_{k-1}$ and $A_{k-1}=A_k\cap X_{k-1}$ we obtain the indicated isomorphisms. Now the induction with respect to $k$ and 5-lemma completes the proof that $\pi_i(A_{m+n+1},C)\to \pi_i(X_{m+n+1},B)$ is an isomorphism for $i<m+n$ and an epimorphism for $i=m+n$.

\vskip 2mm
\noindent
{\bf 4.} Consider a general case. Then according to Corrolary 12.6 %\ref{HCWcwa} 
there is a CW-pair
$(A',C)$ homotopy equivalent to $(A,C)$  and a CW-pair $(B',C)$
homotopy equivalent to $(B,C)$ such that $A'-C$ contains only
cells of dimension $\ge m+1$ and $B'-C$ contains only cells of
dimension $\ge n+1$. Then $X'=A'\cup B'$ is homotopy
equivalent to $X=A\cup B$. According to the previous case
$j':(A',C)\to (X',B')$ is an $(m+n)$-equivalence, consequently
$j:(A,C)\to (X,B)$ is an $(m+n)$-equivalence as well.
\end{proof}

\begin{cor*} If a CW-pair $(X,A)$ is $r$-connected and $A$ is
$s$-connected with $r,s\ge 0$, then the homomorphism
$$\pi_i(X,A)\to \pi_i(X/A)$$
induced by the quotient map
$X\to X/A$ is an isomorphism for $i\le r+s$ and an epimorphism for
$i\le r+s+1$.
\end{cor*}

\begin{proof} Consider the diagram:
$$
\xymatrix{
\pi_i(X,A)\ar[r]
& \pi_i(X\cup CA,CA) \ar[d]\ar[r]
& \pi_i(X\cup CA/CA)\ar[r]^-{\cong}
& \pi_i(X/A)\\
& \pi_i(X\cup CA)\ar[u]_{\cong}\ar[ru]_-{\cong}
& &
}
$$
The first homomorphism is $(r+s+1)$-equivalence by the homotopy
excision theorem for $(s+1)$-connected pair $(CA,A)$ and
$r$-connected pair $(X,A)$. The vertical isomorphism comes from
the long exact sequence for the pair $(X\cup CA,CA)$ and the
remaining isomorphisms are induced by a homotopy equivalence and
the identity $X\cup CA/CA=X/A$.
\end{proof}
\end{cislo}

\begin{cislo} \label{EHfst}{\bf Freudenthal suspension theorem.} We have defined the suspension of a space in 1.5 and the reduced suspension of a space with distinquished point 
in 1.6. In 4.3 we have introduced the suspension of a map. In a similar way we can define
the reduced suspension of a map which preserves distinquished points. This notion  defines
so called suspension homomorphism $\pi_i(X)\to \pi_{i+1}(X), \ [f]\mapsto [\Sigma f]$ for every space $X$.
 
\begin{thm*} [Freudenthal suspension theorem] \index{Freudenthal suspension theorem}
Let $X$ be $(n-1)$-connected CW-complex, $n\ge 1$. Then the suspension homomorphism 
$\pi_i(X)\to \pi_{i+1}(\Sigma X)$ is an
isomorphism for $i\le 2n-2$ and an epimorphism  for $i\le 2n-1$.
\end{thm*}

\begin{proof}
The suspension $\Sigma X$ is a union of two
reduced cones $\widetilde C_{+}X$ and $\widetilde C_{-}X$ with intersection $X$. Now, we get
$$\pi_i(X)\cong\pi_{i+1}(\widetilde C_{+}X,X)\to\pi_{i+1}(\Sigma X,\widetilde C_{-}X)\cong
\pi_{i+1}(\Sigma X)$$
where the first and the last isomorphisms come from the long
exact sequences for pairs $(\widetilde C_{+}X,X)$ and $(\Sigma X,\widetilde C_{-}X)$,
respectively, and the middle homomorphism comes from homotopy
excision theorem for $n$-connected pairs $(\widetilde C_{+}X,X)$ and
$(\widetilde C_{-}X,X)$. What remains is to show that the induced map
on the level of homotopy groups is the same as suspension homomorphism which is left 
to the reader.
\end{proof}
\end{cislo}

\begin{cislo}\label{EHshg}{\bf Stable homotopy groups.} The
Freudenthal suspension theorem enables us to define \emph{stable homotopy
groups}. \index{stable homotopy group} Consider a based space $X$
and an integer $j$. The $n$-times iterated reduced suspension $\Sigma^nX$ is at least
$(n-1)$-connected. If $n\geq j+2$, then $i=j+n\le 2n-2$, so the assumptions of the Freudenthal suspension theorem are satisfied and we get
$$\pi_{j+(j+2)}(\Sigma^{j+2}X)\cong
\pi_{j+(j+3)}(\Sigma^{j+3}X)\cong\pi_{j+(j+4)}(\Sigma^{j+4}X)
\cong\dots$$
Hence we define the $j$-th stable homotopy group of the space $X$ as
$$\pi_j^s(X)=\lim_{n\to\infty}\pi_{j+n}(\Sigma^nX).$$
We will write $\pi_j^s$ for the $j$-th stable homotopy group of
$S^0$.
\end{cislo}


\begin{cislo}\label{EHcom}{\bf Computations.}
In this paragraph we compute $n$-th homotopy groups of
$(n-1)$-connected CW-complexes.

\begin{thma}$\pi_n(S^n)\cong \Z$ generated by the identity map for
all $n\ge 1$. Moreover, this isomorphism is given by the degree map
$\pi_n(S^n)\to \Z$.
\end{thma}

\begin{proof} Consider the diagram
$$\xymatrix{
\pi_1(S^1)\ar[r]^-{\text{epi}}\ar[d]^-{\cong}_-{\deg}
&\pi_2(S^2)\ar[r]^-{\cong}\ar[d]_{\deg}
& \pi_3(S^3)\ar[r]^-{\cong}\ar[d]_{\deg}
& \dots\\
\Z \ar[r]^-{=}& \Z\ar[r]^-{=}&\Z\ar[r]^-{=} &\dots
}$$
where the horizontal homomorphisms are suspension homomorphisms
and the left vertical isomorphism is known from Section 11 and
determined by degree. The statement follows now from the fact that
$\deg f=\deg \Sigma f$.
\end{proof}

\begin{ex*} Prove that $\pi_n(\prod_{\alpha\in A}
X_{\alpha})=\prod_{\alpha\in A}\pi_n(X_{\alpha})$.
\end{ex*}

\begin{thma} $\pi_n(\bigvee_{\alpha\in
A}S^n_{\alpha})=\bigoplus_{\alpha\in A}\Z$ for $n\ge 2$.
\end{thma}

\begin{proof}Suppose first that $A$ is finite. Then CW-complex
$\bigvee_{\alpha\in A}S^n_{\alpha}$ is a subcomplex of CW-complex
$\prod_{\alpha\in A}S^n_{\alpha}$. The pair
$$\big(\prod_{\alpha\in A}S^n_{\alpha},\bigvee_{\alpha\in
A}S^n_{\alpha}\big)$$
is $(2n-1)$-connected since $\prod_{\alpha\in A}S^n_{\alpha}$ is
obtained from $\bigvee_{\alpha\in A}S^n_{\alpha}$ by attaching
cells of dimension $\ge 2n$. Hence
$$\pi_n(\bigvee_{\alpha\in A}S^n_{\alpha})=
\pi_n(\prod_{\alpha\in A}S^n_{\alpha})=\prod_{\alpha\in
A}\pi_n(S^n_{\alpha})=\bigoplus_{\alpha\in
A}\pi_n(S^n_{\alpha})=\bigoplus_{\alpha\in A}\Z.$$

If $A$ is infinite, consider homomorphism $\phi:\bigoplus_{\alpha\in
A}\pi_n(S^n_{\alpha})\to \pi_n(\bigvee_{\alpha\in
A}S^n_{\alpha})$ induced by inclusions $\pi_n(S^n_{\alpha})\to
\bigvee_{\alpha\in A}S^n_{\alpha}$. $\phi$ is surjective since
any $f:S^n\to \bigvee_{\alpha\in A}S^n_{\alpha}$ has a compact
image and meets only finitely many $S^n_{\alpha}$'s. Similarly,
if $h:S^n\times I\to \bigvee_{\alpha\in A}S^n_{\alpha}$ is
homotopy between $f$ and the constant map, it meets only finitely
many $S^n_{\alpha}$'s, so $\phi^{-1}([f])$ is zero.
\end{proof}

\begin{thma} Suppose $n\ge 2$. If $X$ is obtained from
$\bigvee_{\alpha\in A}S^n_{\alpha}$ by attaching cells
$e^{n+1}_{\beta}$ via base point preserving maps
$\p_{\beta}:S^n\to \bigvee_{\alpha\in A}S^n_{\alpha}$, then
$$
\pi_i(X)=
\begin{cases} 0 &\text{if }i< n,\\
\bigoplus_{\alpha\in A}\pi_n(S^n_{\alpha})/N & \text{if
}i=n.
\end{cases}
$$
where $N$ is a subgroup of $\bigoplus_{\alpha\in A}\pi_n(S^n_{\alpha})$
generated by $[\p_{\beta}]$.
\end{thma}

\begin{proof} The first equality is clear from the cellular
approximation theorem.  Consider the long exact sequence for the pair
$(X,X^n=\bigvee_{\alpha\in A}S^n_{\alpha})$
$$\pi_{n+1}(X,X^n)\xrightarrow{\partial}\pi_n(X^n)\to \pi_n(X)\to
0.$$
The pair $(X,X^n)$ is $n$-connected, $X^n$ is
$(n-1)$-connected, hence by Corollary \ref{EHex}
$$\pi_{n+1}(X,X^n)\to
\pi_{n+1}(X/X^n)=\pi_{n+1}(\bigvee_{\beta\in
B}S^{n+1}_{\beta})=\bigoplus_{\beta\in B}\Z$$
is an isomorphism. Hence
$$\pi_n(X)=\pi_n(X^n)/\im\partial=\pi_n(\bigvee_{\alpha\in A}S^n_{\alpha})
/N$$
since $\im\partial$ is generated by $[\p_{\beta}]$.
\end{proof}
\end{cislo}

\begin{cislo}\label{EHhh}{\bf Hurewicz homomorphism.} The
\emph{Hurewicz map}\index{Hurewicz homomorphism}
$h:\pi_n(X,A,x_0)\to H_n(X,A)$  assigns to every element in
$\pi_n(X,A,x_0)$ represented by $f:(D^n,\partial D^n,s_0)\to
(X,A,x_0)$ the element $f_*(\iota)\in H_n(X,A)$ where
$\iota\in H_n(D^n,\partial D^n)=H_n(\Delta^n,\partial \Delta^n)$
is the generator induced by the identity map $\Delta^n\to \Delta^n$.
In the same way we can define the Hurewicz map $h:\pi_n(X)\to
H_n(X)$.

\begin{prop} The Hurewicz map is a homomorphism.
\end{prop}

\begin{proof} Let $c:D^n\to D^n\vee D^n$ be the map collapsing
equatorial $D^{n-1}$ into a point, $q_1,q_2:D^n\vee D^n\to D^n$
quotient maps and $i_1,i_2:D^n\to D^n\vee D^n$ inclusions.
We have the diagram
$$
\xymatrix{
H_n(D^n,\partial D^n)\ar[r]^-{c_*}
& H_n(D^n\vee D^n,\partial D^n\vee\partial D^n)\ar[r]^-{f\vee g}
\ar@<1ex>[d]^{{q_1}_*\oplus{q_2}_*}
& H_n(X,A)\\
& H_n(D^n,\partial D^n)\oplus H_n(D^n,\partial
D^n)\ar@<1ex>[u]^{{i_1}_*+{i_2}_*} &
}
$$
Since ${i_1}_*+{i_2}_*$ is an inverse to ${q_1}_*\oplus {q_2}_*$,
we get
\begin{align*}
h([f]+[g])&=(f+g)_*(\iota)=(f\vee g)_*c_*(\iota)\\
&=\big((f\vee
g)_*({i_1}_*+{i_2}_*)\big)\big(({q_1}_*\oplus{q_2}_*)c_*\big)
(\iota)=(f_*+g_*)(\iota\oplus\iota)\\
&=f_*(\iota)+g_*(\iota)=h([f])+h([g]).
\end{align*}
\end{proof}

We leave the reader to prove  the following
properties of the Hurewicz homomorphism directly from the definition:

\begin{prop} The Hurewicz homomorphism is natural, i.~e. the
diagram
$$
\xymatrix{
\pi_n(X,A)\ar[r]^-{f_*}\ar[d]_{h_X}
& \pi_n(Y,B) \ar[d]^{h_Y}\\
H_n(X,A)\ar[r]_-{f_*}
& H_n(Y,B)
}$$
commutes for any $f:(X,A)\to (Y,B)$.

The Hurewicz homomorphisms make commutative also the following
diagram with long exact sequences of a pair $(X,A)$:
$$
\xymatrix{
\pi_n(A)\ar[r]\ar[d]^{h_A}
& \pi_n(X)\ar[r]\ar[d]^{h_X}
& \pi_n(X,A)\ar[r]^-{\partial}\ar[d]^{h_{(X,A)}}
& \pi_{n-1}(A)\ar[d]^{h_A}\\
H_n(A)\ar[r]
& H_n(X)\ar[r]
& H_n(X,A)\ar[r]^-{\partial}
& H_{n-1}(A)
}$$
\end{prop}
\end{cislo}

\begin{cislo} \label{EHht} {\bf Hurewicz theorem.}
The previous calculations of $\pi_n(\bigvee_{\alpha\in A} S^n_{\alpha})$
enable us to compare homotopy and
homology groups of $(n-1)$-connected CW-complexes via the
Hurewicz homomorphism.

\begin{thma}[Absolute version of the Hurewicz theorem] Let $n\ge 2$.
If $X$ is a $(n-1)$-connected, then
$\tilde H_i(X)=0$ for $i< n$ and $h: \pi_n(X)\to H_n(X)$
is an isomorphism.
\end{thma}

For the case $n=1$ see Theorem 11.5. %\ref{FGfgh}.

\begin{proof} We will carry out the proof only for CW-complexes $X$.
For general method which enables us to enlarge the result to all
spaces  see [Hatcher], %\cite{Ha}, 
Proposition 4.21.

First, realize that $h:\pi_n(S^n)\to H_n(S^n)$ is an isomorphism.
It follows from the characterization of $\pi_n(S^n)$ by
degree in Theorem \ref{EHcom}.

According to Corollary 12.6 %\ref{HCWcwa} 
every $(n-1)$-connected CW-complex $X$ is homotopy
equivalent to a CW-complex obtained by attaching cells of
dimension $\ge n$ to a point. Moreover cells of dimension
$\ge n+2$ do not play any role in computing $\pi_i$ and $H_i$ for $i\leq n$.
Hence we may suppose that
$$X=\bigvee_{\alpha\in
A}S^n_{\alpha}\cup_{\p_{\beta}}\bigcup_{\beta\in
B}e^{n+1}_{\beta}=X^{n+1}$$
where $\p_{\beta}$ are base point preserving maps. Then $\tilde
H_i(X)=0$ for $i<n$.

Using the long exact sequences for the pair $(X,X^n)$ and the
Hurewicz homomorphisms between them we get
$$
\xymatrix{
\pi_{n+1}(X,X^n)\ar[r]^-{\partial}\ar[d]^-{h}
& \pi_n(X^n)\ar[r]\ar[d]^-{h}
&\pi_n(X)\ar[r]\ar[d]^-{h}
& 0\\
H_{n+1}(X,X^n)\ar[r]^-{\partial}
& H_n(X^n)\ar[r]
& H_n(X)\ar[r]
& 0
}$$
Since $\pi_{n+1}(X,X^n)$ is isomorphic to
$\pi_{n+1}(X/X^n)=\bigoplus \pi_{n+1}(S^{n+1}_{\beta})$
and $\pi_n(X^n)=\bigoplus \pi_{n}(S^{n}_{\alpha})$, the first
and the second Hurewicz homomorphisms are isomorphisms. According
to the 5-lemma so is $h:\pi_n(X)\to H_n(X)$.
\end{proof}

Let $[\gamma]\in \pi_1(A,x_0)$, $[f]\in \pi_n(X,A,x_0)$. Then
$\gamma\cdot f$ and $f$ are homotopic (although the homotopy does
not keep the base point $x_0$ fixed), and consequently,
$$(\gamma\cdot f)_*(\iota)=f_*(\iota)$$
for $\iota\in H_n(D^n,\partial D^n)$. Hence
$h([\gamma]\cdot[f])=h([f])$.

Let  $\pi'_n(X,A,x_0)$  be the factor of $\pi_n(X,A,x_0)$ by the
normal subgroup generated by $[\gamma]\cdot[f]-[f]$. Let
$h':\pi'_n(X,A,x_0)\to H_n(X,A)$ be the map induced by the
Hurewicz homomorphism $h$.

\begin{thma} [Relative version of the Hurewicz theorem]
Let $n\ge 2$. If a pair (X,A) of the path connected
spaces is $(n-1)$-connected, then
$H_i(X,A)=0$ for $i< n$ and $h':\pi'_n(X,A,x_0)\to H_n(X,A)$ is
an isomorphism.
\end{thma}

\begin{proof} We will prove the theorem for a pair $(X,A)$
of CW-complexes where $A$ is supposed  to be simply
connected. In this case $\pi'_n(X,A,x_0)=\pi_n(X,A,x_0)$ and
$h'=h$. For general proof see [Hatcher], %\cite{Ha}, 
Theorem 4.37, pages 371--373.

Since $(X,A)$ is $(n-1)$-connected and $A$ is 1-connected,
Corollary \ref{EHex} implies that the quotient map
$\pi_n(X,A)\to\pi_n(X/A)$ is an isomorphism and $X/A$ is
$(n-1)$-connected. The absolute
version of the Hurewicz theorem  and the commutativity of the diagram
$$
\xymatrix{
\pi_n(X,A)\ar[r]^-{\cong}\ar[d]_h
& \pi_n(X/A)\ar[d]^{h}_{\cong}\\
H_n(X,A)\ar[r]^-{\cong}
& H_n(X/A)
}
$$
imply immediately the required statement.
\end{proof}
\end{cislo}

\begin{cislo}\label{EHwt}{\bf Homology version of Whitehead
theorem.} Since computations in homology are much easier that in
homotopy, the following homology version of the Whitehead theorem
gives a very useful method how to prove that two spaces are homotopy
equivalent.

\begin{thm*} [Whitehead theorem] A map $f:X\to Y$ between two
simply connected CW-complexes is homotopy equivalence if
$f_*:H_n(X)\to H_n(Y)$ is an isomorphism for all $n$.
\end{thm*}

\begin{proof} Replacing $Y$ by the mapping cylinder $M_f$ we can
consider $f$ to be an inclusion $X\hookrightarrow Y$.
Since $X$ and $Y$ are simply connected, we have $\pi_1(Y,X)=0$.
Using the relative version of the Hurewicz theorem and the induction with respect to $n$,
we get successively that $$\pi_n(Y,X)=H_n(Y,X)=0.$$
The long exact sequence of homotopy groups for the pair $(Y,X)$ yields that 
$f_*:\pi_n(X)\to \pi_n(Y)$  is an isomorphism for all $n$.
Applying now the Whitehead theorem 12.3 %\ref{HCWwt} 
we get that $f$ is a
homotopy equivalence.
\end{proof}
\end{cislo}