\section{Short overview of some further methods in homotopy
theory}

We start this sections with two examples of computations of
homotopy groups. These computations demonstrate the fact
that the possibilities of the methods we have learnt so far are
very restricted. Hence we outline some further (still very
classical) methods which enable us to prove and compute more.

\begin{cislo}\label{SOsm}{\bf Homotopy groups of Stiefel
manifolds.} Let $n\ge 3$ and $n>k\ge 1$. The Stiefel manifold
$V_{n,k}$ is $(n-k-1)$-connected and
$$
\pi_{n-k}(V_{n,k})=
\begin{cases}\Z &\text{for }k=1,\\
             \Z &\text{for }k\ne 1 \text{ and }n-k\text{ even},\\
           \Z_2 &\text{for }k\ne 1 \text{ and }n-k\text{ odd}.
\end{cases}
$$

\begin{proof} The statement about connectivity follows from the
long exact sequence for the fibration
$$V_{n-1,k-1}\to V_{n,k}\to V_{n,1}=S^{n-1}$$
by induction.

As for the second statement, it is sufficient to prove that
$$
\pi_{n-2}(V_{n,2})=
\begin{cases}\Z &\text{for } n \text{ even},\\
           \Z_2 &\text{for } n \text{ odd}
\end{cases}
$$
and to use the induction in the long exact sequence for the
fibration above.

We have the fibration
$$S^{n-2}=V_{n-1,1}\to V_{n,2}\xrightarrow{p} V_{n,1}=S^{n-1}$$
which corresponds to the tangent vector bundle of the sphere
$S^{n-1}$. If $n$ is even, there is a nonzero vector field on
$S^{n-1}$. This field is  a map  $s:S^{n-1}\to V_{n,2}$ such that
$ps=\id_{S^{n-1}}$. Such a map is called a section and its
existence ensures that the map $p_*:\pi_{n-1}(V_{n,2})\to
\pi_{n-1}(S^{n-1})$ is an epimorphism. Hence we get the following
part of the long exact sequence
$$\pi_{n-1}(V_{n,2})\xrightarrow{\text{\ epi\ }}\pi_{n-1}(S^{n-1})
\xrightarrow{\ 0\ }\pi_{n-2}(S^{n-2})\xrightarrow{\ \cong\ }\pi_{n-2}
(V_{n,2})\to 0.$$
Consequently, $\pi_{n-2}(V_{n,2})=\Z$.

The case $n$ odd is more complicated. We need the fact that the
Euler class of tangent bundle of $S^{n-1}$ is twice a generator
$\iota\in H^{n-1}(S^{n-1})$. We obtain the following part of the Gysin
exact sequence  for cohomology groups with integer coefficients
$$0\to
H^{n-2}(V_{n,2})\xrightarrow{\ 0\ }H^0(S^{n-1})\xrightarrow{\ \cup
2\iota\ }H^{n-1}(S^{n-1})\to H^{n-1}(V_{n,2})\to 0.
$$
From this sequence and  the universal coefficient
theorem we get that
\begin{align*}
0=H^{n-2}(V_{n,2};\Z)&\cong\hom(H_{n-2}(V_{n,2}),\Z)\\
\Z_2\cong
H^{n-1}(V_{n,2})&\cong\hom(H_{n-1}(V_{n,2}),\Z)\oplus\ext
(H_{n-2}(V_{n,2}),\Z)
\end{align*}
which implies that $H_{n-2}(V_{n,2};\Z)\cong\Z_2$. The Hurewicz
theorem now yields  $\pi_{n-1}(V_{n,2})\cong \Z_2$.
\end{proof}
\end{cislo}

\begin{cislo}\label{SOhf}{\bf Hopf fibration.} Consider the Hopf
fibration
$$S^1\to S^3\xrightarrow{\ \eta\ }S^2$$
defined in 10.5.  %\ref{HOMfib}. 
From the long exact sequence for this fibration we get
$$\pi_i(S^2)\cong \pi_i(S^3)\quad\text{for }i\ge 2.$$
Particularly,
$$\pi_3(S^2)\cong\Z$$
with $[\eta]$ as a generator (since $[\operatorname{id}]$ is a generator of $\pi_3(S^3)$). By the Freudenthal theorem
$\Z\cong\pi_3(S^2)\xrightarrow{\text{epi}}\pi_4(S^3)\xrightarrow
{\cong}\pi_1^s.$
The methods we have learnt so far give us only that
$\pi_4(S^3)\cong\pi_1^s$ is a factor of $\Z$ with  $\Sigma\eta$ as a generator.

\begin{ex*} Try to compute as much as possible
from the long exact sequences for the other two Hopf fibrations
in 10.5. %\ref{HOMfib}.
\end{ex*}
\end{cislo}

\begin{cislo}\label{SOcom}{\bf Composition methods} were developed
in works of I. James and the Japanese school of H. Toda in the
1950-ies and are described in the monograph [Toda]. %\cite{To}. 
They enable us to find maps which determine the generators of homotopy groups
$\pi_{n+k}(S^n)$ for $k$ not very big (approximately $k\le 20$).
For these purposes various types of compositions and products are
used.

Having two maps $f:S^i\to S^n$ and $g:S^n\to S^m$ their
composition $gf:S^i\to S^m$ determines an element
$[gf]\in\pi_{i}(S^m)$ which depends only on $[f]$ and $[g]$. If
the target of $f$ is different from the source of $g$, we can use
suitable multiple suspensions to be able to make compositions.
For instance, if $f:S^6\to S^4$ and $g:S^7\to S^3$ we can make
composition $g\circ(\Sigma^3f):S^9\to S^3$. (Here $\Sigma$ stands for reduced suspension.) In this way we get a bilinear map $\pi_a^s\times\pi_b^s\to \pi_{a+b}^s$.

More complicated tool is the \emph{Toda bracket}\index{Toda
bracket}. Consider three maps
$$W\xrightarrow{f}X\xrightarrow{g}Y\xrightarrow{h}Z$$
preserving distinquished points such that $gf\sim 0$ and $hg\sim 0$. Then $gf$  can be extended
to a map $F:\widetilde CW\to Y$ and $hg$ can be extended to a map $G:\widetilde CX\to
Z$. ($\widetilde C$ stands for reduced cone.) Define $\langle f,g,h\rangle:\Sigma W=\widetilde C_+W\cup \widetilde C_-W\to Z$ as
$G\circ \widetilde Cf$ on $\widetilde C_+W$ and $h\circ F$ on $\widetilde C_-W$. 

\begin{figure}[htb]
  \centering
  \def\svgwidth{14cm}
  \input{img/fig-14_1.pdf_tex}
  \caption{Definition of Toda bracket $<f,g,h>$.}
\end{figure}

This definition
depends on homotopies $gf\sim 0$ and $hg\sim 0$. So it defines a
map from $\pi_i^s\times\pi_j^s\times\pi_k^s$ to cosets of
$\pi_{i+j+k+1}^s$. See [Toda] and also Exercise 39 in [Hatcher], Chapter 4.2.

The \emph{Whitehead product}\index{Whitehead product} $[\ ,\
]:\pi_i(X)\times\pi_j(X)\to\pi_{i+j-1}(X)$ is defined as follows:
$f:I^i\to X$ and $g:I^j\to X$ define the map $f\times
g:I^{i+j}=I^i\times I^j\to X$ and we put $[f,g]=(f\times g)/\partial I^{i+j}$.

Having a map $f:S^{2n-1}\to S^n$, $n\ge 2$, we can construct a
CW-complex $C_f=S^n\cup_{f}e^{2n}$ with just one cell in the
dimensions $0$, $n$ and $2n$. Denote the generators of
$H^n(C_f;\Z)$ and $H^{2n}(C_f;\Z)$ by $\alpha$  and $\beta$,
respectively. Then the \emph{Hopf invariant} of $f$ is the number
$H(f)$ such that
$$\alpha^2=H(f)\beta.$$
The Hopf invariant determines a homomorphism
$H:\pi_{2n-1}(S^n)\to \Z$.

For the Hopf map $\eta:S^3\to S^2$ we have $C_{\eta}\cong
\mathbb{CP}^2$, consequently
$$H(\eta)=1.$$
For $\id:S^2\to S^2$
we can make the Whitehead product $[\id,\id]:S^3\to S^2$ and
compute (see [Hatcher], %\cite{Ha}, 
page 474) that $$H([\id,\id])=\pm 2.$$
Since $\pi_3(S^2)\cong \Z$, we get $[\id,\id]=\pm 2\eta$.
One can show (see [Hatcher], %\cite{Ha}, 
page 474 and Corollary 4J.4) that
the kernel of the suspension homomorphism $\pi_3(S^2)\to\pi_4(S^3)$
is generated just by $[\id,\id]$. By the Freudental theorem
this suspension homomorphism is an epimorphism which implies that
$$\pi_4(S^3)\cong \Z_2.$$
Consequently, $\pi_1^s\cong\Z_2$.

\begin{remark*} J. F. Adams proved in [Adams1] %\cite{Ad1} 
that the only maps with the odd Hopf invariant are the maps coming from the
Hopf fibrations $S^3\to S^2$, $S^7\to S^4$ and $S^{15}\to S^8$.
\end{remark*}

Another important tool for composition methods is the EHP exact
sequence for the homotopy groups of $S^n$, $S^{n+1}$ and
$S^{2n}$:
\begin{multline*}
\pi_{3n-2}(S^n)\xrightarrow{E}\pi_{3n-1}(S^{n+1})
\xrightarrow{H}\pi_{3n-2}(S^{2n})\xrightarrow{P}\pi_{3n-3}(S^n)
\to\dots\\
\dots\to \pi_{i}(S^n)\xrightarrow{E}\pi_{i+1}(S^{n+1})
\xrightarrow{H}\pi_{i}(S^{2n})\xrightarrow{P}\pi_{i-1}(S^n)
\to\dots
\end{multline*}
Here $E$ stands for suspension, $H$ refers to a
generalized Hopf invariant and $P$ is defined with connection to
the Whitehead product. See [Whitehead], %\cite{Wh}, 
Chapter XII or [Hatcher], %\cite{Ha},
page 474.

For $n=2$ the EHP exact sequence yields
$$\pi_4(S^2)\xrightarrow{E}\pi_5(S^3)\xrightarrow{H}\pi_4(S^4)
\xrightarrow{P}\pi_3(S^2)\xrightarrow{E}\pi_4(S^3)\to 0.$$
Since $\pi_4(S^3)\cong\Z_2$, $\pi_3(S^2)\cong \Z$ and
$\pi_4(S^4)\cong \Z$, we obtain that $P$ is a multiplication by 2
and $H=0$. From the long exact sequence for the Hopf fibration (see \ref{SOhf})
we get that $\pi_4(S^2)\cong\pi_4(S^3)\cong\Z_2$
with the generator $\eta(\Sigma\eta)$. So $\pi_5(S^3)$ is either
$\Z_2$ or $0$. By a different methods one can show that
$$\pi_5(S^3)\cong\Z_2$$
with the generator $(\Sigma\eta)(\Sigma^2\eta)$.
\end{cislo}

\begin{cislo}\label{SOcoh}{\bf Cohomological methods} have been
playing an important role in homotopy theory since they were
introduced in the 1950-ies.

By the methods used in proofs in Section 12 we can construct so
called \emph{Eilenberg-McLane spaces}\index{Eilenberg-McLane
space} $K(G,n)$ for any $n\ge 0$ and any
group $G$, Abelian if $n\ge 2$. These spaces are up to homotopy
equivalence uniquely determined by their homotopy groups
$$
\pi_i(K(G,n))=
\begin{cases} 0 &\text{for }i\ne n,\\
              G &\text{for }i=n.
\end{cases}
$$
Moreover, these spaces provide the following homotopy description of reduced singular
cohomology groups
$$[(X,*),(K(G,n),*)]\xrightarrow{\ \cong\ } \widetilde H^n(X;G).$$
To each $[f]\in [(X,*),(K(G,n),*)]$ we assign
$$f^*(\iota)\in  \widetilde H^n(X;G)$$
where $\iota$ is the generator of
$$\widetilde H^n(K(G,n);G)\cong \hom(\widetilde H_n(K(G,n);\Z),G)\cong\hom(G,G)$$
corresponding to $\id_{G}$.

A system of homomorphisms $\theta_X:\widetilde H^n(X;G_1)\to \widetilde H^m(X;G_2)$ which is
natural, i.~e. $f^*\theta_Y=\theta_X f^*$ for all $f:X\to Y$,
is called a \emph{cohomology operation}.\index{cohomology
operation} A system of cohomology operations $\theta_j:\widetilde H^{n+j}\to
\widetilde H^{m+j}$ is called \emph{stable}\index{stable cohomology
operation} if it commutes with suspensions
$\Sigma\theta_j=\theta_{j+1}\Sigma$.

The most important stable cohomology operations for singular
cohomology are the Steenrod squares and the Steenrod powers:
\begin{align*}
Sq^i:&\ \widetilde H^n(X;\Z_2)\to \widetilde H^{n+i}(X;\Z_2)\\
P^i_p:&\ \widetilde H^n(X;\Z_p)\to  \widetilde H^{n+2i(p-1)}(X;\Z_p) \quad\text{for }p\ne 2
\text{ a prime}.
\end{align*}
For their definition and properties see [SE] %\cite{SE} 
or [Hatcher], %\cite{Ha}, 
Section 4.L. These operations can be also interpreted as homotopy classes
of maps between  Eilenberg-McLane spaces, for instance
$$Sq^i:K(\Z_2,n)\to K(\Z_2,n+i).$$

\begin{example} We show how the Steenrod squares can be
used to prove that some maps are not homotopic to  a trivial one.
Consider the Hopf map $\eta:S^3\to S^2$. We know that $C_{\eta}=\mathbb{CP}^2$
and $H^2(\mathbb{CP}^2;\Z_2)$ and $H^4(\mathbb{CP}^2;\Z_2)$
have generators $\alpha$ and $\alpha^2$. Since one of the
properties of the Steenrod squares is
$$Sq^nx=x^2\quad \text{for }x\in H^n(X;\Z_2),$$
we get that $Sq^2\alpha=\alpha^2\ne 0$. Using this fact we show that $[\Sigma\eta]
\in\pi_4(S^3)$ is nontrivial.

For reduced cones and reduced suspensions one can prove  that 
$$\widetilde C_{\Sigma\eta}=\Sigma \widetilde C_{\eta}\simeq\Sigma\mathbb{CP}^2.$$
Then $\Sigma\alpha:\Sigma\mathbb{CP}^2\to K(\Z_2,3)$ and $\Sigma\alpha^2:\Sigma\mathbb{CP}^2\to k(\Z_2;5)$ represent generators in
$H^3(\Sigma\mathbb{CP}^2;\Z_2)$ and
$H^5(\Sigma\mathbb{CP}^2;\Z_2)$,
respectively. Now
$$Sq^2(\Sigma\alpha)=\Sigma(Sq^2\alpha)=\Sigma\alpha^2\ne 0.$$
If $\Sigma\eta$ were homotopic to a constant map, we would have
$\widetilde C_{\Sigma\eta}=S^3\vee S^5$, and consequently,
$Sq^2(\Sigma\alpha)=0$ since $Sq^2$ is trivial on $S^3$.
\end{example}

\begin{example} We outline how to compute $\pi_{n+1}(S^n)$
using
cohomological methods. A generator $\alpha\in H^n(S^n)$ induces
up to homotopy a map $S^n\to K(\Z,n)$. Further,
$H^n(K(\Z,n);\Z)\cong\Z$ with a generator $\iota$
and $H^{n+2}(K(\Z,n);\Z_2)$ $\cong\Z_2$
with the generator  $Sq^2\rho\iota$ where $\rho:H^n(X;\Z)\to
H^n(X;\Z_2)$ is induced by reduction mod 2. $Sq^2\rho\iota$
induces up to homotopy a map
$$K(\Z,n)\xrightarrow{Sq^2\rho\iota} K(\Z_2,n+2).$$
Consider the fibration
$$\Omega K(\Z_2,n+2)\to PK(\Z_2,n+2)\to K(\Z_2,n+2)$$
where $PX$ is the space of all maps $p:I\to X$, $p(1)=x_0$
and $\Omega X$ is the space of all maps $\omega: I\to X$,
$\omega(0)=\omega(1)=x_0$. (These maps are called loops in $X$.)
One can show that $\Omega K(\Z_2,n+2)$
has a homotopy type of $K(\Z_2,n+1)$. The pullback of the
fibration above by the map $Sq^2\rho\iota:K(\Z,n)\to K(\Z_2,n+2)$
is the fibration
$$K(\Z_2,n+1)\to E\xrightarrow{p} K(\Z,n).$$
Since $Sq^2\rho\alpha=0$ in $H^{n+2}(S^n;\Z)$, one can show that
the map $\alpha:S^n\to K(\Z,n)$  can be lifted to a map $f:S^n\to
E$.
$$
\xymatrix{
& E\ar[d]^p\\
S^n\ar[r]_-{\alpha} \ar[ur]^-{f} & K(\Z,n)
}
$$
One can compute $f^*$ in cohomology (using so called long Serre exact
sequence) and then also $f_*$ in homology. A modified version of
the homology Whitehead theorem implies that $f$ is an
$(n+2)$-equivalence. Hence $f_*:\pi_{n+1}(S^n)\to \pi_{n+1}(E)$
is an isomorphism. Using the long exact sequence for the
fibration $(E, K(\Z,n),p)$ we get
$$\Z_2\cong \pi_{n+1}(K(\Z_2,n+1))\xrightarrow{\ \cong\ }\pi_{n+1}(E)\cong \pi_{n+1}(S^n).$$
For more details see [MT]. %\cite{MT}.
\end{example}

The Steenrod operations form a beginning for the second course in
algebraic topology which should contain spectral sequences, other
homology and cohomology theories, spectra. We refer the reader to
[Adams2], [Kochman], [MT], [Switzer], [Whitehead] 
%\cite{Ad2}, \cite{Ko}, \cite{MT}, \cite{Sw}, \cite{Wh} 
or to the last sections of [Hatcher]. %\cite{Ha}.
\end{cislo}