Hopf generator and mapping-cone coefficient

The Hopf map is a locally trivial circle bundle: in the chart $z_0\mathrm{\ne }0$ with $w=z_1\mathrm{/}{z}_0$, the section is $(1,w)\mathrm{/}\sqrt{1+\mathrm{\mid }w{\mathrm{\mid }}^2}$ and the fiber coordinate is $z_0\mathrm{/}\mathrm{\mid }{z}_0\mathrm{\mid }$; use $(w,1)\mathrm{/}\sqrt{1+\mathrm{\mid }w{\mathrm{\mid }}^2}$ and $z_1\mathrm{/}\mathrm{\mid }{z}_1\mathrm{\mid }$ in the other chart. These formulas give continuous inverse bundle charts. Put $a_i([z])=\mathrm{\mid }{z}_i{\mathrm{\mid }}^2\mathrm{/}(\mathrm{\mid }{z}_0{\mathrm{\mid }}^2+\mathrm{\mid }{z}_1{\mathrm{\mid }}^2)$, $b_i=\max (0,a_i-1\mathrm{/}4)$ and ${\rho }_i=b_i\mathrm{/}(b_0+b_1)$. At least one $a_i\ge 1\mathrm{/}2$, so the denominator is positive, and $\mathrm{supp}{\rho }_i\subseteq \{a_i\ge 1\mathrm{/}4\}\subseteq \{z_i\mathrm{\ne }0\}$. This is a finite numeration with closed supports subordinate to the charts. By [F6] the Hopf bundle is Hurewicz and hence Serre. By [F1], ${\pi }_2(S^1)={\pi }_3(S^1)=0$; the LES [F2] therefore makes $h_{\ast }:{\pi }_3(S^3)\to {\pi }_3(S^2)$ an isomorphism. By [F3] its source is generated by the identity, so $[h]$ generates ${\pi }_3(S^2)\cong \mathbb{Z}$.

Write ${\mathbb{C}\mathbb{P}}^2=\{[z_0:z_1:z_2]\}$ and use the open cell $z_2\mathrm{\ne }0$ with representative $(w_0,w_1,1)$, $w\in {\mathbb{C}}^2$. Its characteristic disk can be taken as the unit ball in ${\mathbb{C}}^2$ via $w\mapsto [w_0:w_1:\sqrt{1-\mathrm{\mid }w{\mathrm{\mid }}^2}]$; the interior parametrizes the complement of ${\mathbb{C}\mathbb{P}}^1$, and on $\mathrm{\mid }w\mathrm{\mid }=1$ the boundary map is $[w_0:w_1]=h(w)$. Thus ${\mathbb{C}\mathbb{P}}^2$ is the mapping cone $C_h$. By [F4], the degree-two generator squares to the degree-four generator, so [F5] gives $H(h)=1$.

Let $d_m:S^3\to S^3$ have degree $m$. The radial cone extension $D^4\to D^4$, $ru\mapsto r{d}_m(u)$, has relative degree $m$ and boundary $d_m$: on collapsing the disk boundary its map is $\mathrm{\Sigma }{d}_m$, whose degree is $m$ by the natural homology suspension in [F6]. Together with the identity on $S^2$, it defines a map $C_{h\circ d_m}\to C_h$, since the boundary attaching maps agree. It pulls the degree-two generator of $C_h$ back to that of $C_{h\circ d_m}$ and the degree-four generator back to $m$ times the latter. Pulling back $x_h^2=y_h$ therefore gives $x_{h\circ d_m}^2=m{y}_{h\circ d_m}$, which is exactly $H(h\circ d_m)=m$ by [F5].

The coefficient $H(f)$ is invariant under a based homotopy $H_t$ of attaching maps. A comparison $C_{f_0}\to C_{f_1}$ is the identity on $S^2$; on a cone point $[x,t]$ it equals $H_{2t}(x)$ for $0\le t\le 1\mathrm{/}2$ and the target cone point $[x,2t-1]$ for $1\mathrm{/}2\le t\le 1$. The formulas agree at the seam, respect the tip, and are continuous by quotient descent. Reversing the homotopy gives its homotopy inverse: the composites insert a path followed by its reverse, which contracts by shrinking its traversed homotopy parameter, and then a linear cone reparametrization removes the constant collar. Both maps restrict to the identity on the bottom sphere and have relative top-cell degree $+1$, so cup-product naturality preserves $H$. Since step 1.1 identifies every homotopy class with $[h\circ d_m]$ for a unique integer $m$, step 2.1 implies that any attachment with $H(f)=m$ has class $m[h]$.