Attaching a single cell kills the represented homotopy class

The map $f$ need not be cellular. By [F7] homotope it to a cellular $f^{\mathrm{\prime }}:S^p\to X$ and form $Y^{\mathrm{\prime }}=X{\cup }_{f^{\mathrm{\prime }}}{D}^{p+1}$, an actual CW pair relative to $X$ with one new $(p+1)$-cell. The homotopy from $f$ to $f^{\mathrm{\prime }}$ gives a homotopy equivalence $Y\simeq Y^{\mathrm{\prime }}$ relative to $X$: map a concentric inner disk to the other characteristic disk and use the homotopy on the outer boundary annulus; the reverse homotopy gives the inverse, and the two annuli in each composite contract to give homotopies fixing $X$. Thus the required relative CW model is supplied rather than assumed. The equivalence fixes the basepoint in $X$, even when the cellular approximation homotopy was unbased.

The characteristic map $\mathrm{\Phi }:(D^{p+1},S^p)\to (Y,X)$ of [F1] is a relative representative in the sense of [F3] once its marked boundary value in $X$ is moved to the base point along $X$; by the disk model [F5] it determines a class $\iota \in {\pi }_{p+1}(Y,X,x_0)$, using this specified basepoint data, and it is the based characteristic-disk class appearing in the statement. Since f is based, choose its specified marked boundary point; arbitrary transport paths are not asserted to give the same class when ${\pi }_1(X)$ acts nontrivially.

Apply [F2] to the CW pair $(Y^{\mathrm{\prime }},X)$ of step 1.1 and transport across its homotopy equivalence relative to $X$ with $n=p+1\ge 2$: there are no cells outside $X$ of dimension below $p+1$, so at $x_0$ the homomorphism ${\pi }_i(X,x_0)\to {\pi }_i(Y,x_0)$ is an isomorphism for $1\le i\le p-1$ and a surjection for $i=p$. This is assertion (i).

The connecting homomorphism sends $\iota$ to $\pm \alpha$. Indeed $\mathrm{\Phi }$ restricts on the boundary sphere $S^p$ to the composite $f$ followed by the inclusion $X\hookrightarrow Y$, and the boundary sphere $\mathrm{\partial }{D}^{p+1}=S^p$ is identified with the image of $f$ by the pushout, so the restriction of $\mathrm{\Phi }$ to the distinguished face of the disk model is a based representative of $\alpha$; the connecting homomorphism is induced by exactly that restriction, so $\mathrm{\partial }\iota =\pm \alpha$, the sign being the boundary-orientation sign of the disk model.

Now suppose $X$ is simply connected. Then [F6] applies to the CW model of step 1.1 with $A=X$, $k=p+1$ and its single attached cell. The relative equivalence transports its characteristic-disk class to the original one (the homotopy sits on its boundary annulus), and simple connectivity removes the transport-path ambiguity. Thus ${\pi }_{p+1}(Y,X,x_0)$ is free abelian on the class $\iota$ of the characteristic cell, hence infinite cyclic generated by $\iota$.

Consequently $\alpha \in \mathrm{im}(\mathrm{\partial })$, and exactness of the sequence of [F4] at ${\pi }_p(X)$ identifies $\mathrm{im}(\mathrm{\partial })=\ker ({\pi }_p(X)\to {\pi }_p(Y))$: the class $\alpha$ maps to zero in ${\pi }_p(Y)$, and the kernel of the induced map on ${\pi }_p$ is a subgroup of ${\pi }_p(X)$ containing $\alpha$. This is assertion (ii).

The connecting homomorphism restricted to this free cyclic group is determined by $\mathrm{\partial }\iota =\pm \alpha$, so its image is the subgroup generated by $\alpha$: the trivial subgroup when $\alpha =0$, and the cyclic subgroup generated by $\alpha$ otherwise (which may be finite when $\alpha$ has finite order). By step 3.1 the kernel of ${\pi }_p(X)\to {\pi }_p(Y)$ equals that image, and the first isomorphism theorem gives ${\pi }_p(Y)\cong {\pi }_p(X)\mathrm{/}⟨\alpha ⟩$. This is assertion (iii).