p-surgery kills the represented pi-p class below the middle dimension

To supply the CW prerequisite under the stated choice assumption, use [F7] to replace each face by an open Euclidean tube $U$ of the same homotopy type. Cover $U$ by all balls with rational centres and rational positive radii whose closures lie in $U$. This is a countable open cover with convex finite intersections; contract each nonempty intersection to its first rational point in a fixed enumeration. A supplied subordinate partition makes the projection from its Čech realization to $U$ a homotopy equivalence: its section is the partition barycentre and each fibre contracts linearly to that section. Collapsing the convex intersection factors identifies this realization up to homotopy with the nerve, by the simplex-by-simplex mapping-cylinder argument of Hatcher, section 4G, Propositions 4G.1–4G.2 and Corollary 4G.3. There are countably many simplices, so at most countable choice is spent by that argument. The nerve is a CW complex (its simplex cells are closure-finite with the weak realization topology). This supplies a based CW model of each face; a homotopy inverse carries the attaching sphere to a map into that model. Homotopic attaching maps have equivalent adjunction spaces: a homotopy is inserted on a boundary annulus of the attached disk, and its reverse gives the inverse; their composites contract the two annuli, fixing the base. The same construction transports attachments along the model equivalence. Thus the two cell models of [F3] may be replaced by actual relative CW pairs, preserving the indicated face groups and characteristic-disk boundary classes. By [F3] and [F1] the pair $(W_{\phi },M)$ is homotopy equivalent, relative to $M$, to the pair obtained from $M$ by attaching one $(p+1)$-cell along the core embedding ${\phi }_0$, so the pair ${\pi }_i(M)\to {\pi }_i(W_{\phi })$ is the map of [F4] for the attachment along ${\phi }_0$. Hence ${\pi }_i(M)\to {\pi }_i(W_{\phi })$ is an isomorphism for $1\le i\le p-1$ and a surjection for $i=p$, and with ${\mathrm{\partial }}_M:{\pi }_{p+1}(W_{\phi },M)\to {\pi }_p(M)$ the connecting homomorphism of the pair, the class $z$ of the underlying sphere lies in $\ker ({\pi }_p(M)\to {\pi }_p(W_{\phi }))=\mathrm{im}({\mathrm{\partial }}_M)$.

The $q$-cell cannot join or create components because $q\ge p+2\ge 3$, so the outgoing face is connected since the trace is connected. Fix a basepoint in each face and a path between them through the trace; all comparisons use that specified path (there is no canonical homomorphism independent of basepoint transport). Dually, [F3] and [F1] express $(W_{\phi },M_{\phi })$ as the pair obtained from $M_{\phi }$ by attaching one $q$-cell along the belt-sphere embedding $\beta$, relative to $M_{\phi }$. Since $q\ge p+2$, the new cell has dimension at least $p+2$, so [F5] with $n=q$ gives that ${\pi }_i(M_{\phi })\to {\pi }_i(W_{\phi })$ is an isomorphism for $1\le i\le q-2$, in particular for $1\le i\le p$, and a surjection for $i=q-1$.

Combination. For $1\le i\le p-1$ both maps ${\pi }_i(M)\to {\pi }_i(W_{\phi })$ and ${\pi }_i(M_{\phi })\to {\pi }_i(W_{\phi })$ are isomorphisms, so ${\pi }_i(M)\to {\pi }_i(M_{\phi })$ is an isomorphism; for $i=p$ the first map is a surjection and the second an isomorphism, so the composite ${\pi }_p(M)\to {\pi }_p(M_{\phi })$ is a surjection. This proves (i).

Kernel in degree $p$. Since ${\pi }_p(M_{\phi })\to {\pi }_p(W_{\phi })$ is injective, the kernel of ${\pi }_p(M)\to {\pi }_p(M_{\phi })$ equals the kernel of ${\pi }_p(M)\to {\pi }_p(W_{\phi })$, which is $\mathrm{im}({\mathrm{\partial }}_M)$ by [F6]; this is a subgroup of ${\pi }_p(M)$ containing $z$, and consequently ${\pi }_p(M_{\phi })\cong {\pi }_p(M)\mathrm{/}\mathrm{im}({\mathrm{\partial }}_M)$ by the first isomorphism theorem. This proves (ii).

Simply connected case. If $M$ is simply connected, apply the third clause of [F4] to the cell attachment model of step 1.1: the relative group ${\pi }_{p+1}(M{\cup }_{{\phi }_0}{D}^{p+1},M)$ is infinite cyclic on the class of the characteristic disk, and the connecting homomorphism has image the subgroup generated by $z$, which is cyclic and can be finite when $z$ has finite order. Transporting along the homotopy equivalence of pairs of step 1.1, the kernel $\mathrm{im}({\mathrm{\partial }}_M)$ is the subgroup generated by $z$, cyclic, possibly finite, and trivial when $z=0$, and step 2.2 gives ${\pi }_p(M_{\phi })\cong {\pi }_p(M)\mathrm{/}⟨z⟩$. This proves (iii).

The hypothesis $p\le q-2$ is exactly $2p+2\le m$; it is used in step 1.2 to make the dual inclusion an isomorphism in degree $p$. Beyond that range the dual cell can meet degree $p$ and the conclusion of (i) may fail, so no statement is made there.