Kernel classes are represented by embedded spheres below the middle dimension

By the relative disk definition, $x$ has a boundary map $g:S^p\to M$ and a target disk filling $f\circ g$. For $p\ge 1$ keep its marked source point at the basepoint of $M$ as follows. Choose small source and target coordinate balls about these points, with $g$ of the source ball landing in the target ball and both marked coordinates zero. Multiplying the target coordinates of $g$ by $1-t\chi$ for a source bump $\chi =1$ near the marked point makes $g$ constant on a smaller ball by a based homotopy. Apply [F3] relative to that smaller closed ball. In its constant region insert, with a further cutoff, a small linear $p$-dimensional coordinate cap through zero; this is a based smooth homotopy and gives injective ambient derivative at the marked point. Now [F4] applies relative to that singleton because $m\ge 2p+2>2p$, giving an embedded sphere through the same basepoint. Glue the resulting based boundary homotopy cylinder to the target filling, so the represented relative class remains $x$. The target map is never smoothed when $X$ is merely a CW complex. For $p=0$ keep the marked endpoint fixed and move the other to a distinct nearby point in a source chart, appending the image of that path to its target path.

For finitely many classes choose these representatives successively. Relative to a small marked-point neighbourhood, perturb each sphere transversely to the previous finitely many spheres. Their expected intersection dimension is $2p-m\le -2$, so there are no intersections outside the protected neighbourhood. Their homotopies again attach to their target nullhomotopies. If unbased disjoint surgery representatives are needed, move their marked points along short distinct source paths and include these paths in their whiskers; a finite family then has entirely disjoint embedded spheres. This changes basepoint representatives by the recorded action and does not change the corresponding generated kernel.

The normal structure gives a stable isomorphism ${\nu }_M\cong f^{\ast }\xi$. Pull it back along $g$. The supplied nullhomotopy makes $(f\circ g{)}^{\ast }\xi$ stably trivial: a bundle pulled back over a compact ball can be framed directly. To see this without a strong-AC homotopy-invariance theorem, take finitely many trivializing charts of the pulled-back bundle and a finite support-subordinate partition, constructed from small balls with closures inside these charts. Their weighted coordinate maps embed the bundle into a finite trivial bundle, giving a continuous orthogonal projection of constant rank. The finite radial projection-and-Gram–Schmidt construction of the sphere-bundle cancellation lemma trivializes it over the ball. Restriction to the boundary gives the required stable trivialization. Thus the stable class of $g^{\ast }{\nu }_M$ is zero. In classifying notation this is $({\nu }_M{)}_{\ast }(\mathrm{\partial }x)=0$, with the target normal datum $\xi$, even when $X$ has no smooth normal bundle.