Surgery below the middle dimension improves connectivity

In the module range $p\ge 2$, the representation lemma gives embedded representatives of a finite generating family, and the stable-normal lemma gives compatible actual framings. Use unbased disjoint surgery representatives with their recorded whiskers, as in that lemma. Each normal-map surgery is normally bordant to the original map. The trace comparison kills exactly the generated submodule and preserves all lower relative groups, because $f$ is $p$-connected and hence induces a fundamental-group isomorphism. After finitely many surgeries the relative group in degree $p+1$ is zero and all lower ones stay zero. Therefore the endpoint is $(p+1)$-connected. The identical argument works at $p=1$ once the fundamental groups have been identified, using the abelian relative degree-two formulation of the trace lemma.

We justify existence of a finite generating family rather than assume a Noetherian group ring. Under countable choice embed the compact $M$ in Euclidean space. A generic height is Morse; it has finitely many critical points. Choose finitely many disjoint neighbourhoods of them and bumps constant near each point. Small finite shifts of the height values make them distinct while creating no new critical point: outside the protected neighbourhoods $\parallel dh\parallel$ has a positive minimum, and inside them sufficiently small $C^2$ shifts preserve the nondegenerate critical germs. The earlier handle-to-CW suppliers then give $M$ finite CW homotopy type without using the strong-AC excellent-function existence theorem. Replace $f$ by a cellular map on this finite model, so its mapping cylinder is a finite CW pair.

In the module cases $p\ge 1$, put $n=p+1\ge 2$. With the common fundamental group $\pi$, lift this mapping-cylinder pair to universal covers. Lifting disk maps and homotopies identifies its relative homotopy in degrees at least two with that of the covered pair, with the deck action. The covered subspace is simply connected, and $p$-connectivity makes the pair $p$-connected. The published choice-free relative Hurewicz comparison identifies its first relative homotopy with $H_{p+1}$ and makes lower relative homology vanish. Its cellular relative chain complex is finite free over $\mathbb{Z}[\pi ]$, with one lift for each finite cell orbit. Acyclicity below degree $p+1$ permits cancelling split differential pairs from the bottom upward: a surjection onto the bottom free module splits, and inductively the remaining bottom module is finitely generated projective. Thus the cycle module in degree $p+1$ is finitely generated projective after these cancellations, and its quotient by boundaries is finitely generated. The Hurewicz identification is natural under deck maps, so ${\pi }_{p+1}(f)$ is finitely generated over $\mathbb{Z}[\pi ]$. This argument uses split projectivity, not a general claim that submodules of finite free group-ring modules are finitely generated.

If $p=1$ and $f$ is only surjective on fundamental groups, both source and target have finite presentations from their finite CW models. The kernel of a surjection between finitely presented groups is finitely normally generated: use finitely many source generators, write target generators as their images, and add finitely many target relators in those generators; their normal closure is the kernel modulo the source relators. Represent those finitely many kernel loops by embedded circles with target nullhomotopies. The stable-normal framing lemma supplies their compatible framings. The corresponding $1$-surgeries kill these normal generators, and the dual trace cells, of dimension $m-1\ge 3$, preserve the outgoing fundamental group. Hence the resulting map has a fundamental-group isomorphism. It remains $1$-connected. Apply steps 1.2–2.1 with $p=1$ and the choice-free degree-two relative Hurewicz theorem, then step 1.1 to its finite abelian relative module. The endpoint is $2$-connected.

If $p=0$, the connected finite target has a finitely generated fundamental group. Choose core paths representing finitely many generators not already in the source-image subgroup, and represent them by $0$-sphere surgery data with distinct endpoints and the target paths. The orientation of $\xi$ makes determinant transport along every core agree with the incoming normal orientation. The zero-sphere clause of the framing lemma therefore supplies compatible normal frames and an oriented normal trace. The trace lemma enlarges the image subgroup by these generators; after finitely many $0$-surgeries it is all of ${\pi }_1(X)$. Both manifolds remain connected, so the relative fundamental pointed set is trivial and the new map is $1$-connected. Concatenating the finite normal bordisms gives the asserted normal bordism in every case. The complementary trace index is always $m-p\ge p+2$, exactly the range preserving the required relative map groups. Iteration stops at the first failed inequality.