A relative map cell kills its class with the correct fundamental-group action

The inclusion of the original mapping cylinder of $f$ into $Z$ fixes $M$ and is a homotopy equivalence: both cylinders retract onto the same $X$, and their maps onto $X$ commute. Its inverse up to homotopy relative to $M$ can be constructed by extending the cylinder contraction across the characteristic disk with its specified map $h$; equivalently use the mapping-cylinder homotopy-extension property. Thus ${\pi }_i(Z,M)={\pi }_i(f)$, with the disk of $W$ mapping to the specified class $x$. By definition ${\pi }_i(Z,W)={\pi }_i(F)$.

We verify the only relative-cell input directly. A disk map into $(W,M)$ has all its boundary in $M$. In the open attached cell smooth its Euclidean coordinate map near the inverse image of a small closed ball about the cell centre, using finitely many interior source bumps and Euclidean relative approximation. Choose a regular point in that small ball by Sard. For a source dimension less than $n$ its preimage is empty, since a transverse inverse image would have negative dimension. Radially retract the punctured cell from that point onto its boundary and leave $M$ fixed; this retracts $W$ minus that point onto $M$. Consequently ${\pi }_i(W,M)=0$ for $i<n$. No smoothness of the attaching map or of $X$ is needed; all smoothing takes place inside the open Euclidean cell and off the source boundary.

For a source $n$-disk the regular-point preimage in step 1.2 is finite, since it is discrete and contained in a compact interior set. Small disjoint disks about these points map locally diffeomorphically to the cell, with orientation degree $+1$ or $-1$. On their complement retract the map into $M$ by the same punctured-cell retraction. Join these small disks to the marked outer boundary point by a finite embedded tree of thin tubes in the source, arranged disjointly away from their common endpoint. Cutting along this tree is the standard disk description of relative multiplication: each small disk contributes the characteristic relative disk or its inverse, and the image of its tube, after retraction, supplies a path in $M$ carrying its marked value to the basepoint. Comparing that path with the fixed characteristic-disk path gives the ${\pi }_1(M)$ action. For $n\ge 3$ the relative group is abelian, so this is a sum of signed translates. For $n=2$ it is an ordered product; changing the cutting order or tube produces conjugates, so the image in any target relative group is the normal subgroup generated by those translates. Conversely every translate and its inverse is represented by the corresponding whiskered characteristic disk, and every conjugate maps to that normal subgroup. This proves the exact image description needed below, not freeness of a degree-two relative group.

The natural exact sequence of the triple $M\subset W\subset Z$ reads ${\pi }_n(W,M)\to {\pi }_n(Z,M)\to {\pi }_n(Z,W)\to {\pi }_{n-1}(W,M)$. The last term vanishes by step 1.2, hence the middle map is surjective and its kernel is the image computed in step 2.1, namely the stated orbit-generated subgroup, with normal closure in degree two. Both neighbouring groups ${\pi }_i(W,M),{\pi }_{i-1}(W,M)$ vanish in lower group degrees, so the triple sequence gives the asserted lower isomorphisms. In degree one, the image of ${\pi }_1(M)$ in ${\pi }_1(X)$ is unchanged: attaching a cell of dimension at least two only adds relations represented by loops whose image under $f$ is already null by the disk extension. The relative degree-one coset description therefore gives the same pointed set. If $f_{\ast }$ is an isomorphism, identify the action group with ${\pi }_1(X)$; for $n\ge 3$ the group is abelian and its action is precisely a group-ring module action.

For a 1-cell, van Kampen says the new core adjoins a loop to the fundamental group of the connected $M$. Under $F$ its image is $\mathrm{\ell }$, so the image subgroup in ${\pi }_1(X)$ becomes $⟨H,\mathrm{\ell }⟩$. Paths from the basepoint classify the relative fundamental pointed set by cosets of that image, yielding the stated low-degree formulation. This uses no nonexistent relative degree-zero group or abelian module structure. The case $x=0$ in higher degrees gives a zero orbit-generated kernel, and all degree ranges and choices were specified.