Surgery on a normal map preserves its normal bordism class

Contract the disk factor to see that $f\circ \phi$ on $S^p\times D^q$ is homotopic to $f\circ {\phi }_0\circ {\mathrm{pr}}_{S^p}$, which the supplied disk $h$ makes null-homotopic. The attaching region is a subcomplex of the product handle. Start with the constant map on the handle and apply [F9] to the reversed null-homotopy on its attaching region. At the end this gives a handle map extending $f\circ \phi$, and its union with $f\circ {\mathrm{pr}}_M$ on the cylinder gives a continuous $F:W_{\phi }\to X$. This proves the continuous assertion in (i).

By hypothesis the stable bundle isomorphism $B:{\nu }_{W_{\phi }}\to F^{\ast }{\nu }_X$ extends $b$, and by [F3] the outgoing face is identified with $M_{\phi }$, whose stable normal bundle restricts to ${\nu }_{W_{\phi }}$ along the face; hence $b_{\phi }:=B{\mathrm{\mid }}_{M_{\phi }}$ is a stable isomorphism ${\nu }_{M_{\phi }}\to f_{\phi }^{\ast }{\nu }_X$ over the outgoing face, and $B$ restricts to $b$ over the incoming face, with the identifications of [F4]. This proves (ii).

If $f$ is smooth, first prescribe the extension on a small collar on both sides of the attaching region by $f(\phi (x,y))$, constant in the transverse collar coordinate; the boundary-local extension convention and a slightly extended disk factor give a smooth map on a neighbourhood of the cylinder in the rounded trace. Its restriction to the inner collar boundary is homotopic to $f\circ \phi$, hence null-homotopic, so the reversed-HEP argument of step 1.1 extends it continuously over the remaining handle. Apply [F5] relative to the closed cylinder, where this map is now smooth on a neighbourhood, to obtain a smooth $F$ with the same cylinder values. Supply $B$ for this chosen $F$, as required by the datum; the proof does not keep a fixed bundle map while changing its covered base map. If $f$ is merely continuous, use the continuous $F$ of step 1.1. This proves the remaining assertion in (i).

Degree of $f_{\phi }$. The compact oriented $(m+1)$-manifold $W_{\phi }$ has boundary $M\bigsqcup M_{\phi }$, and by [F6] the boundary class pushes forward to zero in $H_m(W_{\phi };\mathbb{Z})$. Applying $F_{\ast }$ and using that $F$ restricts to $f$ and $f_{\phi }$ gives $f_{\ast }[M]-(f_{\phi }{)}_{\ast }[M_{\phi }]=0$, with the signs fixed by the orientation convention of [F7]; since $f_{\ast }[M]=[X]$ by hypothesis, $(f_{\phi }{)}_{\ast }[M_{\phi }]=[X]$, so $f_{\phi }$ has degree one in the total-fundamental-class sense, even if $M_{\phi }$ is disconnected (which can happen when $q=1$).

The data $(W_{\phi },F,B)$ are an oriented bordism from $M$ to $M_{\phi }$ together with a map to $X$ restricting to $f$ and $f_{\phi }$ on the faces and a stable isomorphism ${\nu }_{W_{\phi }}\to F^{\ast }{\nu }_X$ restricting to $b$ and $b_{\phi }$: this is exactly a normal bordism in the sense of [F1]. Hence $(f_{\phi },b_{\phi })$ is a degree-one normal map normally bordant to $(f,b)$, and the normal bordism class is preserved. This proves (iii).

If $p\ge 1$ and $p\le q-2$, the class $z=[{\phi }_0]\in {\pi }_p(M)$ represented by the underlying sphere lies in the kernel of ${\pi }_p(M)\to {\pi }_p(M_{\phi })$ by [F8], independently of the bundle data. This proves (iv).