Handle Cancellation Slides and Elementary Moves

Extend the isotopy by ${\phi }_t:={\phi }_0$ for $t\le 0$ and ${\phi }_t:={\phi }_1$ for $t\ge 1$; the constancy hypotheses make this a smooth map $M\times \mathbb{R}\to \mathrm{\partial }W$ whose every time slice is an embedding. Define $\mathrm{\Psi }(x,t):=({\phi }_t(x),t)$ on $M\times \mathbb{R}$; it is injective, since $({\phi }_t(x),t)=({\phi }_s(y),s)$ forces $t=s$ and then $x=y$. Its differential is $d\mathrm{\Psi }(v,s)=(d{\phi }_t(v)+s{\mathrm{\partial }}_t{\phi }_t,s)$, which is injective: $s=0$ from the second component and then $v=0$ from injectivity of $d{\phi }_t$. At points with $x\in \mathrm{int}M$ the source $\mathrm{int}M\times \mathbb{R}$ and the target $\mathrm{\partial }W\times \mathbb{R}$ both have dimension $n$, so $d\mathrm{\Psi }$ is an isomorphism there, and the image $S^{\circ }:=\mathrm{\Psi }(\mathrm{int}M\times \mathbb{R})$ is an open subset of $\mathrm{\partial }W\times \mathbb{R}$.

Apply the compact full-dimensional graph-velocity construction of [F8] to the graph $\mathrm{\Psi }$ of step 1.1. At a source-boundary point, extend the smooth graph map to an open coordinate neighborhood; its derivative is invertible, so [F3] gives the required local inverse and extends the velocity there. The finite partition and bump construction in [F8] gives a smooth compactly supported field $\widehat{Y}(p,t)=(u_t(p),0)$ on $\mathrm{\partial }W\times \mathbb{R}$ with $u_t({\phi }_t(x))={\mathrm{\partial }}_t{\phi }_t(x)$ for every $x\in M$, including $\mathrm{\partial }M$.

Since the isotopy is stationary for $t$ near its endpoints, multiply this extension by a smooth temporal cutoff that is one wherever the prescribed velocity is nonzero and vanishes on smaller endpoint neighborhoods. This preserves its values on the graph and compact support and makes $u_t=0$ near $t=0,1$. Thus the boundary of the disk factor may move; no stationary spatial collar is required.

Writing $(u_t(p),0):=\widehat{Y}(p,t)$ defines a smooth time-dependent vector field $u_t$ on $\mathrm{\partial }W$ whose support over the compact interval $[0,1]$ is a compact subset of $\mathrm{\partial }W$. By [F4] it has a global evolution $H_t:={\mathrm{\Psi }}_{t,0}$, $t\in [0,1]$, with $H_0={\mathrm{id}}_{\mathrm{\partial }W}$; each $H_t$ is a diffeomorphism with inverse ${\mathrm{\Psi }}_{0,t}$, $H_t$ is the identity outside the compact support of the family, and $H_t=\mathrm{id}$ for $t\le \epsilon$ and $H_t=H_1$ for $t\ge 1-\epsilon$ because $u_t$ vanishes there.

Extend the diffeotopy over the interior by a collar deformation. By [F2] fix a collar $c:\mathrm{\partial }W\times [0,\eta )\to W$ and a smooth function $\chi :[0,\eta )\to [0,1]$ with $\chi \equiv 1$ on $[0,\eta \mathrm{/}4]$ and $\chi \equiv 0$ on $[\eta \mathrm{/}2,\eta )$. Define $K:W\to W$ by $K(c(p,s)):=c(H_{\chi (s)}(p),s)$ on the collar image and $K:={\mathrm{id}}_W$ outside. The collar image $V:=c(\mathrm{\partial }W\times [0,\eta ))$ is open, $Z:=c(\mathrm{\partial }W\times [0,\eta \mathrm{/}2])$ is compact with $Z\subseteq V$, and on $V\setminus Z$ one has $\chi (s)=0$, so the collar formula is the identity there and agrees with the outside definition; hence $K$ is smooth. The same computation with $H_{\chi (s)}^{-1}={\mathrm{\Psi }}_{0,\chi (s)}$ in place of $H_{\chi (s)}$ gives a smooth inverse, so $K$ is a diffeomorphism of $W$ that is the identity outside the compact set $Z$ and satisfies $K{\mathrm{\mid }}_{\mathrm{\partial }W}=H_1$, since $\chi (0)=1$.

For every $x\in M$, including its boundary, step 2.1 gives $\frac{d}{dt}{\phi }_t(x)=u_t({\phi }_t(x))$. Uniqueness in [F4] therefore yields $H_t({\phi }_0(x))={\phi }_t(x)$ on the whole attaching region. With step 5.1, $K({\phi }_0(x))={\phi }_1(x)$.

Define $\mathrm{\Phi }:W{\cup }_{{\phi }_0}H\to W{\cup }_{{\phi }_1}H$ by $\mathrm{\Phi }{\mathrm{\mid }}_W:=K$ and $\mathrm{\Phi }{\mathrm{\mid }}_H:={\mathrm{id}}_H$. It is compatible with the two identifications: for $z\in M$ one has $\mathrm{\Phi }(z)=z$ and $\mathrm{\Phi }({\phi }_0(z))=K({\phi }_0(z))={\phi }_1(z)$, and in the target $z$ is identified with ${\phi }_1(z)$. In the seam charts given by the collar data of [F1] and the collar $c$, a source point $c({\phi }_0(z),s)$ with $s\le \eta \mathrm{/}4$ is glued to the handle point $(z,s)$ and is mapped to $c(H_{\chi (s)}({\phi }_0(z)),s)=c({\phi }_1(z),s)$, which is glued to $(z,s)$ in the target; thus $\mathrm{\Phi }$ reads as the identity on a neighbourhood of the seam. Hence $\mathrm{\Phi }$ is smooth with smooth inverse $K^{-1}\bigsqcup {\mathrm{id}}_H$, and it is the identity on $H$ and outside a collar of the region swept by the isotopy.

The attachments in the statement are formed with corners rounded. The map $\mathrm{\Phi }$ preserves the collar data of the corner and therefore carries a compatible rounding of the first presentation to a compatible rounding of the second; by [F5] the rounded attachments are diffeomorphic, and the resulting diffeomorphism is still the identity outside a collar of the swept attaching region in $W$, which is what the swept region corresponds to under the two gluings.

If further handles are attached to the outgoing boundary of the two presentations, then gluing the same handles along attaching data that correspond under $\mathrm{\Phi }$ gives diffeomorphic total manifolds: the map $\mathrm{\Phi }$ on the base together with the identity on the additional handles is compatible with the identifications, exactly as in step 7.1. In particular, attaching data carried into the swept region by the isotopy are transported by $\mathrm{\Phi }$, so the two total manifolds are diffeomorphic.

By [F2] choose a chart at $p$ for $S_1$ and let $f_1,\dots ,f_b$ be the last $b=m-\dim S_1$ coordinate functions; they are smooth, vanish on $S_1$ near $p$, and their differentials at $p$ are linearly independent and span the annihilator of $T_p{S}_1$. Similarly choose $g_1,\dots ,g_a$ from a slice chart of $S_2$, $a=m-\dim S_2$, spanning the annihilator of $T_p{S}_2$. Restricting to a common smaller neighbourhood of $p$, all these functions are defined there and still have the same properties at $p$.

Since $\dim S_1+\dim S_2=m$ and $T_p{S}_1+T_p{S}_2=T_{p}N$ by [F1], the sum is direct, $T_{p}N=T_p{S}_1\oplus T_p{S}_2$. Hence the annihilator of $T_p{S}_1$ and the annihilator of $T_p{S}_2$ meet only in $0$, and the $a+b=m$ covectors $d{g}_1(p),\dots ,d{g}_a(p),d{f}_1(p),\dots ,d{f}_b(p)$ are linearly independent: a linear relation splits into a combination of the $d{g}_j$ equal to minus a combination of the $d{f}_i$, which lies in the intersection of the two annihilators and hence vanishes, forcing all coefficients to vanish by independence in each family.

Within the slice chart of $S_1$ used in [F2], the functions $f_1,\dots ,f_b$ are $b$ of the coordinate functions, namely the last $b$, so their common zero locus inside that chart is exactly $S_1$ intersected with the chart domain; after restriction to the common smaller neighbourhood of step 1.1 this remains true there. The same holds for $g_1,\dots ,g_a$ and $S_2$.

Put $F:=(g_1,\dots ,g_a,f_1,\dots ,f_b)$ on the common neighbourhood of $p$, regarded as a smooth map into ${\mathbb{R}}^m$. By step 2.1 its differential at $p$ is an isomorphism, so by [F3] there is an open neighbourhood $U$ of $p$ such that $F{\mathrm{\mid }}_U$ is a diffeomorphism onto an open subset of ${\mathbb{R}}^m$; composing with a translation, $F{\mathrm{\mid }}_U$ is a chart $\varphi :(U,p)\to ({\mathbb{R}}^m,0)$ of the smooth manifold $N$ of [F4] at $p$.

In the chart $\varphi$ of step 3.1 the coordinates are the ordered functions $g_1,\dots ,g_a,f_1,\dots ,f_b$; by step 2.2, $\varphi (S_2\cap U)=\{x_1=\cdots =x_a=0\}\cap \varphi (U)$ and $\varphi (S_1\cap U)=\{x_{a+1}=\cdots =x_m=0\}\cap \varphi (U)$. Writing $a=\dim S_1$ and $b=\dim S_2$ and identifying the last $b$ coordinates with ${\mathbb{R}}^b$ and the first $a$ with ${\mathbb{R}}^a$ gives exactly $\varphi (S_1\cap U)={\mathbb{R}}^a\times \{0\}\cap \varphi (U)$ and $\varphi (S_2\cap U)=\{0\}\times {\mathbb{R}}^b\cap \varphi (U)$.

For the general clause use the sum map instead of defining functions. Identify a neighbourhood of $p$ with ${\mathbb{R}}^m$ by a chart [F4] carrying $p$ to $0$, and consider $H:S_1\times \cdots \times S_r\to {\mathbb{R}}^m$, $H(s_1,\dots ,s_r)=s_1+\cdots +s_r$. Its differential at $(p,\dots ,p)$ sends $(v_1,\dots ,v_r)$ to $v_1+\cdots +v_r$, which is invertible exactly because $T_{p}N=T_p{S}_1\oplus \cdots \oplus T_p{S}_r$; by [F3] $H$ restricts to a diffeomorphism from a neighbourhood of $(p,\dots ,p)$ onto an open neighbourhood $U$ of $p$, with smooth inverse $H^{-1}(q)=(a_1(q),\dots ,a_r(q))$, $a_j(q)\in S_j$. Shrink $U$ so that for each $i$, every $q\in S_i\cap U$ lies in the chosen factor neighbourhood and $(p,\dots ,q,\dots ,p)$ lies in the inverse-function domain. Since $H$ is injective there and $H(p,\dots ,q,\dots ,p)=q$, a point $q\in U$ lies in $S_i$ exactly when $a_j(q)=p$ for every $j\mathrm{\ne }i$: if $q\in S_i$ then $(p,\dots ,q,\dots ,p)$ and $(a_1(q),\dots ,a_r(q))$ are two preimages of $q$, hence equal, and conversely $a_j(q)=p$ for $j\mathrm{\ne }i$ gives $q=a_i(q)\in S_i$. Composing $H^{-1}$ with charts of the $S_j$ at $p$ given by [F2] produces a chart $\varphi =({\varphi }^{(1)},\dots ,{\varphi }^{(r)})$ of $N$ at $p$, ${\varphi }^{(j)}$ taking values in ${\mathbb{R}}^{\dim S_j}$, and the criterion just proved says $\varphi (S_i\cap U)=\{{\varphi }^{(j)}=0,j\mathrm{\ne }i\}\cap \varphi (U)$, a coordinate subspace.

For (ii), use the rounded product $D^k\times D^{n-k}$ as the standard $n$-disc and attach a second copy along $S^{k-1}\times D^{n-k}$. The first factors glue as two hemispheres of $S^k$, with a smooth seam in their collar coordinates. Taking the product with $D^{n-k}$ and rounding the remaining corners gives $S^k\times D^{n-k}$; the lower handle's belt sphere becomes $\{p\}\times S^{n-k-1}$, for a point $p$ in its core hemisphere. For $k=0$ the seam is empty and this says that adding a disjoint disc gives $S^0\times D^n$.

Put $p=n-k-1\ge 0$. Model the disk factor $D^{p+1}$ by $D^p\times [0,1]$, rounding its bottom and side corners, and retain the top face $D^p\times \{1\}$ as the attaching hemisphere. This is the usual disk with a corner introduced along the equator: in meridian coordinates a smooth monotone rounding identifies it with $D^{p+1}$, carrying the top face onto the upper hemisphere. Its disk parametrization is chosen to be ${\phi }_0$. The same product charts on the attaching seam are used on the handle side. Thus the rounded attachment in (i) is represented by the rounded product $((S^k\times [0,1]){\cup }_{S^k\times \{1\}}{D}^{k+1})\times D^p\mathrm{.}$ For $p=0$ the disk factor is simply an interval and the attaching hemisphere is one endpoint.

The first factor of step 1.2 is a disk with an extra boundary collar. Identify its cap with the unit disk in ${\mathbb{R}}^{k+1}$ and send $(u,t)\in S^k\times [0,1]$ to $(2-t)u$. The cap boundary and $t=1$ have the same radial collar coordinate, so this is a smooth identification, across the seam, with the disk of radius $2$. Consequently the product in step 1.2 is a product of disks, whose compatible rounding is a standard $n$-disc (round the convex product boundary and use its smooth radial parametrization). The rounding-independence supplier makes this conclusion independent of the compatible profiles. This proves (i), including $k=0$ and $k=n-1$.

In the two-hemisphere identification of step 1.1 choose the disk-factor hemisphere and its framing exactly as in step 1.2. The standard $(k+1)$-handle is then attached by $1\times {\phi }_0$, so step 2.1 returns an $n$-disc. These are the required compatible identifications for the consecutive standard pair; Countable Choice is inherited only from the attachment and rounding conventions.

Write $L={\mathrm{\partial }}_{+}{W}_0$, $q=n-k$, and $R=D^k\times S^{q-1}$ for the lower handle's outgoing region in $M$. A small disk-fibre neighbourhood of $B=\{0\}\times S^{q-1}$ meets $A$ only near the unique crossing: the compact part of $A$ outside a crossing chart is disjoint from the compact sphere $B$. In that chart [F2] makes the crossing a single transverse disk. For $k>0$, a radial diffeotopy in the fibres of $R$, extended across its boundary into a collar in the old-boundary summand, carries the portion away from this small neighbourhood out of $R$. Concretely choose an increasing radial map with $h(0)=0$, taking the chosen small radius to radius $1$, and equal to the identity beyond radius $1+\delta$ in the extended collar; interpolate increasing maps to obtain the diffeotopy. It fixes $B$ and the region beyond that collar, but it does move points in the old-boundary collar. For $k=0$ the single crossing already means one upper attaching point on the new sphere and one in $L$.

Normalize the crossing disk and its normal disk coordinates. In the product chart the disk is a graph over the transverse $D^k$ factor; interpolate its graph to the constant graph, on a smaller disk and with a cutoff on a surrounding annulus. After shrinking, the graph projections remain invertible and the adjustment is an isotopy of embeddings. Fibre contraction makes the upper attaching neighbourhood as small as needed. The corresponding normal-coordinate adjustment is obtained by rescaling its transition germ: for a transition fixing the zero section, replace the fibre variable by $tz$ and divide the output fibre variable by $t$. This extends smoothly at $t=0$ to the induced normal linear map. Choose the normal coordinates with that same linear map; after shrinking the compact disk, all these maps are embeddings by the inverse-function theorem. [F5] extends these full-dimensional attaching-region isotopies and transports the later data. Expanding the small transverse disk by the radial map of step 1.1 now makes one upper hemisphere the fibre $D^k\times \{X\}$, with its framed neighbourhood $D^k\times D^{q-1}$, where $D^{q-1}$ is a small hemisphere patch at $X\in S^{q-1}$. The other hemisphere and its neighbourhood lie in $L$ outside the lower attaching region. This is the crossing normalization in Wall's cancellation proof, printed pp. 146–147, and Lück's Cancellation Lemma, printed p. 6.

The lower hemisphere gives an embedded $D^k\times D^{q-1}$ in $L$, whose boundary is the part $\phi (S^{k-1}\times D^{q-1})$ of the lower attaching data. Shrink and move the lower normal disk inside its extended disk coordinates to a small disk near $X$: the maps $y\mapsto c_t+{\lambda }_{t}y$, with ${\lambda }_t>0$, give the isotopy, using the stipulated extension beyond the disk boundary at the end. Transport the upper data by [F5]. The resulting lower attaching region lies in a collar of the boundary of that embedded lower-hemisphere disk. Their union, with corners rounded, is therefore a single disk $E\subset L$. In coordinates it is $(S^{k-1}\times D^q)\cup (D^k\times D^{q-1})$ glued along the chosen normal hemisphere. This is the collar-and-cap product model of the standard-pair lemma with dimensions $n-1$ and $k-1$; for $k=0$ only the $D^0\times D^{n-1}$ piece remains.

In these coordinates the upper hemisphere is the fibre patch on the outgoing lower handle, and its complementary hemisphere is the disk patch in $E$. The framing is carried in the same normal coordinates throughout, so the collar of $E$ and the two handle bodies have exactly the standard complementary attachment. The argument also covers $q=1$: the normal hemisphere is $D^0$, the upper attaching sphere is a boundary component, and precisely one of the two belt points lies on it. All changes are isotopies of the framed attaching regions, supported in the crossing charts, radial collar and swept disk neighbourhoods; [F5] converts them into comparisons of the attached manifolds.

By [F2] the affected region of the outgoing boundary ${\mathrm{\partial }}_{1}W$ is an embedded closed disc $E$ to which the pair is attached in the standard way, and the attaching data are isotopic to the given ones through embeddings fixed outside a compact neighbourhood of the two attaching regions. By [F3] the total manifolds obtained from the given data and from the standard data are diffeomorphic relative to ${\mathrm{\partial }}_{0}W$, with a diffeomorphism supported near the swept region; it therefore suffices to prove the claim for the standard pair attached to $E$.

Localize the normalization before composing diffeomorphisms. In the construction of [F2], put $q=n-k$. The radial expulsion and crossing-chart adjustment act, on the old-boundary side, only in an extended collar of the original lower attaching region; their other support is in the lower handle. After that expulsion the upper sphere's complementary hemisphere, with its normal coordinates, is an embedded cap $D^k\times D^{q-1}$ in ${\mathrm{\partial }}_{1}W$, outside the lower attaching region and joined to it along $S^{k-1}\times D^{q-1}$. The old-boundary part of the upper attaching region before expulsion is contained in this cap neighbourhood and the seam collar: the radial map is the identity off that collar. Retain the full lower attaching tube, rather than its subsequently shrunken version. Its union with the cap neighbourhood is the rounded product $(S^{k-1}\times D^q){\cup }_{S^{k-1}\times D^{q-1}}(D^k\times D^{q-1})$, a closed $(n-1)$-disk by F4 with dimensions $n-1,k-1$. The attaching patch can be taken smaller than the displayed hemisphere; expanding it in its disk coordinates gives the same product model. Enlarge this disk slightly by its boundary collar to a disk $D$, containing the seam collars and the compact old-boundary sweeps in its interior. The remaining graph, normal-radius and lower-normal-disk adjustments of [F2] may now be made inside this disk neighbourhood and the lower handle: the normal-radius contractions and translations stay in the retained tubes, and the cap coordinates are fixed away from their seam collar. Thus all their old-boundary supports lie in $D$. When $k=0$, the cap is simply the disk about the upper handle's old-boundary foot, and the other support is in the new $0$-handle, giving the same conclusion. Choose the extensions in [F3] in the corresponding collar of $D$ and the handles. The final model disk $E$ lies in this enclosing $D$. This proves the required support containment; it does not shrink the support to the possibly smaller final disk $E$.

For the standard pair, a small $n$-disc with outgoing face $E$, together with the two handle bodies is an $n$-disc attached to $W$ along $E$: the standard complementary pair region fills a ball by [F4], and its attaching boundary is exactly $E$. Hence $W{\cup }_{\phi }{h}^k{\cup }_{\psi }{h}^{k+1}$ is diffeomorphic to the boundary connected sum $W\mathrm{♮}{D}^n$ formed along $E$.

Apply [F5] to the connected component of $W$ containing $E$, with $D:=E$, and leave every other component fixed. This identifies $W\mathrm{♮}{D}^n$ with $W$ by a diffeomorphism equal to the identity outside a collar of $E$. The diffeomorphism may be taken to be the identity on ${\mathrm{\partial }}_{0}W$, because the connected sum is formed in a collar of ${\mathrm{\partial }}_{1}W$ disjoint from a collar of ${\mathrm{\partial }}_{0}W$, and both collars are part of the given collar data.

By step 2.1 the normalization comparison is the identity off a collar of $D$ and the two handle bodies. The ball replacement and absorption in steps 3.1–4.1 take place in a collar of $E\subset \mathrm{int}D$ and the handles. Choose that collar inside the same collar neighbourhood of $D$; these maps then fix its complement and carry that neighbourhood onto the corresponding neighbourhood, so their composition has the asserted support. The affected boundary disk in the support clause is this enclosing disk $D$. Every later attaching embedding is carried by the composed boundary diffeomorphism; attaching those handles by the transported embeddings extends the comparison relative to ${\mathrm{\partial }}_{0}W$.

Consequently a geometrically cancelling consecutive pair may be deleted from, or added to, any handle presentation of the same manifold, relative to the incoming boundary. The argument covers the endpoints $k=0$ and $k=n-1$, where the intersection is read by the endpoint conventions of [F1] and the standard model is the corresponding endpoint case of [F4].

Choose an embedded closed disc $E\subseteq V$ with $x\in \mathrm{int}E$ and attach an $n$-disc $D^n$ to $W$ along $E$; by [F1], applied to $N=W$ and $D=E$, the boundary connected sum $W\mathrm{♮}{D}^n$ is diffeomorphic to $W$ relative to ${\mathrm{\partial }}_{0}W$, by a diffeomorphism equal to the identity outside a collar of $E$.

By [F2] the standard $n$-disc admits the decomposition $D^n=D^n\cup h^k\cup h^{k+1}$ with the two handles attached in the standard complementary way along a disc of its boundary: the standard $k$-handle is attached along the equatorial embedding and the standard $(k+1)$-handle fills the resulting $S^k\times D^{n-k}$ back to a disc. Hence the attached disc in step 1.1 can be decomposed into the two standard handles supported over $E$ (the upper region lies in the boundary after the lower attachment).

Transporting this decomposition along the absorption diffeomorphism of step 1.1 and adjusting the attaching data by an isotopy inside $V$ using [F3], we obtain attaching embeddings of a $k$-handle $h^k$ and, after it, a $(k+1)$-handle $h^{k+1}$ whose lower attaching region lies in $V$ and whose upper attaching region lies in the boundary region obtained from $V$ after the lower attachment, forming the standard complementary pair on an embedded disc $E\subseteq V$, with $W\cup h^k\cup h^{k+1}$ diffeomorphic to $W$ relative to ${\mathrm{\partial }}_{0}W$.