Smooth Surgery Traces and Handle Trading

The product $V\times I$ is compact: for an open cover and each time, compactness of $V$ supplies finitely many product neighbourhoods covering that time slice; intersect their time intervals to obtain a neighbourhood of that time, and compactness of $I$ supplies finitely many such neighbourhoods. Thus a finite subcover exists. The image $K=F(V\times I)$ is compact, being the continuous image of the compact space $V\times I$, and $K\subseteq W$; every point of $K$ has by [F6] an open neighbourhood with compact closure contained in $W$, finitely many of these cover $K$, and their union $V^{\mathrm{\prime }}$ is an open neighbourhood of $K$ with $\overline{V^{\mathrm{\prime }}}$ compact and $\overline{V^{\mathrm{\prime }}}\subseteq W$.

Extend $F$ in the time direction by $\tilde{F}(t,x)=F(x,0)$ for $t\le 0$, $\tilde{F}(t,x)=F(x,t)$ for $0\le t\le 1$ and $\tilde{F}(t,x)=F(x,1)$ for $t\ge 1$: the prescriptions agree on the overlaps because $F$ is constant for $t\le \epsilon$ and for $t\ge 1-\epsilon$, so $\tilde{F}:\mathbb{R}\times V\to N$ is smooth and each ${\tilde{F}}_t$ is a smooth embedding; let $\mathrm{\Theta }:\mathbb{R}\times V\to \mathbb{R}\times N$, $\mathrm{\Theta }(t,x)=(t,\tilde{F}(t,x))$, be the graph map.

The graph map is an injective immersion with invertible differential at every point: injectivity is immediate from the first coordinate, and at $(t_0,x_0)$ a boundary chart of $V$ at $x_0$ and a chart of $N$ at $\tilde{F}(t_0,x_0)$ present the coordinate representative of $\mathrm{\Theta }$ as a map smooth on a relatively open subset of a half-space in the sense of [F2], hence as the restriction of a smooth map $\mathrm{\Phi }$ defined near $(t_0,u(x_0))$ in an open set; the differential of ${\tilde{F}}_{t_0}$ at $x_0$ is invertible by [F1] because ${\tilde{F}}_{t_0}$ is an embedding between $n$-manifolds, so the differential of $\mathrm{\Phi }$ there is invertible and [F3] restricts $\mathrm{\Phi }$ to a local diffeomorphism, exhibiting $\mathrm{\Theta }$ locally as the restriction of an ambient diffeomorphism to the source half-space. At a boundary point its image is a half-space neighbourhood, not an ambient open set; in the interior it is open.

The graph map is proper: for a compact $L\subseteq \mathbb{R}\times N$ the time projection ${\pi }_1(L)$ is compact, ${\mathrm{\Theta }}^{-1}(L)$ is closed in the compact set ${\pi }_1(L)\times V$ by continuity and closedness of $L$, hence ${\mathrm{\Theta }}^{-1}(L)$ is compact; a proper continuous map into this locally compact Hausdorff target is closed: for a closed source subset $A$ and $y$ outside its image, choose a compact target neighbourhood $L$ of $y$; the image of $A\cap {\mathrm{\Theta }}^{-1}(L)$ is compact and hence closed in the Hausdorff target, and deleting it from the interior of $L$ gives a neighbourhood of $y$ missing the image of $A$. Therefore $\mathrm{\Theta }$ is closed, its image $S:=\mathrm{\Theta }(\mathbb{R}\times V)$ is closed in $\mathbb{R}\times N$ and $\mathrm{\Theta }$ is a homeomorphism onto $S$ whose inverse is smooth by step 2.1, so $S$ is an embedded smooth submanifold with boundary of $\mathbb{R}\times N$ in the sense of [F11] with $\mathrm{\partial }S=\mathrm{\Theta }(\mathbb{R}\times \mathrm{\partial }V)$.

Define the horizontal velocity along the graph by placing $Y(\mathrm{\Theta }(t,x)):=(0,{\mathrm{\partial }}_t\tilde{F}(t,x))$ in $\{0\}\oplus T_{\tilde{F}(t,x)}N\subseteq T_{(t,\tilde{F}(t,x))}(\mathbb{R}\times N)$: the assignment is well defined because $\mathrm{\Theta }$ is injective and smooth because ${\mathrm{\Theta }}^{-1}$ is smooth by step 3.1, it is a horizontal smooth field along $S$ in the sense of [F10], and $Y=0$ at every point of $S$ whose first coordinate lies outside $[0,1]$, because $\tilde{F}$ is constant in $t$ there.

At every point $q\in S$ the field $Y$ extends over an open neighbourhood in $\mathbb{R}\times N$ to a smooth horizontal field: choose $(t_0,x_0)={\mathrm{\Theta }}^{-1}(q)$ and, by step 2.1, an open neighbourhood $U$ of $(t_0,x_0)$ in $\mathbb{R}\times V$ mapped diffeomorphically onto ${\mathrm{\Omega }}_0:=\mathrm{\Theta }(U)$; shrink $U$ so that $\tilde{F}(t,x)\in V^{\mathrm{\prime }}$ for all $(t,x)\in U$, possible by continuity because $\tilde{F}(t_0,x_0)\in K\subseteq V^{\mathrm{\prime }}$. If $x_0\in \mathrm{int}V$ then ${\mathrm{\Omega }}_0$ is open in $\mathbb{R}\times N$ and ${\tilde{Y}}_q(\mathrm{\Theta }(t,x)):=(0,{\mathrm{\partial }}_t\tilde{F}(t,x))$ defines on it a smooth horizontal field restricting to $Y$ on $S\cap {\mathrm{\Omega }}_0$. If $x_0\in \mathrm{\partial }V$ then ${\mathrm{\Omega }}_0$ is only a half-space neighbourhood of $q$, but in boundary charts of $V$ the horizontal components of $Y$ are smooth functions on that half-space model, so by the local-extension convention of [F2] they extend smoothly to an open neighbourhood of $q$ in $\mathbb{R}\times N$ while the zero first component extends by zero, giving a smooth horizontal field on a neighbourhood $\mathrm{\Omega }$ that restricts to $Y$ on $S\cap \mathrm{\Omega }$; in both cases $\mathrm{\Omega }$ may be shrunk to lie in $\mathbb{R}\times V^{\mathrm{\prime }}$.

The compact set $S_0:=\mathrm{\Theta }([0,1]\times V)\subseteq S$ is covered by finitely many neighbourhoods ${\mathrm{\Omega }}_{q_1},\dots ,{\mathrm{\Omega }}_{q_m}$ from step 5.1, each contained in $\mathbb{R}\times V^{\mathrm{\prime }}$; let $({\psi }_0,{\psi }_1,\dots ,{\psi }_m)$ be a smooth partition of unity on $\mathbb{R}\times N$ subordinate to the open cover $\{\mathbb{R}\times N\setminus S_0,{\mathrm{\Omega }}_{q_1},\dots ,{\mathrm{\Omega }}_{q_m}\}$, which exists by [F4], and define $\tilde{Y}:={\sum }_{i=1}^m{\psi }_i{\tilde{Y}}_{q_i}$ with each term extended by zero outside ${\mathrm{\Omega }}_{q_i}$; the sum is smooth because $\mathrm{supp}{\psi }_i\subseteq {\mathrm{\Omega }}_{q_i}$, it takes values in the horizontal subbundle and so is a time-dependent vector field $\tilde{Y}(t,y)=X(t,y)\in T_{y}N$ on $N$ over $\mathbb{R}$ in the sense of [F10], and for $q\in S$ one has $\tilde{Y}(q)={\sum }_i{\psi }_i(q)Y(q)=(1-{\psi }_0(q))Y(q)=Y(q)$, because ${\psi }_0(q)\mathrm{\ne }0$ forces $q\mathrm{\notin }{S}_0$ and then $Y(q)=0$ by step 4.1; finally $\mathrm{supp}\tilde{Y}\subseteq \mathbb{R}\times \overline{V^{\mathrm{\prime }}}$ because each ${\mathrm{\Omega }}_{q_i}\subseteq \mathbb{R}\times V^{\mathrm{\prime }}$.

By [F5] choose a smooth function $\beta :\mathbb{R}\to [0,1]$ with $\beta =1$ on $[\epsilon ,1-\epsilon ]$ and $\beta =0$ outside $(\epsilon \mathrm{/}2,1-\epsilon \mathrm{/}2)$, and put $X_t^{\mathrm{\prime }}:=\beta (t)X_t$ for $t\in [0,1]$; then ${\bigcup }_{t\in [0,1]}\mathrm{supp}{X}_t^{\mathrm{\prime }}$ is contained in the compact subset $\overline{V^{\mathrm{\prime }}}\subseteq N$, so [F9] provides a global evolution operator ${\mathrm{\Psi }}_{t,s}:N\to N$ for $s,t\in [0,1]$.

The isotopy identity: fix $x\in V$ and put $\gamma (t):=F_t(x)$ for $t\in [0,1]$; then $\gamma (0)=F_0(x)$ and ${\gamma }^{\mathrm{\prime }}(t)={\mathrm{\partial }}_{t}F(t,x)$ equals $X_t^{\mathrm{\prime }}(\gamma (t))$ for every $t$, because on $[\epsilon ,1-\epsilon ]$ one has $\beta =1$ and $X_t(\gamma (t))={\mathrm{\partial }}_t\tilde{F}(t,x)={\mathrm{\partial }}_{t}F(t,x)$ by step 6.1, while off $[\epsilon ,1-\epsilon ]$ the derivative ${\mathrm{\partial }}_{t}F(t,x)$ vanishes and is multiplied by $\beta (t)\in [0,1]$; the curve $t\mapsto {\mathrm{\Psi }}_{t,0}(F_0(x))$ solves the same equation with the same initial value by the defining property of the evolution operator in [F10], both curves are defined on all of $[0,1]$, and the local uniqueness in [F7] makes them agree near every point of the connected interval, so $H_t\circ F_0=F_t$ for $H_t:={\mathrm{\Psi }}_{t,0}$.

The remaining properties: $H_0={\mathrm{\Psi }}_{0,0}={\mathrm{id}}_N$ and each $H_t$ is a diffeomorphism with inverse ${\mathrm{\Psi }}_{0,t}$, since the cocycle law of [F8] gives ${\mathrm{\Psi }}_{0,t}\circ {\mathrm{\Psi }}_{t,0}={\mathrm{\Psi }}_{0,0}={\mathrm{id}}_N$ and ${\mathrm{\Psi }}_{t,0}\circ {\mathrm{\Psi }}_{0,t}={\mathrm{\Psi }}_{t,t}={\mathrm{id}}_N$; the map $(t,y)\mapsto H_t(y)$ is smooth because near every $(t_0,y_0)$ it agrees by [F7] with the local smooth evolution map of $X^{\mathrm{\prime }}$ through the point $H_{t_0}(y_0)$ at time $t_0$; if $y\mathrm{\notin }\overline{V^{\mathrm{\prime }}}$ then $X_t^{\mathrm{\prime }}(y)=0$ for all $t$ by step 6.1, so the constant curve at $y$ solves the equation of $X^{\mathrm{\prime }}$, [F7] gives $H_t(y)=y$, and hence $H_t={\mathrm{id}}_N$ outside $W$ because $\overline{V^{\mathrm{\prime }}}\subseteq W$; finally $X_t^{\mathrm{\prime }}=0$ for $t\le \epsilon \mathrm{/}2$ and for $t\ge 1-\epsilon \mathrm{/}2$, so the cocycle law gives $H_t={\mathrm{id}}_N$ for $t\le \epsilon \mathrm{/}2$ and $H_t={\mathrm{\Psi }}_{t,1-\epsilon \mathrm{/}2}\circ H_{1-\epsilon \mathrm{/}2}=H_{1-\epsilon \mathrm{/}2}=H_1$ for $t\ge 1-\epsilon \mathrm{/}2$.

In the product normal form of $\phi$, a point of $\phi (S^p\times \mathrm{\partial }{D}^q)$ has a chart in which $M$ is an open subset of ${\mathbb{R}}^m$ and the removed piece is the open half-space-product ${\mathbb{R}}^p\times \mathrm{int}{D}^q$; the complement there is locally a closed half-space, so $N$ is a smooth manifold with boundary $\mathrm{\partial }M\bigsqcup \phi (S^p\times S^{q-1})$, and the new boundary component is a closed embedded $(m-1)$-manifold.

Put $B=S^p\times S^{q-1}$ and $P^{\mathrm{\prime }}=D^{p+1}\times S^{q-1}$. The latter has boundary $\mathrm{\partial }{P}^{\mathrm{\prime }}=B$, and $\phi {\mathrm{\mid }}_B$ is a diffeomorphism from $B$ onto the new boundary part of $N$. This uses the supplied embedding on its boundary, rather than identifying that restriction with its derivative framing along the core.

Choose collars $c_N:B\times [0,\epsilon )\to N$ and $c_{P^{\mathrm{\prime }}}:B\times [0,\epsilon )\to P^{\mathrm{\prime }}$, with $c_N(b,0)=\phi (b)$ and $c_{P^{\mathrm{\prime }}}(b,0)=b$. On the quotient $N{\cup }_{\phi {\mathrm{\mid }}_B}{P}^{\mathrm{\prime }}$, define $C(b,s)=[c_N(b,-s)]$ for $s\le 0$ and $C(b,s)=[c_{P^{\mathrm{\prime }}}(b,s)]$ for $s\ge 0$. This is a homeomorphism onto an open seam neighbourhood: each half is a collar homeomorphism, and their relatively open half-images together are saturated in the disjoint union. For each boundary chart $y$ on $B$, use $(y(b),s)$ as a seam chart. Two such charts have transition $(y_2\circ y_1^{-1},s)$; an overlap with a chart away from the seam lies in $s<0$ or $s>0$, where it is a smooth collar-coordinate change. These charts and the original charts away from $B$ generate a smooth atlas. Every seam point is interior, and the remaining boundary is exactly $\mathrm{\partial }M$. A compatible seam smoothing is this signed collar presentation after straightening its collar coordinate.

The quotient is Hausdorff and second countable. Its quotient map from $N\bigsqcup P^{\mathrm{\prime }}$ is closed: saturating a closed set adds only images of its intersections with the compact seam, which are compact and closed in the opposite Hausdorff piece. Distinct quotient points have disjoint finite fibres; finite Hausdorff separation gives disjoint open sets about those fibres, and the complements of the quotient images of their closed complements give disjoint quotient neighbourhoods. The open cover by the two pieces minus their seam and the signed collar has a countable base, by the countable bases of the pieces and of $B\times (-\epsilon ,\epsilon )$. Thus the atlas defines a smooth $m$-manifold. If $M$ is closed, $N$ is compact, so the quotient is compact with empty boundary. This proves (i).

Fix an open seam neighbourhood $U$ in the quotient. On either piece $Q=N$ or $Q=P^{\mathrm{\prime }}$, write $c_0,c_1$ for the old and new collars of its compact boundary part $B$. Extend $c_0$ to the other boundary parts using [F3], and form the smooth boundaryless double $DQ$ using [F5]. Near $B$ in its positive half put $X_i=(c_i{)}_{\ast }{\mathrm{\partial }}_t$. Extend their coordinate components locally across $B$ and combine them by [F9]; this retains the original fields on a smaller positive-side neighbourhood. In the signed $c_0$ coordinate $r$, both $dr(X_i)>0$ along $B$, and hence on a smaller neighbourhood. Choose a smooth $\theta :[0,1]\to [0,1]$ equal to $0$ near $0$ and $1$ near $1$, and flow $X_s=(1-\theta (s))X_0+\theta (s)X_1$ from $b\in B$ for time $t\ge 0$, writing the result as $C_s(b,t)$. Local flow existence, uniqueness and smooth parameter dependence give a jointly smooth family. Its differential in $(b,t)$ at $t=0$ is $(v,a)\mapsto v+a{X}_s(b)$, an isomorphism. Local inverses exist by [F6]; uniqueness and strict increase of $r$ prevent two such trajectories from meeting with different boundary initial points or different flow times. Compactness of $[0,1]\times B$ therefore permits one $\delta >0$ for which all $C_s:B\times [0,\delta ]\to DQ$ are embeddings, constant in $s$ near its endpoints, with $C_0=c_0$ and $C_1=c_1$ on this band. This is the local collar-family construction of the double theorem's proof, rather than an additional assertion of its statement.

Choose an open neighbourhood $W$ of $B$ in $DQ$ disjoint from its other seam parts, with its intersection with the positive half contained in the inverse image of $U$ in $Q$. Shrink $\delta$ so the whole compact swept collar band of step 4.1 lies in $W$; this is possible because $C_s(b,0)=b$ uniformly on the compact set $[0,1]\times B$. Apply [F8] to the compact smooth $m$-manifold with boundary $V=B\times [0,\delta ]$, the boundaryless $m$-manifold $DQ$, and the isotopy $F_s(b,t)=C_s(b,t)$. It gives an ambient isotopy $A_s$ equal to the identity outside $W$ and satisfying $A_s\circ c_0=C_s$ on the band. Every $A_s$ fixes $B$ pointwise, and fixes all other seam parts because they lie outside $W$. A point off the seam cannot cross it during this isotopy: bijectivity and pointwise seam fixing imply $A_s^{-1}(\mathrm{\partial }Q)=\mathrm{\partial }Q$. Thus $A_s$ preserves the positive half. Its time-one restriction $H_Q$ is a boundary-fixing diffeomorphism of $Q$, equal to the identity outside the inverse image of $U$, and $H_Q(c_0(b,t))=c_1(b,t)$ near $B$.

The maps $H_N,H_{P^{\mathrm{\prime }}}$ descend to a bijection of the quotients because they fix the identified boundary points. In the old source and new target seam coordinates it is $(b,s)\mapsto (b,s)$; away from the seam it and its inverse are the smooth maps on the pieces. Consequently it is a diffeomorphism equal to the identity outside $U$. A compatible seam smoothing is straightened into its signed collar presentation as in step 2.1, so the same comparison applies. Since $U$ was arbitrary, this proves (ii).

Finally let ${\phi }_t$ be a smooth isotopy of framed embeddings, constant near the ends, with images in the interior of $M$, and put $F(x,y,t):={\phi }_t(x,y)$ on the compact manifold with boundary $S^p\times D^q$ with values in the boundaryless manifold $N:=\mathrm{int}M$; choose an open neighbourhood $W\subseteq \mathrm{int}M$ of the swept image with $\bar{W}$ compact in $\mathrm{int}M$. Apply [F8] with $V=S^p\times D^q$ and $N=\mathrm{int}M$: there is a smooth $H:\mathrm{int}M\times I\to \mathrm{int}M$ with $H_t\circ {\phi }_0={\phi }_t$ for every $t$, every $H_t$ a diffeomorphism, and $H_t$ the identity outside $W$. Since $H_1$ is the identity outside the compact set $\bar{W}\subseteq \mathrm{int}M$, it extends by the identity across $\mathrm{\partial }M$ to a diffeomorphism $\widehat{H}:M\to M$ agreeing with ${\phi }_1\circ {\phi }_0^{-1}$ on ${\phi }_0(S^p\times D^q)$; hence $\widehat{H}$ carries the complement of ${\phi }_0(S^p\times \mathrm{int}{D}^q)$ onto the complement of ${\phi }_1(S^p\times \mathrm{int}{D}^q)$ and satisfies $\widehat{H}\circ {\phi }_0={\phi }_1$ on the whole product, in particular on the common boundary sphere, so together with the identity on the glued piece $D^{p+1}\times S^{q-1}$ it descends, using collars on the second complement transported by $\widehat{H}$, to the required diffeomorphism $M_{{\phi }_0}\to M_{{\phi }_1}$ supported near the swept region. Other collar choices are compared by (ii); no uniqueness of the resulting diffeomorphism is asserted. This proves (iii), and with it the concluding isotopy-invariance of the construction.

For the product $D^k\times D^{n-k}$ the boundary is the union of the two products with the boundary of one factor, $\mathrm{\partial }(D^k\times D^{n-k})=\mathrm{\partial }{D}^k\times D^{n-k}\cup D^k\times \mathrm{\partial }{D}^{n-k}$, overlapping exactly in $\mathrm{\partial }{D}^k\times \mathrm{\partial }{D}^{n-k}$; writing $\mathrm{\partial }{D}^k=S^{k-1}$ and $\mathrm{\partial }{D}^{n-k}=S^{n-k-1}$ gives claim (i). The degenerate cases are included: for $k=0$ the first term is $S^{-1}\times D^n=\mathrm{\varnothing }$ and the second is $D^0\times S^{n-1}=S^{n-1}$.

The glued manifold $N^{\mathrm{\prime }}$ is covered by the interior of $N\setminus \psi (S^{k-1}\times \mathrm{int}{D}^{n-k})$, the interior of the handle, and a collar neighbourhood of the seam supplied by [F4], in which the two pieces are presented as half-spaces meeting along the seam; the seam has the product model recorded in [F5], so the union is a smooth manifold with boundary.

A point of $N^{\mathrm{\prime }}$ is a boundary point exactly when it lies in $\mathrm{\partial }N$ outside the open attaching region or in the outgoing region $D^k\times S^{n-k-1}$ of the handle: points of the attaching region and of the seam that lie over its interior are interior points of $N^{\mathrm{\prime }}$ by the collar model of step 1.2, and the remaining boundary points of the handle are precisely its outgoing region by claim (i). The two parts meet exactly along $\psi (S^{k-1}\times S^{n-k-1})$, where $\psi$ and the boundary identification of the handle agree. This proves claim (ii); the rounding enters only through the smooth structure of the seam, and changing it changes the result at most by a diffeomorphism equal to the identity outside the collar by [F3].

The belt sphere $\{0\}\times S^{n-k-1}$ lies in the outgoing region $D^k\times S^{n-k-1}\subseteq \mathrm{\partial }{N}^{\mathrm{\prime }}$ and is closed there because $S^{n-k-1}$ is closed in $D^k\times S^{n-k-1}$. Near a point $(0,y)$ the outgoing region is an open subset of $D^k\times S^{n-k-1}$ with the product smooth structure, and the tangent directions of $\{0\}\times S^{n-k-1}$ are the $S^{n-k-1}$-directions, so the complementary normal directions inside $\mathrm{\partial }{N}^{\mathrm{\prime }}$ are the $D^k$-factor directions; the product trivialization identifies this normal bundle with the trivial bundle of rank $k$. This is claim (iii), and it makes the belt sphere a closed embedded submanifold of $\mathrm{\partial }{N}^{\mathrm{\prime }}$ in the sense of Embedded smooth submanifolds with boundary.