Sublevel Deformation and the Handle Attachment Theorem

Use the closed-band convention and choose a metric. If $K=\mathrm{\varnothing }$, the zero field suffices and all trajectory assertions are vacuous. Otherwise the metric exists under the stated choice axiom.

The open set $\{df\mathrm{\ne }0\}$ contains $K$. Cover $K$ by finitely many coordinate neighborhoods with compact closures in this open set; their union $W$ is relatively compact. Choose $\rho =1$ near $K$ with support in $W$.

On $W$ set $Y=\rho \mathrm{grad}f\mathrm{/}\mathrm{\parallel }\mathrm{grad}f{\mathrm{\parallel }}^2$, and set it to zero outside $W$. The support condition makes this smooth with compact support. Since $df(\mathrm{grad}f)=\mathrm{\parallel }\mathrm{grad}f{\mathrm{\parallel }}^2$, $df(Y)=1$ near $K$.

The field is complete. On every trajectory segment contained in $K$, differentiation gives $d(f\circ {\mathrm{\Phi }}_t)\mathrm{/}dt=1$. Starting at an endpoint the same identity holds on its open neighborhood, so the trajectory enters the band in the required time direction. A first exit before the claimed level would have value strictly between $a$ and $b$, contradicting continuity. Integrating gives the identity through both endpoints; strict unit speed gives the asserted hitting time.

The controlled-time lemma defines $T$ on the entire closed product and gives $f(T(x,t))=t$. The inverse candidate is $S(y)=({\mathrm{\Phi }}_{a-f(y)}(y),f(y))$, whose first coordinate lies in $M_a$.

Uniqueness of flow gives ${\mathrm{\Phi }}_s{\mathrm{\Phi }}_t={\mathrm{\Phi }}_{s+t}$ wherever defined: both sides are integral curves with the same initial point. Consequently $ST$ and $TS$ are identities. Smooth dependence on time and initial point makes both maps smooth, including at endpoints by local extension. Regular level coordinates give the usual boundary structures. If one fiber is empty, the inverse formula forces $K$ empty; the empty map is the required diffeomorphism.

Take the complete normalized cutoff field and put $T=b-a$. Its time-$T$ map is an ambient diffeomorphism with inverse time $-T$, by uniqueness and smoothness of the flow. It maps $M_a$ onto $M_b$, also as seen in the product description.

If a point initially below $a$ first reaches $a$ at time $s\ge 0$, it can reach $b$ only at time $s+T$ by the unit-speed identity. Thus its value at time $T$ cannot exceed $b$. A point that never reaches $a$ stays below $a$. The reversed argument starting below $b$ proves ${\mathrm{\Phi }}_{-T}(M^b)\subseteq M^a$. These two inclusions prove ${\mathrm{\Phi }}_T(M^a)=M^b$. Restrict the ambient diffeomorphism and its inverse. If the band is empty, the two sets coincide and the zero field is sufficient.

For $f(x)\le a$ the time parameter is zero, so $H(s,x)=x$. For $a\le f(x)\le b$ the controlled-time identity gives $f(H(s,x))=(1-s)f(x)+sa\in [a,b]$. Thus $H$ takes values in $M^b$.

The maximum function and the complete flow are continuous, so the formula is continuous even at $f=a$. At $s=0$ it is the identity; at $s=1$ its image lies in $M^a$ and it fixes that set at every time. This proves the strong retraction, including empty sets. No smoothness across $f=a$ is asserted.

In the prescribed corner chart write the profiles as $z=g_0(w)$ and $z=g_1(w)$ along their common transverse direction. They agree outside a compact interval. The graphs $g_t=(1-t)g_0+t{g}_1$ are smooth embedded profiles with the same fixed ends. Use these graphs over the compact corner locus.

Choose a smooth cutoff $\chi$ supported in the collar and equal to one near the compact union of these graphs where $g_1-g_0\mathrm{\ne }0$. The time-dependent field $V_t=\chi (w,z)(g_1(w)-g_0(w)){\mathrm{\partial }}_z$ is smooth. Along the moving graph it has exactly its velocity.

Apply the flow theorem to ${\mathrm{\partial }}_t+V_t$ on an open time interval times the doubled collar. Its solutions exist for $0\le t\le 1$: spatial motion is in a fixed compact set, and any finite endpoint is extendible in a coordinate neighborhood. Uniqueness gives inverse evolution and carries the initial graph to the final graph, preserving the specified side. Extend by the identity. If the corner locus is empty, the identity is already the answer.

Use the closed-band convention. Choose pairwise disjoint Morse neighborhoods of the finitely many critical points, and smaller neighborhoods with compact closure in them. Compactness allows a neighborhood of the band with no other critical points outside these charts. In each chart the field $(2u,-2v)$ has derivative $-4(\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2)$.

Choose a metric; away from the critical points the field $-\mathrm{grad}f$ has strictly negative derivative. Cover the band neighborhood by the Morse neighborhoods and a regular open set avoiding the closures of the smaller charts. A subordinate partition of unity patches these fields. At a regular point the derivative is a convex combination of strictly negative numbers; on a smaller chart only its local field is present.

Choose a relatively compact neighborhood of the compact band within the field domain and a bump equal to one near the band. Multiply by it and extend by zero. This leaves the required local formulas intact and gives a complete field. With no critical points the regular field alone is used; with an empty band use zero. In dimension zero each local field is zero and there are no regular points to test.

The Morse lemma supplies the displayed coordinates after shrinking $\epsilon >0$. Such cutoffs exist: take a smooth function $0\le \eta <1$ with support compactly inside $(0,2\epsilon )$ and integral greater than $\epsilon$, and put $\mu (t)={\int }_t^{2\epsilon }\eta (s)ds$. A bump equal to a constant less than one on a sufficiently long closed subinterval gives $\eta$. The perturbation has support compactly inside the chart, so gluing by zero is smooth.

Put $x=\mathrm{\mid }u{\mathrm{\mid }}^2$, $y=\mathrm{\mid }v{\mathrm{\mid }}^2$. Then $dF=-2(1+{\mu }^{\mathrm{\prime }})u\cdot du+2(1-2{\mu }^{\mathrm{\prime }})v\cdot dv$. Both scalar magnitudes are positive, so its only chart critical point is $(0,0)$; its Hessian there has the same index, including empty coordinate blocks. Its value is $c-\mu (0)<c-\epsilon$. All other critical points and their values are unchanged.

Since $F\le f$, one inclusion of upper sublevels holds. Wherever $F\mathrm{\ne }f$, $x+2y<2\epsilon$, hence $f=c-x+y<c+\epsilon$; elsewhere the two functions coincide. This proves the reverse inclusion. If $F\ge c-\epsilon$, then $f\ge F\ge c-\epsilon$. Thus the modified closed band is a closed subset of the original compact band. Its only candidate critical point has been lowered out of it.

Use the Morse chart and choose $\epsilon$ small enough that the ball of squared radius $6\epsilon$ is contained in it. Apply the lowering construction, put $x=\mathrm{\mid }u{\mathrm{\mid }}^2$, $y=\mathrm{\mid }v{\mathrm{\mid }}^2$, and subtract $c$ from both functions. Thus $F=-x+y-\mu (x+2y)$. Put $r=\sup \{t:\mu (t)>0\}<2\epsilon$. Since $\mu (0)>\epsilon$ and ${\mu }^{\mathrm{\prime }}>-1$, $r>\epsilon$.

First let $0<k<n$. For each $x\ge 0$ the equation $-x+s-\mu (x+2s)=-\epsilon$ has a unique positive solution $s=s(x)$: its left side is strictly increasing in $s$, is below $-\epsilon$ at zero because $x+\mu (x)\ge \mu (0)>\epsilon$, and tends to infinity. Its derivative in $s$ is $1-2{\mu }^{\mathrm{\prime }}>0$, so $s$ is smooth and $s^{\mathrm{\prime }}=(1+{\mu }^{\mathrm{\prime }})\mathrm{/}(1-2{\mu }^{\mathrm{\prime }})>0$. Also $s(x)\ge x-\epsilon$, with equality for $x\ge r$ (in fact it holds earlier). Choose $d>0$ with $d<s(0)$ and $\epsilon +3d<r$. The compact region $H_0=\{y\le d,x\le \epsilon +y\}$ is parametrized by $(U,V)\mapsto (\sqrt{\epsilon +d\mathrm{\mid }V{\mathrm{\mid }}^2}U,\sqrt{d}V)$ on $D^k\times D^{n-k}$. Its inverse is $(u,v)\mapsto (u\mathrm{/}\sqrt{\epsilon +\mathrm{\mid }v{\mathrm{\mid }}^2},v\mathrm{/}\sqrt{d})$. These formulas are smooth on the axes. Its attaching face is $\mathrm{\mid }U\mathrm{\mid }=1$, on $f=-\epsilon$, and its core is $v=0$.

The union of the lower sublevel with $H_0$ has local boundary $y=\max (d,x-\epsilon )$. Round this single corner by a smooth nondecreasing function $j(x)$ equal to that maximum off a small neighborhood of $x=\epsilon +d$. Choose the rounding above the maximum and below $s(x)$; the strict gap at the corner permits this, for example by smoothing the absolute-value formula for the maximum on a sufficiently short interval. Then $j>0$ and $j=s=x-\epsilon$ for $x\ge r$. This is precisely the compatible rounding of the attached product handle.

The graphs $y=j_t(x)=(1-t)j(x)+ts(x)$ stay strictly positive. Near them use the smooth field $V_t=(s(x)-j(x))v\mathrm{/}(2j_t(x))$ in the $v$ coordinates and zero in the $u$ coordinates. Then $dy(V_t)=s-j$ on the graph. Multiply this field by a bump that is one on the moving graphs where they differ and zero near $v=0$ and outside the chart. The graph difference has compact support in $x$, and all these graph points lie in the chosen chart. Integrate ${\mathrm{\partial }}_t+V_t$ with this cutoff on time times the chart. Smooth dependence and uniqueness give inverse evolution; compact spatial support gives continuation throughout $0\le t\le 1$. Since $dy\mathrm{/}dt=s-j$ on the moving graph, this isotopy carries $\{y\le j(x)\}$ onto $\{y\le s(x)\}$, is identity away from the chart, and fixes the core. Hence the rounded attachment is diffeomorphic to $\{F\le -\epsilon \}$. This uses positive smooth radii, never an inverse of a flat cutoff at its endpoint.

The lowering lemma gives $\{F\le \epsilon \}=\{f\le \epsilon \}$ and a compact regular $F$-band from $-\epsilon$ to $\epsilon$. Its product description supplies the complementary collar. An extra finite collar does not change the diffeomorphism type: join it to an inner collar and reparametrize the collar interval by a smooth increasing map fixed near its inner end. Thus the smooth handle change reaches the upper sublevel.

For $k=0<n$, the lower local sublevel is empty and $\{F\le -\epsilon \}$ is the disk $\mathrm{\mid }v{\mathrm{\mid }}^2\le s(0)$, so it is a disjoint zero-handle; the same regular collar finishes. For $k=n>0$, there is no $v$ variable: $x+\mu (x)>\epsilon$ for every $x\ge 0$, so the missing disk $x\le \epsilon$ is filled along its whole sphere; outside that disk the lower sublevel was already present locally. For $n=0$, the chart is a single point and crossing its value adds that point. These descriptions require no corner or angular coordinate.

Choose the local attaching tube and the adapted complete descending field $X$. If the regular band $f^{-1}([a,c-\epsilon ])$ is empty, both levels and the attaching region are empty and transport is vacuous. Otherwise, on that compact band $-df(X)>0$ has a positive minimum. Normalize to $Z=X\mathrm{/}(-df(X))$ near that band and cut off outside a relatively compact regular neighborhood. Its flow $\mathrm{\Psi }$ has $df(Z)=-1$ throughout the band; finite-time continuation follows exactly as for the regular product.

Let $h$ be the local tube embedding. Define $h_a(z)={\mathrm{\Psi }}_{c-\epsilon -a}(h(z))$. Reverse flow for the same time is its inverse onto the image. Smooth dependence and uniqueness show that this is an embedding, including its disk boundary. The derivative sends the chosen normal disk coordinates to linearly independent normal coordinates, since the level-to-level map is a diffeomorphism. It therefore transports the framing, not just the sphere.

The same flow gives the collar between $M_a$ and $M_{c-\epsilon }$. Near the critical model its complementary collar is the regular modified-function collar from the local lemma; outside the compact modification support the modified function equals $f$. On that common regular region choose its descending collar field to agree with $Z$: patch $Z$ with any descending field for the modified function using a smooth partition, as in the adapted-field construction, and normalize by the negative derivative of the modified function. The convex combination remains descending. Choose the partition to retain $Z$ on a smaller neighborhood of the chart interface. Uniqueness of flow then makes the collar charts agree on their common flow neighborhoods there. If the attaching region is empty there is nothing to transport. These arguments use neither orientation nor Morse–Smale transversality.

Put $c=f(p)$. Regularity of the endpoints gives $a<c<b$. Choose $\epsilon >0$ with $[c-\epsilon ,c+\epsilon ]\subset (a,b)$ and a sufficiently large relative Morse chart for the local lemma. All closed subbands are compact, and the two outer bands have no critical points.

The local lemma attaches one compact product handle to $M^{c-\epsilon }$, rounds it, and identifies the resulting smooth manifold with the modified lower sublevel. Its complement in $M^{c+\epsilon }$ is the regular modified-function collar. The modification has compact chart support, so all maps glue to the unchanged exterior using the common collars. Absorbing the final collar yields the smooth attachment description of $M^{c+\epsilon }$.

Transport the attaching tube and its framing to $M_a$ along the lower regular band. The lower and upper regular sublevels are diffeomorphic, and their product collars allow the attachments to be glued under these identifications. Hence the same handle attached to $M^a$ gives $M^b$. Compatible corner choices give diffeomorphic answers.

For later pair calculations, the comparison can retain a pushed-in copy $A_0$ of the lower sublevel. Indeed all adjustments occur in compact boundary collars or the attaching chart: choose the inner edge of the lower collar below their support, and compress $M^a$ to that inner edge. Both the original lower sublevel and the lower sublevel in the attachment retract to this same copy by collar compression. Thus their inclusions into the compared upper spaces agree up to homotopy of pairs. This does not assert that an ambient diffeomorphism sends the original lower boundary to the attachment seam. Empty lower sublevels and indices $0,n$ are exactly the cases proved in the local lemma.

In the adapted chart the equations are $u(t)=e^{2t}u(0)$ and $v(t)=e^{-2t}v(0)$. A trajectory that stays in this chart for all sufficiently negative time converges to $p$ if and only if $v=0$. On this plane $f=c-\mathrm{\mid }u{\mathrm{\mid }}^2$, so the closed disk $\mathrm{\mid }u\mathrm{\mid }\le \sqrt{\epsilon }$ ends at $M_{c-\epsilon }$. This is the local core of the constructed handle.