Finite comparison of parametrized elementary relative handle bands

Exact consequences for handles; further work is distinguished. Near the normalized birth, choose the descending field to be the negative Euclidean gradient of $a(s)+b(s)u+c_0{u}^3+Q(y)$ in the smaller box. For parameters close enough to the event the two critical points and the segment between them lie in this box. The line $y=0$ is invariant; its scalar equation is $\dot{u}=-b(s)-3c_0{u}^2$. For the parameter side with two roots it has exactly one local connecting trajectory between them. No additional connecting trajectory can leave the chart: choose a fixed compact annulus between two nested balls inside this model chart. At $s=0$ the gradient has no zero on the annulus, and thus its norm has a positive lower bound $m$, uniform for all sufficiently small parameters. A negative-gradient trajectory crossing the annulus from its inner to its outer boundary has Euclidean length at least the difference of the two radii and loses function value at least $m$ times that difference, since $dF\mathrm{/}d\mathrm{\ell }=-\mathrm{\mid }\mathrm{\nabla }F\mathrm{\mid }$ along arc length. But the two critical values converge to one another at the event. For small parameters their difference is less than this fixed positive loss, so a trajectory joining them cannot cross the annulus. Inside the chart the diagonal linear $Q$ equations force every transverse coordinate to be zero on a trajectory tending to both critical points: a growing coordinate cannot tend to zero at its positive-time end and a decaying coordinate cannot tend to zero at its negative-time end. Thus the local trajectory is also the unique global trajectory. The positive and negative $Q$ directions give transverse complementary normal directions to that trajectory. Consequently its attaching sphere and belt sphere intersect geometrically once at an intermediate level, and the actual [F6] applies. This proves that this particular elementary birth inserts a geometrically cancelling adjacent-index handle pair. It does not merely infer cancellation from the two index counts. The local coordinate model gives the framing: the negative eigenspaces supply the lower handle's core disk, the positive eigenspaces supply the cocore normal coordinates, and flow transports the attaching thickening. Hence the one intersection is a framed handle cancellation in the sense of the existing supplier. It is supported away from the fixed boundary collars. Later handles must be transported by the supported cancellation diffeomorphism; simply deleting the pair while forgetting the terminal parametrization is not the asserted move.

Regular-parameter intervals: a relative left-right trivialization proved by a vector field. Suppose $s$ varies in a sufficiently short closed interval containing no exceptional event. The finitely many critical points have smooth disjoint branches $p_j(s)$ by the parameter implicit function theorem, and their distinct critical values $c_j(s)$ stay separated. Successive square completion, with $s$ a parameter, gives smooth local embeddings ${\psi }_{j,s}$ with $F_s({\psi }_{j,s}(z))=c_j(s)+Q_j(z)\mathrm{.}$ Put $V_{j,s}={\mathrm{\partial }}_s{\psi }_{j,s}\circ {\psi }_{j,s}^{-1}$. Differentiating the last equation gives ${\mathrm{\partial }}_s{F}_s+d{F}_s(V_{j,s})=c_j^{\mathrm{\prime }}(s)$ near that critical branch. Choose a smooth scalar field $Z_s(t)$ on the value interval, supported away from $0,1$ and the fixed collar value ranges, equal to $c_j^{\mathrm{\prime }}(s)$ on disjoint neighborhoods of the values $c_j(s)$. Such a field is a finite sum of disjoint bumps following those values. Near a critical point the equation $d{F}_s(Y_s)=Z_s(F_s)-{\mathrm{\partial }}_s{F}_s$ is therefore solved by $Y_s=V_{j,s}$. Away from the critical branches it is solved by $Y_s=(Z_s(F_s)-{\mathrm{\partial }}_s{F}_s)\frac{\mathrm{grad}{F}_s}{\mathrm{\mid }\mathrm{grad}{F}_s{\mathrm{\mid }}^2}\mathrm{.}$ A partition of unity patches these solutions because the equation is linear in $Y_s$. Take $Y_s=0$ on the fixed collar: both sides vanish there. These vector fields have compact support in the interior of $W$, so their time-dependent flow ${\mathrm{\Phi }}_s$ exists throughout the short parameter interval, fixes the boundary collar, and is a diffeomorphism (use the ordinary flow theorem on the extended space with field ${\mathrm{\partial }}_s+Y_s$). Let ${\eta }_s$ be the scalar flow of $Z_s$, initially the identity. Uniqueness for the scalar ODE gives $F_s\circ {\mathrm{\Phi }}_s={\eta }_s\circ F_{s_0}\mathrm{.}$ Thus regular level surfaces, handle bands, and both incoming and outgoing identifications are carried by actual collar-fixed diffeomorphisms. The scalar flow is increasing and is a translation near each critical value, since $Z_s$ is locally constant in $t$ there; hence a transported Euclidean critical-point field retains its exact Morse model. There is no appeal to “structural stability” in this argument. A finite subdivision covers any compact parameter interval without events. If an attaching embedding at one slice is $S:S^{k-1}\times D^{n-k}\to F_{s_0}^{-1}(a)$, its transported embedding is ${\mathrm{\Phi }}_s\circ S$, considered in the level $F_s^{-1}({\eta }_s(a))$. If its elementary outgoing identification was $d$, it becomes ${\mathrm{\Phi }}_s{\mathrm{\mid }}_{\mathrm{o}\mathrm{u}\mathrm{t}}\circ d\circ ({\mathrm{\Phi }}_s{\mathrm{\mid }}_{\mathrm{i}\mathrm{n}}{)}^{-1}$, with the handle part transported by the same local coordinates. These formulas are necessary: writing only that the attaching sphere is isotopic loses the terminal boundary parametrization.

Changing the descending field inside one elementary band. This step is local to a band with at most one critical point; it does not require a global Morse–Smale theorem. The actual local stable-manifold supplier [F4] and its complete contraction proof were read. Its displayed contraction proof extends to a smooth compact parameter interval as follows. First any given standard downward gradient-like field $X$ on such a band can be written as $-{\mathrm{grad}}_{g}f$ for a smooth metric $g$. Near its critical point use the Euclidean metric from its defining chart. Off the critical point split the tangent bundle as $\mathbb{R}X\oplus \ker df$ and prescribe $g(X,X)=-df(X)>0$, $g(X,v)=0$ for $v\in \ker df$, and any positive metric on $\ker df$. Then $g(X,\cdot )=-df$; convex patching of such metrics preserves that linear equation and agrees with the Euclidean metric on a smaller critical chart. Use a product metric on the fixed end collars. Thus two specified fields can be joined by the negative gradients of the convex metric path $g_{\tau }=(1-\tau )g_0+\tau g_1$, retaining the collar condition. For completeness, smooth dependence of the unstable disks is justified rather than assumed. The linearizations are $A_{\tau }=-g_{\tau }^{-1}\mathrm{Hess}f$, smoothly parameterized, self-adjoint for $g_{\tau }$, and uniformly hyperbolic on the compact parameter interval. Their positive/negative spectral subbundles vary smoothly. One elementary formula is obtained by conjugating to an ordinary symmetric matrix and taking its matrix sign $A(A^2{)}^{-1\mathrm{/}2}$; the positive square root of a positive definite matrix is smooth by the implicit function theorem, since the derivative $H\mapsto RH+HR$ has eigenvalues equal to positive sums of eigenvalues of $R$ and is invertible. The projectors are $(I\pm \mathrm{sign}A)\mathrm{/}2$. A smooth family of projectors $P_{\tau }$ has smooth trivializations: solve $U^{\mathrm{\prime }}=[P^{\mathrm{\prime }},P]U$, which differentiating $P_{\tau }{U}_{\tau }-U_{\tau }{P}_0$ shows transports the ranges and kernels. Thus fixed spectral coordinates may be used on a short parameter interval. In those coordinates the supplied proof's weighted integral operator for stable paths is ${\mathcal{T}}_{\tau ,a}z(t)=e^{t{A}_{\tau }}a+{\int }_0^t{e}^{(t-r)A_{\tau }}{P}_{s,\tau }{R}_{\tau }(z(r))dr-{\int }_t^{\mathrm{\infty }}{e}^{(t-r)A_{\tau }}{P}_{u,\tau }{R}_{\tau }(z(r))dr\mathrm{.}$ Choose a common spectral gap, a smaller positive decay exponent, and a uniform cutoff radius making the contraction constant strictly less than one for all nearby $\tau$. This is possible because $R_{\tau }(0)=D{R}_{\tau }(0)=0$ uniformly on the compact parameter interval. The operator is smooth in $(\tau ,a,z)$ on the weighted path space. To check this, each spatial derivative of the cut-off $R_{\tau }$ is bounded; products of decaying path variations remain bounded in the same weighted norm. Parameter derivatives of the matrix exponentials add polynomial factors in $t$ or $\mathrm{\mid }t-r\mathrm{\mid }$, which remain integrable after reserving a smaller exponential gap. Repeated differentiation therefore gives uniformly dominated integral operators of every finite order. Its path derivative has norm below one. Differentiating the fixed-point equation and solving by the convergent Neumann series proves smooth parameter and initial-point dependence; repeated differentiation proves every higher derivative, since the highest derivative always occurs in the same invertible linear operator $I-D_z\mathcal{T}$. Time reversal gives the same result for unstable disks. This is the parameter version of the supplier's actual proof, with its uniform estimates and differentiability step supplied here. Intersect the smoothly varying local unstable disk with a fixed nearby regular level below its critical value. The intersection is transverse because $df(X_{\tau })<0$, so it is a smooth family of framed attaching sphere sections. Its normal coordinates come from the complementary local stable directions and vary smoothly as well. Flow their sufficiently thin thickenings down to the incoming level of the elementary band. There is no intervening critical point. On the compact part outside the local critical chart the strict descending inequality has a uniform negative bound, so these trajectories reach the incoming level in uniformly bounded time. Their hitting time is smooth by the implicit function theorem, since the derivative of $f$ along the trajectory is nonzero at the hit. Consequently the full framed attaching embeddings vary by a smooth isotopy. The same flow-plus-local-handle construction gives a smooth family of outgoing identification maps. These can be extended to ambient isotopies on the compact level surface by the actual compact-source isotopy-extension supplier, or retained as the explicit smoothly varying embeddings and outgoing maps. This proves that changing fields on an elementary band introduces only isotopy of the framed attaching embedding with the corresponding change of the outgoing identification. Constant changes of the chosen frame at a critical point are changes of the standard handle coordinates; if a frame component is reversed, retain that handle-coordinate automorphism explicitly. It is not a new Dehn-surgery framing integer. No classification of globally broken trajectories or same-index handle slides was used.

A critical-value crossing gives independent framed attachments. Take a short parameter interval with just one crossing, between nondegenerate critical branches $p(s),q(s)$, and no other critical points in the chosen height band. Use parameter Morse coordinates at the two disjoint branches and choose one smooth descending field agreeing with the Euclidean models there; partition of unity constructs it exactly as in the read supplier [F5]. At the crossing parameter their values agree, say at $c$. Choose the same fixed positive model height drop $ϵ$ at each critical point, so that each local unstable sphere section has height $F_s(p(s))-ϵ$ or $F_s(q(s))-ϵ$. For $s$ close enough to the crossing both sections lie strictly below both critical values. All flow from those sections to a lower regular level $a$ therefore takes place in a region without critical points. The corresponding unstable disks cannot hit the other critical point before reaching their sections: the function drop to each section is $ϵ$, while the separation of the two critical values is less than $ϵ$; choose the local disk in its own disjoint Morse chart. Thus the global attaching spheres in $F_s^{-1}(a)$ are disjoint. Equivalently, distinct points on them have backward limits at different critical points, so flow uniqueness forbids an intersection. By compactness, choose disjoint sufficiently thin framed attaching neighborhoods for both. The whole band is obtained by attaching these two disjoint framed handles and then a regular product to an upper level $b$. This follows from the same simultaneous local-handle construction and compact regular-flow comparison as the read [F7]; it does not require their indices to agree, since the charts, local function modifications, and all attaching neighborhoods are disjoint. One can also cross the two distinct critical heights in either order. Disjoint quotient attachments commute: both iterated quotients identify exactly the same two disjoint attaching subsets with the same handle subsets. Their corner roundings agree outside disjoint seam collars and are compared by the existing corner-rounding isotopies inside them. The normal coordinates are unchanged; hence these are independent framed attachments, not merely independent core spheres. For exact parametrization bookkeeping, write $A,B$ for the two framed attachments in the common lower surface. Let $d_A$ and $d_B$ be the identifications to the respective first intermediate surfaces, and let $e_{B\mathrm{\mid }A}$ and $e_{A\mathrm{\mid }B}$ identify the second surgered surfaces with the common upper surface. Denote by $d_A^B$ and $d_B^A$ the maps extended over the other, disjoint handle. The actual identification from the simultaneous two-handle model to the upper level gives $e_{B\mathrm{\mid }A}\circ d_A^B\simeq e_{A\mathrm{\mid }B}\circ d_B^A$ under the canonical identification of the two simultaneous quotient surfaces. To verify this equality, on the complement of the two attaching regions both maps are the same normalized flow from the lower to the upper level. On either handle part they are obtained from the same local model coordinates followed by the same outgoing regular flow. Their collar adjustments differ only by the supported corner-rounding isotopies just described. This proves the displayed isotopy and supplies the outgoing-map rule at the crossing. It is precisely the extra condition missing from an unparametrized “critical heights interchange” statement.

What finite event reduction now establishes. Choose parameter values immediately before and after each of the finitely many events, and divide the remainder into finitely many regular intervals. Retain a regular level between each pair of critical values on each regular slice. Steps 2.1 and 3.1 compare the induced elementary handle bands along regular intervals by relative diffeomorphism, framed attaching isotopy, changes of handle-coordinate frame, and insertion/removal of empty product bands. Step 4.1 compares the two sides of each crossing by interchange of two independent framed attachments, with its displayed outgoing-map rule. The cubic normalization in [F1] and step 1.1 compare the two sides of each birth by a geometrically cancelling adjacent-index pair, carrying all later attaching data and the final boundary map by the actual supported cancellation comparison. In particular, this proves finite reduction of the Morse-induced elementary handle descriptions along the constructed path. Empty ends, disconnected manifolds, and index $0\mathrm{/}n$ births are included: the formulas allow $\mathrm{ind}Q=0$ or $n-1$, and no connected-level or nonempty-end assumption occurred. The supplied Morse/handle correspondence realizes any specified finite handle description by such Morse data. To use the reduction as a move theorem for presentations where the cancellation boundary identification has been suppressed, one must still prove that the chosen suppression preserves the actual end parametrization; the explicit move above instead retains that identification throughout. It is not a proof that arbitrary relative surgery diagrams are related by ordinary Kirby moves.