A counterexample to the nearby Lagrangian conjecture

1. Start with a nonzero tube class having zero spherical class

Let $T_{k,l}$ be the component of the standard form $q_{k,l}(x,y)=-\|x\|^2+\|y\|^2$ among regular angular functions. The homogeneous extension of $h$ is $g(ru)=r^2h(u)$, and its negative domain is a smooth tube of type $S^{k-1}\times D^l$. Adding positive and negative squares gives $\mathcal T=\operatorname{hocolim}_mT_{m,m}$.

The manuscript's Theorem 2.1 uses a fiber sequence $H^s(*)\to\mathcal T\xrightarrow{\gamma}BG$ and identifies the composite from quadratic forms $BO\to\mathcal T\to BG$ with $BJ$. For the boundary map $\partial:\pi_{10}BG\to\pi_9H^s(*)$, the group $\pi_{10}BG=\pi_9^S\cong(\mathbb Z/2)^3$ contains the order-two image of $J_9$ in the kernel, so $|\operatorname{im}\partial|\le4$. The two-primary part of $\pi_9H^s(*)\cong\pi_{10}\operatorname{Wh}^{\mathrm{diff}}(*)$ is $\mathbb Z/8\oplus(\mathbb Z/2)^2$, of order 32. The boundary map is therefore not surjective, producing $$ 0\ne a\in\pi_9\mathcal T,\qquad\gamma_*a=0. $$ The separate injectivity of $J_8$ will trivialize the negative eigenbundle of a one-critical-point family.

Theorem 2.1; Proposition 2.2 · pp. 5–6

2. Separate realization of the class from the embedding argument

Lemma 2.3 represents $a$ at a sufficiently large finite stage $T_{m,m}$, $m\ge20$, by a family $g_b$ that is standard quadratic near both critical points of a height function $\theta:S^9\to\mathbb R$. Set $N=2m$ and add the auxiliary sphere $S^{N-1}$ to the base. Proposition 5.2 parametrizes the critical locus by $S^9\times S^{N-1}$ and successively establishes Morse-family regularity, injectivity using momenta, and compactness. These give a smooth embedding, closedness, and a diffeomorphism with the base; $\Gamma_F^*\lambda=d(F|_{\operatorname{Crit}(F)})$ gives exactness.

Lemma 2.3 · pp. 6–7 Proposition 5.2 · pp. 18–20

3. Hypothetical isotopy forces the same class to vanish

Extend the reverse isotopy to a Euclidean neighborhood and transport the generating family through finitely many small Hamiltonian steps. Each step adds a fixed split quadratic form and performs only a homotopy through regular homogeneous data. At the zero section, the critical-locus projection becomes a diffeomorphism, giving exactly one nondegenerate critical point in each fiber. Restriction to the $S^9$ slice at a fixed $v_0\in S^{N-1}$ retains the nonzero class $a$ and zero spherical class. Proposition 3.1 makes the stable tube class of this same family zero, a contradiction. This argument transports the constructed generating family; it does not require defining a numerical invariant of the Lagrangian from every possible generating family.

§6.4; proof of Theorem 1.1 · p. 26

1. A cutoff linear coupling fixes the critical equation

Write $c\|w\|\le\|\nabla g_b(w)\|\le C\|w\|$ and $\|d_wd_bg_b(w)\|\le C_1\|w\|$. Outside pole neighborhoods $U_0$, where $g_b=q_{m,m}$, put $\delta_0=\min_{B\setminus U_0}\|d\theta\|>0$. First choose $K\delta_0>C_1/c$ and set $R(b)=\Lambda e^{K\theta(b)}$. Let $G_b$ smooth $g_b$ only near the origin, and let $\beta$ equal one on $[0,1]$ and zero on $[2,\infty)$. Use $$ F(b,v,w)=G_b(w)-R(b)\beta(\|w\|/T)\langle v,w\rangle. $$ Put $D_\beta=\sup_{s\ge0}(|\beta(s)|+s|\beta'(s)|)$ and choose $K,\Lambda,T$ in this order: $\Lambda$ excludes critical points with $\|w\|\le1$, and $T>\max\{1,R_{\max}/c,D_\beta R_{\max}/c\}$ excludes the exterior and the cutoff region. Every critical point lies in $1<\|w\|<T$, and the necessary and sufficient equation is $\nabla g_b(w)=R(b)v$.

§5; equations (29)–(32) · p. 19

2. The parametrization is a diffeomorphism; regularity uses all variables

Homogeneity gives the smooth parametrization $$ (b,u)\longmapsto\left(b,\frac{\nabla g_b(u)}{\|\nabla g_b(u)\|}, \frac{R(b)u}{\|\nabla g_b(u)\|}\right),\qquad u\in S^{N-1}, $$ with inverse $u=w/\|w\|$. Variations in $v^\perp$ contribute $-R\delta v$, and radial variation of $w$ contributes $Rv$, making the differential of the full vertical-gradient equation surjective. This is the Morse-family condition; invertibility of every fiber Hessian is not asserted here. Lemma 5.1 gives an exact Lagrangian immersion $\Gamma_F(b,v,w)=(b,v;d_{b,v}F)$.

Lemma 5.1 · p. 18 §5; equations (29)–(32) · p. 19

3. Two momentum tests and compactness give an embedding

Points with the same image first have the same $(b,v)$. Their sphere momenta are $d_vF(\dot v)=-R\langle w,\dot v\rangle$ for $\dot v\perp v$, so equality implies $w-w'=tv$. The segment between these points lies in the radius-$R/c$ ball. The $C^1$ extension of $d_bg_b$ makes the estimate $$ \|d_bg_b(w)-d_bg_b(w')\|\le(C_1/c)R|t| $$ valid even if the segment crosses the origin. The difference of the $B$-momenta is $$ \Delta p_B=d_bg_b(w)-d_bg_b(w')-KRt\,d\theta_b. $$ For $b\notin U_0$, its norm is at least $R|t|(K\delta_0-C_1/c)$, forcing $t=0$. Within $U_0$, the gradient equation for the fixed nondegenerate quadratic form already has a unique solution. Thus the immersion is globally injective.

The critical locus is compact without boundary, so the injective immersion is an embedding. Its image $L$ is closed and diffeomorphic to $Q$, with primitive $F|_{\operatorname{Crit}(F)}=-g_b(w)$. Since $F=g_b$ for $\|w\|\ge2T$, the infinity class is unchanged. A diffeomorphism of abstract manifolds is distinguished here from a Hamiltonian isotopy inside the cotangent bundle.

§5; equation (33) and conclusion · pp. 19–20

1. The spherical hypothesis fixes the local quadratic model

Move the critical point to the origin, make its value zero, and replace the function locally by its Hessian quadratic form. Gradient lower bounds compared with cutoff errors exclude additional critical points. Lemma 3.2 identifies a negative sublevel with the negative domain at infinity and shows that attaching the single Morse handle gives a contractible positive sublevel, forcing Morse index $p$. It also constructs a fiber homotopy equivalence from the sphere bundle of the negative eigenbundle $E^-$ to the family of negative domains at infinity.

Thus zero spherical class implies $BJ[E^-]=0$. Injectivity of $J_8:\pi_9BO\to\pi_9BG$ and the stable range $p\ge20$ make the negative-eigenspace map nullhomotopic even in the finite Grassmannian. A radial cutoff varying slowly on a logarithmic scale fixes the local model as $q_{p,p}$. Add positive squares and use Lemma 3.3 to smooth the homogeneous data; the local model becomes $q_{k,l}$ with $k=p$ and $l>k+9$. Adding the missing negative squares later returns to diagonal stabilization.

Lemma 3.2; Lemma 3.3; §3 · pp. 7–11

2. Distinguish products fiber by fiber from a product family

Between the inner and outer spheres, put $s_b(u,t)=f_b(r(t)u)/r(t)^2$. A regular band around $C_b=\{s_b=0\}$ is $C_b\times J$. Flows in the sign regions embed the fixed inner negative tube into the outer one and identify the complement with $C_b$. Both boundary inclusions are homotopy equivalences, making $C_b$ a family of $h$-cobordisms on $\Sigma=S^{k-1}\times S^{l-1}$.

The ordinary $h$-cobordism theorem on each fiber does not provide product structures varying smoothly over $S^9$. The manuscript performs one interval stabilization and extends it into the fixed cylinder on $Y=S^{k+l-1}$, obtaining $W_b=(Y\times[-1,0])\cup(C_b\times J)$. A flow identifies its complement as a product $E_b\times I$ over its possibly varying outgoing boundary bundle $E\to B$. Absorbing that outgoing product collar makes the total a fixed cylinder, so $[W]=0$.

Lemmas 4.1–4.4 · pp. 11–15

3. Detect the original family in the stable range and straighten its cores

In Proposition 4.5, $\dim\Sigma=k+l-2\ge38$ exceeds the Igusa bound $\max(2r+8,3r+5)=35$ for $r=10$. Connectivity of the collar embedding $\Sigma\times J\hookrightarrow Y$, together with connectivity estimates for stable $h$-cobordisms, makes stabilization followed by extension an isomorphism on $\pi_9$. Therefore $[W]=0$ implies $[C]=0$, and classification of families supplies a product structure relative to the incoming end.

Proposition 4.5; §4.3 · pp. 15–16 [CP25, Definition 2.15; Theorems 2.16–2.17, pp. 8–9] [CP25, Proposition 2.11, p. 7]

Shrink the product collar attached to the outer tube and deform the tube to a small round normal neighborhood of its core. To keep a homotopy of cores fiberwise embedded, the argument uses parameter dimension ten, including time, and $$ 10+2(k-1)<k+l-1, $$ together with the codimension estimate for rank defects. This explains the earlier requirement $l>k+9$. Moving to the fixed standard core and lifting the homotopy of domains to defining functions makes the tube family nullhomotopic.

§4.4; completion of Proposition 3.1 · pp. 16–17

1. Realize the reverse isotopy in a Euclidean neighborhood

Embed $Q$ in $\mathbb R^{10}\times\mathbb R^N=\mathbb R^d$, $d=10+N$, and use two radial coordinates for a neighborhood $U\cong Q\times(-\nu,\nu)^2$. Cylindrize the initial family as $F^{(0)}(x,w)=F(r_Q(x),w)$. Lemma 6.1 extends the hypothetical reverse Hamiltonian isotopy with compact support to $T^*\mathbb R^d$. A sufficiently wide cutoff in radial momenta makes radial positions constant along trajectories from the cylinder. Over a central region $U_c$, the final image is exactly the zero section. Constant radial position also prevents other branches of the cylinder from entering the central region.

Lemma 6.1 · p. 21

2. Add fixed split forms and preserve the infinity class

The graph of a small step has mixed coordinates $(X,p)$ and a compactly supported function $\sigma$ with $x=X+\sigma_p$ and $P=p+\sigma_X$. The local composition family is $F_o(X+z,w)+\sigma(X,p)-z\cdot p$. Proposition 6.3 uses a spatial buffer and logarithmic cutoff, replacing the shift by $X+\kappa z$ to define it globally. It differentiates the full critical equations first; $z=\sigma_p$ and a gradient estimate show that the vertical cutoffs $\beta_1,\chi$ are one near critical points. On $V$, where the spatial cutoff $\zeta_1$ is one, the mixed graph equations are recovered directly. Outside $D$, which contains the spatial support of $\sigma$, one has $z=0$, so terms involving derivatives of $\zeta_1$ vanish even where that cutoff varies. These two regions establish regularity and a diffeomorphism between old and new critical loci.

The homogeneous reference is $G_0(X,w,z,p)=g_o(X,w)-z\cdot p$. Its difference from the new function and the full vertical first derivative of that difference are uniformly bounded. Lemma 3.3 restores exact homogeneity without changing a neighborhood of the critical locus. The form $-z\cdot p$ has signature $(d,d)$, and smoothing is a homotopy through regular homogeneous functions. Hence every fixed compact slice retains its free stable tube class. Iterating through finitely many steps $s$ gives a family in $2p=2m+2ds$ variables.

Lemma 6.2; Proposition 6.3 · pp. 22–25

3. At the zero section the Hessian is nondegenerate, so vanishing applies

Over $U_c$ the generating map is a diffeomorphism onto the zero section, making $\operatorname{Crit}(F^{(s)})|_{U_c}\to U_c$ a diffeomorphism. Each fiber therefore has one critical point, varying smoothly. A nonzero Hessian-kernel vector $v$ would give a tangent vector $(0,v)$ to the regular critical locus in the kernel of the projection differential. This is impossible; the Hessian is invertible.

Restricting to the fixed slice $b\mapsto((b,v_0),0)$ gives a family over $S^9$ satisfying every hypothesis of Proposition 3.1: $p=m+ds\ge20$, type $T_{p,p}$, uniform homogeneous infinity data, exactly one nondegenerate critical point, and zero spherical class. Its free tube class, however, equals the nonzero free class of $a$. Basepoint changes cause no problem, since the fundamental-group action fixes zero and cannot turn a nonzero based class into a freely null class. The contradiction with Proposition 3.1 proves that the constructed $L$ is not Hamiltonian isotopic to the zero section.

§6.4; proof of Theorem 1.1 · p. 26