Disk embedding and manifold realization: boundary data and tensor obstructions

Source and verification

This article explains family 305 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

In four dimensions, algebraically cancelling surface intersections need not provide the embedded disks needed to cancel them geometrically. Family 305 contains two claimed counterexamples to unrestricted disk embedding and a companion claimed counterexample to four-dimensional manifold realization. Their common mechanism retains boundary markings and nilpotent deformation equations until a finite tensor calculation produces a contradiction. (OpenAI, 2026) (OpenAI, 2026b) (OpenAI, 2026c)

🏷️ Preliminaries: disks, duals, and boundary data

A proper disk map sends its boundary into the boundary of a four-manifold and its interior into the interior. A generic immersion has isolated transverse double points. A locally flat embedded disk has neighborhoods homeomorphic to the standard local pair $({\mathbb{R}}^4,{\mathbb{R}}^2)$, with the corresponding half-space model at its boundary. Local flatness is a topological condition and does not assume smoothness.

A framing trivializes the normal two-plane bundle. For an immersed input disk it induces a normal framing along the boundary circle.

Choose paths, called whiskers, from the surfaces to a basepoint. The equivariant intersection number $\lambda (f,g)$ sums intersection signs together with the loops obtained by following those paths. Its coefficients lie in $\mathbb{Z}[{\pi }_{1}M]$, so cancelling ordinary signed counts is weaker than cancelling each group-labelled count.

Sphere maps $g_j$ are algebraic duals to disk maps $f_i$ when

$$\lambda (f_i,g_j)={\delta }_{ij}1.$$

A geometric dual instead meets its designated disk in exactly one transverse point and misses the other disks. The reduced Wall self-intersection invariant $\tilde{\mu }(g)$ records labelled self-intersections modulo the additive relations identifying a label with its inverse and discarding integer multiples of the identity label. This is an additive quotient, not a quotient by an ideal of the group ring.

The Whitney trick attempts to remove a pair of cancelling intersection points by a disk joining them. In dimension four that auxiliary disk can itself intersect the original surfaces, so an iterative construction is needed. The established disk theorem works for good groups, a class defined by the ability to perform appropriate immersed-disk operations with nullhomotopic double-point loops. The manuscripts import that theorem and its closure properties.

🏷️ Three different conclusions

Boundary-only obstruction claimed in the first manuscript

There is a compact connected oriented smooth four-manifold $M$ containing proper immersed disks $f_i$ with disjoint embedded boundary circles and framed sphere maps $g_i$ such that

$$\begin{array}{c}\lambda (f_i,g_j)={\delta }_{ij}1,\\ \lambda (g_i,g_j)=0,\tilde{\mu }(g_i)=0,\end{array}$$

but those boundary circles admit no pairwise disjoint proper locally flat disk fillings. No relative homotopy classes or output framings are prescribed. Each $g_i$ may be individually embedded, although different spheres may intersect. (OpenAI, 2026a)

The vanishing conditions concern the dual spheres. The theorem does not require the original disk intersections to vanish.

The marked-tensor manuscript claims a disk counterexample with the same algebraic input but excludes replacements preserving both boundary maps and the induced boundary framings. It also imposes no relative homotopy constraint. Its markings belong to the fixed construction; they do not add an extra condition on hypothetical replacements. The boundary-only manuscript makes the stronger assertion that discarding the output framing requirement still cannot help. (OpenAI, 2026b) (OpenAI, 2026a)

The third manuscript concerns aspherical spaces, whose universal covers are contractible. A finite classifying space for a group is a finite CW complex with that group and contractible universal cover. An oriented integral Poincaré duality group of dimension four satisfies the finiteness conditions for a projective resolution and

$$H^i(G;\mathbb{Z}G)=\{\begin{array}{ll}\mathbb{Z},& i=4,\\ 0,& i\ne 4,\end{array}$$

with trivial action on the final $\mathbb{Z}$. These are necessary duality properties of an oriented closed aspherical four-manifold.

Manifold-realization obstruction claimed in the companion

There is a finitely presented oriented integral Poincaré duality group $G$ of dimension four with a finite four-dimensional aspherical Poincaré complex, but no closed aspherical topological four-manifold has fundamental group $G$. (OpenAI, 2026c)

A Poincaré complex has the relevant homological duality; it need not be a manifold. This statement already includes a finite classifying space, so its obstruction is not a failure of finite presentation. It gives a counterexample in dimension four, not in every dimension.

🏷️ From hypothetical disks to smooth cuts

The disk constructions start with a smooth graph-type four-manifold and finitely many capped gropes. A grope is a tree of surfaces attached along paired curves; caps close the terminal curves. Its boundary word records an iterated commutator. The constructions install a finite list of triple commutator laws with independently chosen conjugating paths before hypothetical disks are introduced.

Deleting neighborhoods of the lower grope surfaces gives the actual disk problem. Shallow caps become the immersed input disks. Normal-circle tori over the other basis curves, compressed along upper stages, give algebraic dual spheres. Verifying their framings and matching every cancellation label in the exterior group is a substantial geometric part of the manuscripts; it is not replaced here by an ordinary intersection-number calculation.

Suppose the specified boundaries had disjoint embedded fillings. In the boundary-only construction, join each actual disk image back to the grope base by an annulus and alter the ambient map to a graph on this protected set. The boundary loop already lies in a subgraph whose fundamental group injects into that of the full graph. It therefore has a filling in that subgraph. Since the graph has no second homotopy, the old and new graph-valued fillings are homotopic relative boundary.

The graph’s neighborhood extension property extends this prescribed homotopy over the ambient manifold. Regular levels of graph edges can then be chosen away from the protected disk images. These levels are smooth three-dimensional walls with a prescribed torus boundary. Extra closed wall components are allowed in this construction; the cut complement need not be connected. Thus smooth differential forms are used away from the topological disks, without smoothing those disks or identifying them with the original caps. (OpenAI, 2026a)

The marked-tensor construction has a different cut arrangement: two vertex pieces joined by $2m$ connected walls, with $m\ge 5$ active walls. Mayer–Vietoris, after fixing specified loop evaluations, gives complementary tangent data on the two vertices. These distinct geometric arrangements should not be identified merely because their eventual obstructions both use at least five factors. (OpenAI, 2026b)

🏷️ Potentials and actual equations

Holonomy assigns a matrix to a based loop by parallel transport in a flat connection. The manuscripts use $S{L}_2$ holonomy and retain basepoints, so matrices are not quotiented by simultaneous conjugation. A formal deformation is a compatible power-series family near a reference connection.

Each wall supplies a potential, a function whose internal-coordinate gradient gives its flatness equations. A boundary-adjusted Chern–Simons transgression constructs this function. Stokes’ theorem relates the wall values to curvature on a vertex piece.

The equations must be retained as ideals. For example, the ideals $(x)$ and $(x^2)$ have the same zero set but define different quotients: the class of $x$ is nonzero and square-zero in the second. Replacing an ideal by its radical would destroy precisely this information.

In the boundary-only manuscript, let $a_i$ be three boundary holonomy coordinates, $y_i$ the remaining wall coordinates, and $P_i(a_i,y_i)$ its potential. Write $C_i^{+},C_i^{-}$ for independent copies and set

$$\begin{array}{r}\Psi (a,C)=\sum _i{P}_i(a_i,C_i^{+})\\ -\sum _i{P}_i(0,C_i^{-}).\end{array}$$

Restrictions from the cut region give a graph $C=Y(a,t)$. Let $v=({\partial }_C\Psi )(a,Y)$, differentiating before substitution, and let $\mathfrak{r}$ be the region’s deformation-equation ideal. The manuscript establishes

$$(v)=\mathfrak{r},\Psi (a,Y)\in {\mathfrak{r}}^2.$$

The first equality retains all equations, including nilpotents; the second records the curvature-squared term from Stokes. With $p_i(y_i)={\partial }_{a_{i,1}}{P}_i(0,y_i)$, the construction also supplies both cubic laws

$$\begin{array}{c}{a}_{i,1}{a}_{j,1}{a}_{k,1}\in (v),\\ p_i(Y_i^{-})p_j(Y_j^{-})p_k(Y_k^{-})\in (v).\end{array}$$

They are separate ideal memberships, not consequences of merely setting the potential to zero. Algebraic replacement and a correction of $Y$ make its potential vanish exactly while transporting both memberships. The corrected graph need not itself parametrize representations: the geometric information needed later has already been encoded in those identities. (OpenAI, 2026a)

In the marked-tensor manuscript, an algebraic-coordinate theorem instead preserves based holonomy on the full critical quotient. Its proof uses derived and shifted-symplectic constructions; this post treats that theorem and its formal comparison as inputs. The finite obstruction below can be understood after taking the resulting separate algebraic functions as given. (OpenAI, 2026b)

🏷️ The marked finite tensor obstruction

Let $K$ be a characteristic-zero field, $h_i$ separate coordinate blocks, $F_i(h_i)$ their algebraic potentials, and $b_i(h_i)$ perturbations on an active set $I$ of size at least five. The parameter ring is

$$T=K[t_i:i\in I]/(t_i^2:i\in I).$$

Each parameter squares to zero, but mixed products such as $t_i{t}_j{t}_k$ for distinct indices remain nonzero. Define

$$\begin{array}{c}{S}_0=\sum _i{F}_i,\\ S_t=S_0+\sum _{i\in I}{t}_i{b}_i.\end{array}$$

The forbidden data are two formal graphs $H_L(x,t)$ and $H_R(y)$ with $H_L(0,0)=0$ and $H_R(0)=0$, complementary injective central tangent maps, exact identities $S_t(H_L)=S_0(H_R)=0$, and, for every distinct active triple,

$$\begin{array}{rl}{t}_i{t}_j{t}_k& \in J_L,\\ (b_i{b}_j{b}_k)(H_R)& \in J_R.\end{array}$$

The centering permits nilpotent constant displacements of $H_L$ before setting the parameters to zero. Here $J_L$ and $J_R$ are generated by the internal gradient entries of $S_t$ and $S_0$, respectively, restricted after differentiation. There is no regular-sequence or isolated-critical-point assumption, nor a flatness assumption on these gradient quotients. (OpenAI, 2026b)

Algebraicity permits finite approximation and specialization to sufficiently large characteristic $p>2$, preserving the equations, membership witnesses, and nonzero tangent determinant. In that characteristic use the finite ring and form algebra

$$\begin{array}{c}{A}_p=\mathbb{k}[z_1,\dots ,z_N]/(z_1^p,\dots ,z_N^p),\\ {\Omega }_p=A_p\otimes \bigwedge ⟨d{z}_1,\dots ,d{z}_N⟩.\end{array}$$

Ordinary differentiation is well defined because $d(z_{\nu }^p)=0$. Besides $d$, use the Koszul differential $D=d{S}_0\wedge (-)$ and its perturbation $D_t=d{S}_t\wedge (-)$.

A smooth formal graph with equations $c_1,\dots ,c_r$ has normal form

$$U_c=\underset{j=1}{\overset{r}{\bigwedge }}{c}_j^{p-1}d{c}_j.$$

Its graph ideal annihilates it because $c_j^p=0$. Graph vanishing of the potential makes it a Koszul cycle; it is also ordinarily closed. Formal coordinate changes, including nilpotent parameter displacements, preserve the coordinate-power ideal by Frobenius.

Define the residue integral by extracting the coefficient of $z_1^{p-1}\cdots z_N^{p-1}$ in a top-degree form. It vanishes on exact forms. Complementary graphs give normal forms $U_0,V$ with nonzero pairing: concatenating their linear equations gives an invertible matrix $M$, and

$$\int U_0\wedge V=\pm \det (M{)}^p\ne 0.$$

Actual gradient memberships become actual boundaries. Contraction ${\iota }_{\nu }$ with a coordinate vector satisfies

$${\iota }_{\nu }{D}_t+D_t{\iota }_{\nu }=({\partial }_{z_{\nu }}{S}_t)\mathrm{id}.$$

Thus the source membership gives $t_i{t}_j{t}_k{U}_t=D_t{B}_{ijk}$, and the target membership gives $b_i{b}_j{b}_{k}V=D{C}_{ijk}$. The coefficients in a membership witness are not differentiated by $D_t$.

🏷️ Why five active factors suffice

Block separation makes Koszul homology a tensor product ${⨂}_i{H}_i$. Let $B_i$ be multiplication by $b_i$ on $H_i$, and $K_i=\ker B_i$. The target triple laws kill every $B_i{B}_j{B}_k[V]$. Splitting each factor into its kernel and a complement shows that $[V]$ is a sum of tensors having at most two active factors outside their kernels.

There is a second condition: ordinary $d$ induces a differential ${\delta }_i$ on $H_i$. The kernel $K_i$ need not be a subcomplex. A linear projection onto representatives for ${\delta }_i$-homology can nevertheless be chosen to send $K_i$ into $K_i\cap \ker {\delta }_i$, by choosing complements first inside $K_i\cap \ker {\delta }_i$ and then in the full complex.

Applying these projections changes the represented form by $dZ+DE$. Its pairing with the ordinarily closed Koszul cycle $U_0$ is unchanged. Every resulting pure tensor has at least $|I|-2\ge 3$ active factors killed by both $B_i$ and ${\delta }_i$.

For each such factor, the commutator $[{\delta }_i,B_i]$ is induced by $d{b}_i\wedge (-)$. A cycle representative $a_i$ therefore admits $w_i$ with $D_i{w}_i=-d{b}_i\wedge a_i$, giving

$$(D_i+t_{i}d{b}_i\wedge )(a_i+t_i{w}_i)=0$$

over $t_i^2=0$. Tensor three of these lifts. Since each perturbation acts in its own block, the product is an exact $D_t$-cycle over the three-parameter ring, including its mixed terms.

Pairing with $t_i{t}_j{t}_k{U}_t=D_t{B}_{ijk}$ gives

$$t_i{t}_j{t}_{k}P(t)=0.$$

The square-free monomials form a basis, so this forces $P(0)=0$. Applying the argument to each tensor summand contradicts the original nonzero pairing. This step specializes an actual boundary identity; it does not assume that homology commutes with a nonflat parameter specialization.

The boundary-only manuscript has a related Frobenius–Koszul theorem with paired positive and negative wall blocks and two cubic membership laws. Its separate algebraic replacement and geometric hypotheses must be checked for that theorem; the marked tensor theorem is not substituted for them. (OpenAI, 2026a) (OpenAI, 2026b)

🏷️ From a forbidden marked filling to a closed realization problem

The realization companion uses the marked construction’s full geometric package. It builds a smooth chamber $D$ with boundary $S$, free fundamental group $H$, and a specified hyperbolic basis for its middle homotopy module. Attaching three-cells along that basis gives a Poincaré pair $(P,S)$ with $P$ homotopy equivalent to a finite graph $BH$.

The retained marking is a map $S\to BH$. The companion proves that no compact topological four-manifold with identified boundary $S$ can be homotopy equivalent to $BH$ while extending that marking. An unmarked homotopy type alone would discard the boundary information that creates the obstruction. (OpenAI, 2026c)

A hypothetical topological filling is not presumed smooth. The proof smooths away from isolated points, takes cuts away from those points and the protected disks, and uses a form complex computing the filled cohomology: degree-zero functions are separately constant near each puncture, and positive-degree forms vanish near them. Stokes is applied coefficient by coefficient on compact smooth regions. Using the ordinary cohomology of the punctured space instead would change the required vertex data.

Reflection across boundary panels then closes the pair to a finite aspherical Poincaré complex $Q$. A separate nonpositively curved model supplies the external assembly results used later. Assuming $Q$ has a manifold model, a marked stable-realization argument makes the prescribed sphere pairs geometrically standard after connected sums with $S^2\times S^2$. An unspecified stable homeomorphism would not retain enough information.

The final removal argument caps a finite chamber union except at the central boundary and removes the standard inner pairs. Farther-out constructions have small images under a direction map. Upper grope pieces are first grouped over finitely many contractible direction patches, then separated inside disjoint neighborhoods of those groups. This controls the image of an entire resulting loop, even when that loop passes through many intersections. Small individual segments alone would not imply nullhomotopy.

The ensuing disk construction may range through the whole compact manifold. Surgery on the remaining framed Lagrangian spheres preserves the fundamental group because geometric duals kill complement meridians, and the surgery trace removes the remaining homology over $\mathbb{Z}H$. The universal cover is then simply connected and acyclic, hence contractible by the homological Whitehead theorem. This produces exactly the marked graph filling that was excluded. The full reflection, stable-realization, and controlled-removal proofs remain manuscript inputs here.

🏷️ Consequences and verification scope

The boundary-only paper deduces that the free group on two generators is not good, using subgroup and quotient closure of goodness. This does not identify the example’s fundamental group with that free group. It also concludes that unrestricted four-dimensional topological surgery and five-dimensional topological $s$-cobordism cannot both hold; that statement does not decide which assertion fails. The marked-tensor paper does not claim either separate implication without an additional theorem. (OpenAI, 2026a) (OpenAI, 2026b)

The selected-source audit covers all three introductions, the boundary-only cut interface and final assembly, the marked tensor theorem and its finite-complex proof after specialization, and selected chamber and realization interfaces. The detailed grope framing and label calculations, deformation-potential construction, algebraic replacement and formal comparison, specialization proof, and full realization arguments were not independently verified. The posts explain the manuscripts’ claims and proof dependencies; they do not constitute an independent verification of these major topological conclusions.

Links

• Family 304 also separates a local topological construction from a finite algebraic obstruction and tracks exactly which integral information survives a change of coefficients.
• Family 302 uses a different boundary-preserving construction; in both families, retaining the prescribed boundary data is essential to the stated conclusion.

🏷️ Key steps and proof outlines

Three proof transitions deserve extra attention: changing a tensor representative without changing its pairing, retaining mixed infinitesimal terms, and turning the manifold-realization construction into the forbidden marked filling. These are also places where an apparently harmless simplification would lose information.

A homotopy that repairs the kernel condition

For the boundary-only obstruction, work in one finite-dimensional block of Koszul cohomology. Let $\mathcal{D}$ be the differential induced by ordinary differentiation and $S$ multiplication by the perturbing function. A class in $N=\ker S$ does not automatically lift through a perturbation of the Koszul differential: the commutator with ordinary differentiation introduces $\mathcal{D}$ as well. The required subspace is

$$K=N\cap {\mathcal{D}}^{-1}N.$$

This means both $x$ and $\mathcal{D}x$ lie in $N$. It is a subcomplex because ${\mathcal{D}}^2=0$. The source constructs a chain map $r$ satisfying

$$r(N)=K,r{|}_K=1,1-r=\mathcal{D}h+h\mathcal{D}.$$

Here is the linear-algebra proof. Choose $N=K\oplus E$. If $\mathcal{D}e\in N$ for $e\in E$, then $e\in K$, so $e=0$. Thus $\mathcal{D}$ is injective on $E$ and $\mathcal{D}E\cap N=0$. Define $h(\mathcal{D}e)=e$, and define $h$ to vanish on $K$, $E$, and an arbitrary remaining vector-space complement. Then $Q=\mathcal{D}h+h\mathcal{D}$ has image in $E\oplus \mathcal{D}E$, is the identity there, and vanishes on $K$. It is an idempotent commuting with $\mathcal{D}$. Taking $r=1-Q$ proves the assertions. Its full image need not equal $K$; only its action on $N$ is prescribed.

For several blocks, tensor these chain maps. The homotopies telescope:

$$\begin{array}{rl}R& =r_1\otimes \cdots \otimes r_d,\\ \mathcal{H}& =\sum _b{r}_{<b}\otimes h_b\otimes 1_{>b},\\ 1-R& =\mathcal{D}\mathcal{H}+\mathcal{H}\mathcal{D},\end{array}$$

with the usual graded signs. Consequently the corrected tensor has the same pairing with an ordinarily closed normal cycle. The cubic annihilation laws ensure that each surviving tensor has at most two factors outside the individual kernels. Five active factors therefore leave three factors on which the perturbation can be lifted. This is the precise counting reason for the threshold five in this argument. (OpenAI, 2026a)

Why the three-parameter calculation has eight terms

Over $B=\mathbb{k}[s_1,s_2,s_3]/(s_1^2,s_2^2,s_3^2)$, suppose each selected block has a cycle $x_r$ and a correction $y_r$ satisfying

$${\delta }_r{y}_r=-d{p}_r\wedge x_r.$$

The complete lifted tensor is

$${\kappa }_s=(x_1+s_1{y}_1)\otimes (x_2+s_2{y}_2)\otimes (x_3+s_3{y}_3),$$

with unchanged factors kept in their original positions. Its expansion contains the constant term, three linear terms, three quadratic mixed terms, and one cubic term. The coefficient of a square-free monomial $s_A$ in its differential is

$$\delta {\kappa }_A+\sum _{r\in A}(d{p}_r\wedge {)}_r{\kappa }_{A\setminus \{r\}}.$$

Each differentiation of a $y_r$ cancels the corresponding perturbation term. Further perturbation in the same slot vanishes because $s_r^2=0$. This proves the cycle identity for every subset $A$, including the mixed terms. Keeping only the first-order corrections would leave precisely the quadratic and cubic equations unchecked.

The moving graph must also remain in the same finite coordinate quotient. Its coordinate change can have a nilpotent constant displacement. The manuscript first inverts the augmented map including the parameters, then reduces modulo their squares. In characteristic $\ell >2$, every element of $(s_1,s_2,s_3)$ has zero $\ell$-th power by Frobenius. Applying the same calculation to the forward and inverse maps shows that both preserve the ideal generated by the $\ell$-th powers of the coordinates. Thus the moving normal form and the lifted tensor can actually be paired in one ring. The final contradiction extracts the coefficient of $s_1{s}_2{s}_3$ from an actual boundary identity; it does not require homology to commute with specialization. (OpenAI, 2026a) (OpenAI, 2026b)

The final surgery calculation for manifold realization

The realization companion needs a finite geometric construction with the original central boundary marking. Its parameter order is essential. First fix the coordinate cover and the requested grope height; obtain a propagation bound $b$ for finite preparation. Next choose the inner radius $R$ to make directions sufficiently small. Finally choose an outer truncation $L$ containing the full $b$-buffer, and retain complete preparations before inserting complementary fillings. Thus no retained attaching region depends on a chamber beyond the truncation.

The prepared geometric duals kill meridians in the complement of the immersed Lagrangian spheres. General position gives surjectivity onto the ambient fundamental group, while those duals kill the kernel. Hence the complement has the same fundamental group. Null double-point loops are therefore null in the needed complement. Subsequent tower constructions use that fact and the protected duals; the manuscript does not assert a small diameter for their null homotopies or final disks.

After obtaining disjoint framed Lagrangian spheres, perform interior surgery. To see exactly what disappears, let there be $m$ hyperbolic planes and let $W$ be the five-dimensional surgery trace. With regular coefficients $\mathbb{Z}H$, the relative chain module of $(W,Z)$ is $(\mathbb{Z}H{)}^m$ in degree three. Its boundary injects as the Lagrangian half of $H_2(Z;\mathbb{Z}H)$. Therefore $H_3(W;\mathbb{Z}H)=0$, while $H_2(W;\mathbb{Z}H)$ is the remaining dual half.

Read the trace from its outgoing boundary $V$. The relative module is now in degree two, and the map

$$H_2(W;\mathbb{Z}H)⟶H_2(W,V;\mathbb{Z}H)$$

is the intersection pairing of the dual half with the surgered Lagrangian. It is an isomorphism. The relative exact sequence kills every positive-degree homology group of the universal cover of $V$. That cover is simply connected and has CW homotopy type, so it is contractible. All surgeries were interior, preserving the central boundary marking. This produces exactly the marked graph-type filling previously ruled out, which closes the contradiction. (OpenAI, 2026c)

🏷️ Manuscripts and verification scope

• A boundary-only obstruction to four-dimensional disk embedding — pinned manuscript and companion source.
• A marked tensor obstruction to four-dimensional disk embedding — pinned manuscript and companion source.
• A PD4 group without an aspherical manifold model — pinned manuscript and companion source.

No family scope page was present at lean/docs/305.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A boundary-only obstruction to four-dimensional disk embedding.

🐻  OpenAI 2026b. A marked tensor obstruction to four-dimensional disk embedding.

🐻  OpenAI 2026c. A PD4 group without an aspherical manifold model.