# Equations over groups and the Kervaire conjecture

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

Take a nontrivial group \(A\), add one new generator \(t\), and impose one relation \(w=1\), where \(w\) is any word in \(t\) and elements of \(A\). Can the resulting group be trivial? The *Kervaire conjecture* says that it cannot. It goes back to Kervaire's 1965 description of the fundamental groups of knotted spheres in high dimensions. OpenAI proved it in 2026, by showing that when the total exponent of \(t\) in \(w\) is \(\pm1\), no element of \(A\) other than \(1\) becomes trivial [OpenAI-K]. This course proves that theorem. This lesson states it, deduces the conjecture from it, explains what is known around it, and lays out the proof.

- Equations over groups (Section 1).
- The theorem and the Kervaire conjecture (Section 2).
- Examples and the role of the exponent sum (Section 3).
- Context (Section 4).
- The plan of the proof (Section 5).

We use free products of groups and their universal property [Hatcher, Section 1.2]: for groups \(A\) and \(C\) there is a group \(A*C\) with homomorphisms \(A\to A*C\) and \(C\to A*C\) such that every pair of homomorphisms \(A\to B\), \(C\to B\) extends uniquely to a homomorphism \(A*C\to B\). The group \(A*C\) is generated by the images of \(A\) and \(C\).

## 1. Equations over groups

Let \(A\) be a group and \(\langle t\rangle\cong\mathbb Z\) an infinite cyclic group, and put \(H=A*\langle t\rangle\). An element \(w\in H\) is a product of elements of \(A\) and powers of \(t\); we read it as a word in an unknown \(t\) with coefficients in \(A\). For a group \(B\), a homomorphism \(\varphi:A\to B\) and \(b\in B\), write \(w(b)\in B\) for the image of \(w\) under the homomorphism \(H\to B\) that is \(\varphi\) on \(A\) and sends \(t\) to \(b\).

**Definition 1.1.** The equation \(w(t)=1\) is *solvable over* \(A\) if there are a group \(B\), an injective homomorphism \(\varphi:A\to B\) and an element \(b\in B\) with \(w(b)=1\).

A solution need not lie in \(A\) itself: the group may have to be enlarged. Let \(\langle\!\langle w\rangle\!\rangle_H\) be the normal closure of \(w\) in \(H\), the smallest normal subgroup containing \(w\), and put \(Q=H/\langle\!\langle w\rangle\!\rangle_H\). This is the group obtained from \(A\) by adding the generator \(t\) and the relation \(w=1\).

**Lemma 1.2.** The equation \(w(t)=1\) is solvable over \(A\) if and only if the composite \(\kappa:A\to H\to Q\), the *coefficient map*, is injective.

*Proof.* If \(\kappa\) is injective, take \(B=Q\), \(\varphi=\kappa\) and \(b\) the image of \(t\); then \(w(b)\) is the image of \(w\) in \(Q\), which is \(1\). Conversely, let \(\varphi:A\to B\) be injective and \(w(b)=1\). The homomorphism \(\psi:H\to B\) extending \(\varphi\) and \(t\mapsto b\) kills \(w\), hence its normal closure, so it factors through a homomorphism \(\bar\psi:Q\to B\) with \(\bar\psi\circ\kappa=\varphi\). Since \(\varphi\) is injective, so is \(\kappa\). \(\square\)

## 2. The theorem and the Kervaire conjecture

Let \(p:H\to\mathbb Z\) be the homomorphism that is trivial on \(A\) and sends \(t\) to \(1\); \(p(w)\) is the total exponent of \(t\) in \(w\), its *exponent sum*. A word \(w\) is *unimodular* if \(p(w)=\pm1\).

**Theorem 2.1** (OpenAI [OpenAI-K, Theorem 1.1]). For every group \(A\) and every unimodular \(w\in A*\langle t\rangle\), the coefficient map
\[
A\longrightarrow(A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle
\]
is injective. Equivalently, every equation \(w(t)=1\) over \(A\) whose exponent sum is \(\pm1\) is solvable over \(A\).

No assumption is made on \(A\): it may have torsion, be uncountable, or have no finite presentation. OpenAI's release contains a formalization of this statement in Lean [OpenAI-K-Lean].

**Corollary 2.2** (the Kervaire conjecture). If \(A\ne1\), then \((A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle\ne1\) for every \(w\in A*\langle t\rangle\).

*Proof.* If \(p(w)=\pm1\), the nontrivial group \(A\) embeds in the quotient by Theorem 2.1. Otherwise let \(d=|p(w)|\), so \(d=0\) or \(d\ge2\). The homomorphism \(p\) followed by \(\mathbb Z\to\mathbb Z/d\mathbb Z\) kills \(w\), hence induces a homomorphism from the quotient onto \(\mathbb Z/d\mathbb Z\), which is nontrivial; here \(\mathbb Z/0\mathbb Z=\mathbb Z\). \(\square\)

The *weight* of a group is the least number of elements whose normal closure is the whole group. The quotient by \(w\) is trivial exactly when \(w\) normally generates \(A*\langle t\rangle\), so Corollary 2.2 says: *the free product of a nontrivial group with \(\mathbb Z\) has weight at least two*.

## 3. Examples and the role of the exponent sum

**Example 3.1** (exponent sum zero can kill coefficients). Let \(A=\mathbb Z/6\mathbb Z\), written multiplicatively with generator \(c\), and \(w=tc^2t^{-1}c^{-3}\), with \(p(w)=0\). In the quotient \(Q\), conjugation by \(t\) carries \(c^2\), whose cube is \(1\), to \(c^3\), so \((c^3)^3=c^9=c^3\) equals \(1\) in \(Q\). Then \(c^2=t^{-1}c^3t=1\) as well, and \(c=c^3c^{-2}=1\). So the coefficient map kills all of \(A\), and \(Q\cong\mathbb Z\), generated by the image of \(t\). The quotient is still nontrivial, as Corollary 2.2 says, but only because of the exponent sum.

**Example 3.2** (unimodular words). For \(w=at\) the solution \(t=a^{-1}\) lies in \(A\), and \(Q\cong A\) (Exercise 6.1). For \(w=tat^{-1}bt\) with \(a,b\in A\), also with \(p(w)=1\), no formula for a solution is apparent; Theorem 2.1 gives one in some overgroup of \(A\), whatever \(a\) and \(b\) are. Whether a solution exists inside \(A\) itself is a different question [KMO].

**Example 3.3** (larger exponent sums). Coefficient injectivity for every \(w\) with \(p(w)\ne0\) is the *Kervaire–Laudenbach conjecture* [OpenAI-K, Section 1.1]. Theorem 2.1 is its case \(p(w)=\pm1\). The companion manuscript [OpenAI-H, Theorem 1] proves injectivity for systems of equations whose exponent-sum matrix has full rank over \(\mathbb Q\); one equation in one unknown with \(p(w)\ne0\) is such a system, so it settles the Kervaire–Laudenbach conjecture as well. The course [Nonsingular systems of equations over groups](course:nonsingular-systems-over-groups/systems-of-equations-over-groups) proves that theorem with the methods of this course. Corollary 2.2 needs neither: for \(|p(w)|\ge2\) the quotient maps onto \(\mathbb Z/|p(w)|\mathbb Z\).

## 4. Context

Kervaire showed that for \(n\ge3\) a group is the fundamental group of the complement of a smooth \(n\)-sphere in the \((n+2)\)-sphere exactly when it is finitely presented, has abelianization \(\mathbb Z\), trivial second homology, and weight one [Kervaire 1965, Théorème 1.1]. Whether a free product \(A*\mathbb Z\) with \(A\ne1\) can have weight one arose there, and Chen traces the group-theoretic problem to this setting [Chen, Section 1].

Gerstenhaber and Rothaus proved that a system of equations over a finite group whose exponent-sum matrix has nonzero determinant has a solution in a finite overgroup [Gerstenhaber–Rothaus 1962, Theorem 2]; for one equation in one unknown the condition is that the exponent sum is nonzero. Their proof embeds the finite group in a unitary group and computes the degree of a word map on a compact Lie group; [Word maps on unit quaternions and degree](course:eilenberg-ganea-counterexample/word-maps-on-unit-quaternions-and-degree) proves a localized version of this computation for the unit quaternions. A coefficient \(g\ne1\) of a residually finite group survives in a finite quotient, where the equation is solvable, so Theorem 2.1 holds for residually finite \(A\) by this route. The unitary method extends to hyperlinear groups, those embedding in metric ultraproducts of unitary groups [Pestov, Section 10, Corollary 10.4], and Nitsche and Thom gave a criterion for solvability over such groups that goes beyond nonzero exponent sums [Nitsche–Thom]. Klyachko and Thom developed further topological methods of this kind [Klyachko–Thom].

A second line of work uses diagrams. Klyachko proved Theorem 2.1 for torsion-free groups in 1993, and Chen gave a new proof by estimating the complexity of surfaces in HNN extensions [Chen, Section 1]; for every group, injectivity holds when the relator is a proper power \(w^m\), \(m\ge2\), of a unimodular word, by a theorem of Klyachko and Lurye that Chen generalizes [Chen, abstract; Klyachko–Lurye]. Klyachko showed that the Kervaire–Laudenbach conjecture is equivalent to a statement about presentations of simple groups [Klyachko 2004]. Kawauchi proposed a resolution through ribbon sphere-links [Kawauchi]. Osin and Thom had conjectured a bound on the first \(\ell^2\)-Betti number of a torsion-free group by its weight minus one, which would have implied the torsion-free case; Fisher and Lodha disproved that conjecture in 2026 [Fisher–Lodha]. Klyachko, Mikheenko and Olshanskii study when unimodular equations are solvable inside the group itself rather than in an overgroup [KMO].

OpenAI's proof combines the two lines: a diagram, in the form of a planar surface built from disks and bands, is measured by unitary operators, but the unitary group is that of the group von Neumann algebra of \(A\) itself, which exists for every group. No approximation of \(A\) by matrices is needed. A companion manuscript extends the method to systems of equations whose exponent-sum matrix has full rank over \(\mathbb Q\), proving a conjecture of Howie [OpenAI-H].

## 5. The plan of the proof

Suppose that \(g\in A\) becomes trivial in \(Q\). Then \(g\) is a product of conjugates of \(w^{\pm1}\), and this can be drawn on a disk with holes: each hole is bounded by a copy of \(w\) or \(w^{-1}\), and the outer boundary reads \(g\). Joining the occurrences of \(t\) in pairs by bands leaves boundary curves that read words in \(A\). The key statement is the *planar boundary theorem*: on a connected planar surface made of such disks and bands, if all boundary words but one are trivial in \(A\), so is the last one.

- [The regular trace and spectral phase](the-regular-trace-and-spectral-phase.md): a faithful trace on matrices over the group von Neumann algebra of \(A\), and the subadditivity of the trace of the spectral phase, \(\ell(UV)\le\ell(U)+\ell(V)\).
- [A unitary with a large fixed space](a-unitary-with-a-large-fixed-space.md): for signs with sum \(\pm1\), every unitary \(P\) can be corrected by a block-diagonal unitary built from one matrix \(X\) and its inverse so that the product fixes a subspace of the expected dimension; the proof is a degree computation modulo two.
- [Disks, bands and boundary words](disks-bands-and-boundary-words.md): the combinatorics of word disks and bands, the boundary words, and the Euler characteristic count.
- [The planar boundary theorem](the-planar-boundary-theorem.md): an operator that follows the boundary circles is written as a product of two unitaries; subadditivity and the fixed space give an upper bound for its phase, a computation circle by circle gives the same bound plus the phase of the exceptional boundary word, so that word is trivial.
- [From a relation to a planar surface](from-a-relation-to-a-planar-surface.md): a relation \(g\in\langle\!\langle w\rangle\!\rangle\) gives a map of a disk with holes into a space with fundamental group \(A*\langle t\rangle\); a level set of the circle coordinate cuts out the bands, and the planar components are removed from the inside out. This proves Theorem 2.1.

## 6. Exercises

**6.1** (easy). Show that the quotient of \(A*\langle t\rangle\) by \(w=at\) is isomorphic to \(A\), for every \(a\in A\).

**6.2** (easy). Let \(A=\mathbb Z/2\mathbb Z=\{1,c\}\) and \(w=tct^{-1}c\). Find the quotient \(Q\) and decide whether the coefficient map is injective.

**6.3** (medium). Let \(A\) be a finite group and \(w\) unimodular. Using only the statement of [Gerstenhaber–Rothaus 1962, Theorem 2] quoted in Section 4, prove Theorem 2.1 for residually finite groups \(A\).

**6.4** (easy). Show that the conclusion of Corollary 2.2 can be stated as: no free product \(A*\mathbb Z\) with \(A\ne1\) is normally generated by one element.

**6.5** (easy). Show that the coefficient maps for \(w\), for its conjugates \(hwh^{-1}\) with \(h\in A*\langle t\rangle\), and for \(w^{-1}\) are the same map, so that Theorem 2.1 may be applied to any of them.

## 7. Solutions

**6.1.** The homomorphism \(A*\langle t\rangle\to A\) that is the identity on \(A\) and sends \(t\) to \(a^{-1}\) kills \(at\), so it induces \(Q\to A\); composed with the coefficient map it is the identity of \(A\). The coefficient map is also surjective, since \(t=a^{-1}\) in \(Q\). So it is an isomorphism.

**6.2.** The relation says \(tct^{-1}=c^{-1}=c\), so \(t\) commutes with \(c\) and \(Q\cong\mathbb Z/2\mathbb Z\times\mathbb Z\); the coefficient map is injective. Here \(p(w)=0\), so injectivity is not guaranteed in general (Example 3.1), but it holds in this case.

**6.3.** Let \(g\in A\), \(g\ne1\), and let \(N\) be a normal subgroup of finite index with \(g\notin N\). The equation obtained by mapping the coefficients of \(w\) to \(A/N\) has exponent sum \(\pm1\ne0\), so by Gerstenhaber and Rothaus there are a finite group \(B\supseteq A/N\) and \(b\in B\) solving it. The homomorphism \(A*\langle t\rangle\to B\), \(A\to A/N\subseteq B\), \(t\mapsto b\), kills \(w\), so it factors through \(Q\), and it sends \(g\) to \(gN\ne1\). Hence the image of \(g\) in \(Q\) is not \(1\).

**6.4.** \(Q=1\) exactly when \(\langle\!\langle w\rangle\!\rangle=A*\langle t\rangle\), that is, when \(w\) normally generates the free product.

**6.5.** The normal closures of \(w\), \(hwh^{-1}\) and \(w^{-1}\) are the same subgroup, so the quotients and the coefficient maps coincide.

## References

- [OpenAI-K] OpenAI, *The Kervaire theorem for groups*, OpenAI Math Release preprint, 24 September 2026, Section 1. https://github.com/openai/math/tree/main/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026
- [OpenAI-K-Lean] OpenAI, Lean formalization of coefficient injectivity for unimodular one-relator quotients, OpenAI Math Release, scope note `lean/docs/256.md`. https://github.com/openai/math/blob/main/lean/docs/256.md
- [OpenAI-H] OpenAI, *Nonsingular systems of equations over arbitrary groups*, OpenAI Math Release preprint, 5 October 2026. https://github.com/openai/math/tree/main/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026
- [Kervaire 1965] M. A. Kervaire, *Les nœuds de dimensions supérieures*, Bulletin de la Société Mathématique de France 93 (1965), 225–271, Théorème 1.1; Numdam. https://www.numdam.org/item/10.24033/bsmf.1624.pdf
- [Gerstenhaber–Rothaus 1962] M. Gerstenhaber and O. S. Rothaus, *The solution of sets of equations in groups*, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1531–1533, Theorem 2; free on PubMed Central. https://pmc.ncbi.nlm.nih.gov/articles/PMC220992/
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