Tarcha: Braid Theory and the Artin Presentation with Explicit Half-Twist GeneratorsTextbook
## Motivation
A **braid** on $n$ strands is the everyday object it sounds like: $n$ strings hanging between two
horizontal plates, each string descending monotonically, no two strings meeting. Emil Artin turned
this picture into algebra in 1925 by showing that braids form a group under concatenation and that
this group has a finite presentation with $n-1$ generators. The braid groups sit at the crossroads
of low-dimensional topology (they are the mapping class groups of punctured discs, and closures of
braids produce every link), of algebra (they are the prototypical Artin–Tits groups, torsion-free
and orderable), and of representation theory and mathematical physics through the Burau, Lawrence–
Krammer and Temperley–Lieb representations.
This mission formalizes the braid-group development of a 2023 master's dissertation, Alexsander
Andrey Gomes Tarcha's *Um Estudo Introdutório da Teoria de Tranças* (UNESP, Rio Claro), whose
capstone is Teorema 3.15: the braid group on $n$ strands admits Artin's presentation. The
dissertation builds the group structure on equivalence classes of geometric braids (Teorema 3.9),
shows that the Artin generators generate (Teorema 3.11), derives the braid and commutation
relations (Proposição 3.14), establishes the presentation (Teorema 3.15), and closes with two
structural properties: the full twist is central (Proposição 3.16) and $B_m$ embeds in $B_n$ for
$m \le n$ (Proposição 3.17).
## Setting
Work in the plane $E^2 = \mathbb{C}$. The **ordered configuration space**
$$F_{0,n}E^2 = \{(z_1,\dots,z_n) \in \mathbb{C}^n : z_k \neq z_l \text{ for } k \neq l\}$$
carries the subspace topology of $\mathbb{C}^n$, and the symmetric group $\Sigma_n$ acts on it by
permuting coordinates. The **unordered configuration space** $B_{0,n}E^2 = F_{0,n}E^2/\Sigma_n$
carries the quotient topology; its points are the $n$-element subsets of the plane. The base
configuration is $(1,2,\dots,n)$, and $*$ denotes its class in $B_{0,n}E^2$. The **geometric braid
group** is
$$\pi_1\bigl(B_{0,n}E^2, *\bigr),$$
a loop of $n$-point configurations being exactly a geometric braid, and homotopy of loops being
exactly the equivalence by elementary moves used in the dissertation.
The **elementary half-twist** $\sigma_{i+1}$, for $0 \le i \le n-2$, is the loop that rotates the
two base points $i+1$ and $i+2$ by the angle $\pi$ about their midpoint $i + \tfrac32$, leaving the
other $n-2$ points fixed:
$$t \;\longmapsto\; \Bigl\{\, i+\tfrac32 \pm \tfrac12 e^{\pi i t} \,\Bigr\} \;\cup\;
\{\,k+1 : k \neq i,\, i+1 \,\}, \qquad t \in [0,1].$$
It returns to the base configuration at $t = 1$ with the two moving points interchanged, so it is a
loop in $B_{0,n}E^2$ and defines a class in $\pi_1(B_{0,n}E^2,*)$.
The **abstract braid group** $B_n$ is the group presented by generators $\sigma_1,\dots,\sigma_{n-1}$
subject to
$$\sigma_i\sigma_j = \sigma_j\sigma_i \quad (|i-j| \ge 2), \qquad
\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1} \quad (1 \le i \le n-2).$$
## Formalization targets
### Goal — Teorema 3.15, with the isomorphism pinned on generators
$$\exists\, \varphi : B_n \;\xrightarrow{\ \sim\ }\; \pi_1\bigl(B_{0,n}E^2,*\bigr), \qquad
\varphi(\sigma_{i+1}) = \bigl[\text{half-twist}_i\bigr] \ \ (0 \le i \le n-2).$$
This is the statement the dissertation actually proves: the map $\varphi$ of its proof is defined on
generators by $\varphi(x_i) = [\sigma_i]$, and the work consists in showing that it is a
well-defined homomorphism which is surjective and injective. Asking only for an abstract
isomorphism would leave the generators unconstrained; naming their images is what makes the
presentation usable downstream.
### Milestones
$$\langle\,[\text{half-twist}_i]\,\rangle = \pi_1\bigl(B_{0,n}E^2,*\bigr) \qquad \text{(Teorema 3.11)}$$
$$[\text{half-twist}_i][\text{half-twist}_j] = [\text{half-twist}_j][\text{half-twist}_i]\ (|i-j|\ge 2),
\qquad
[\text{ht}_i][\text{ht}_{i+1}][\text{ht}_i] = [\text{ht}_{i+1}][\text{ht}_i][\text{ht}_{i+1}]$$
$$\forall b \in B_2,\ \exists m \in \mathbb{Z},\ b = \sigma_1^m \qquad \text{(Proposição 3.13)}$$
$$\forall b \in B_3,\ b = \sigma_1^{a_1}\sigma_2^{b_1}\cdots\sigma_1^{a_m}\sigma_2^{b_m}
\qquad \text{(Proposição 3.14)}$$
$$(\sigma_1\sigma_2\cdots\sigma_{n-1})^n \in Z(B_n) \qquad \text{(Proposição 3.16)}$$
$$B_m \hookrightarrow B_n \ \ (m \le n) \qquad \text{(Proposição 3.17)}$$
## Significance
Artin's presentation is what makes the braid groups computable: the word problem, the Burau and
Lawrence–Krammer representations, the Markov moves on braid closures, and the Garside normal form
all start from generators and relations, while the topological side supplies the meaning of those
generators. A formalization that only exhibits an abstract isomorphism cannot be used to compute
with a given geometric braid; the version stated here transports each half-twist loop to a
generator word, which is what downstream work needs.
On the formalization side, the geometric model (configuration spaces and their fundamental groups)
and the algebraic model (a presented group) are already available on the platform in this
environment, and the abstract form of Artin's theorem is already stated there as an open problem.
What this mission adds is: the elementary half-twist as an explicit, machine-checked loop in
$B_{0,n}E^2$ — this definition is proved sorry-free here, including the injectivity of the moving
configuration at every time and the continuity of the path; the sharpened goal that fixes the
isomorphism on generators; and the dissertation's supporting results, none of which is currently
on the platform. None of the milestones or the goal has a machine-checked proof yet.
## Difficulty
Surjectivity of $\varphi$ — every braid is a product of half-twists — is a compactness-and-general-
position argument in the dissertation: cut the braid into finitely many slabs in which a single
crossing occurs. Turning that into a formal proof requires the homotopy-theoretic substitute, since
"general position" is not available for free: a loop of configurations must be subdivided and each
piece pushed to a standard crossing.
Injectivity is harder and is the step where a naive approach fails. It is not enough to check that
the relations hold; one has to know that they are *all* the relations, i.e. that a word whose braid
is null-homotopic is a consequence of the braid relations. The dissertation follows the classical
route through elementary moves on braid diagrams (its Figuras 3.22–3.26), which formalizes as a
long case analysis. The standard modern alternative is the Fadell–Neuwirth fibration together with
an induction on $n$; its inductive step needs the exact sequence of the fibration
$F_{0,n}E^2 \to F_{0,n-1}E^2$, which is itself substantial work.
## Formalization scope
The plane is $\mathbb{C}$; configurations are injective tuples indexed by `Fin n`; the unordered
configuration space is the quotient by the coordinate-permutation action with the quotient
topology; the base configuration is $(1,2,\dots,n)$ (not $(0,1,\dots,n-1)$). Braid generators are
indexed by `Fin (n-1)`, the index $i$ standing for the book generator $\sigma_{i+1}$; truncated
natural subtraction means the degenerate values $n = 0, 1$ give the trivial group, and the
statements are asserted for all $n$ including those cases. The abstract braid group is the
presented group on `Fin (n-1)` modulo the normal closure of the commutation and braid relators.
The half-twist rotates counterclockwise. The mirror symmetry $z \mapsto \bar z$ fixes the base
configuration and exchanges the two orientations, so the goal statement does not depend on this
choice; a solver may use either convention internally.
The goal cannot be satisfied trivially: it asks for a group isomorphism whose values on the
generators are the prescribed classes of explicit loops, so neither the identity on a presented
group nor an abstract counting argument suffices.
Contributions welcome beyond the milestones: the Fadell–Neuwirth exact sequence for the plane, the
pure braid group as the kernel of the map to $\Sigma_n$, the exponent-sum homomorphism, and
torsion-freeness of $B_n$. The half-twist definition published with this mission is reusable for
any further work on braids in this environment.
## Selected references
- Alexsander Andrey Gomes Tarcha, *Um Estudo Introdutório da Teoria de Tranças*, master's
dissertation, UNESP Rio Claro, 2023. https://repositorio.unesp.br/items/9d2ffbf0-8bd2-4ec7-9e45-e2cee8a1b202
- Emil Artin, *Theorie der Zöpfe*, Abhandlungen aus dem Mathematischen Seminar der Universität
Hamburg 4 (1925), 47–72. https://doi.org/10.1007/BF02950718
- Joan S. Birman, *Braids, Links, and Mapping Class Groups*, Annals of Mathematics Studies 82,
Princeton University Press, 1974. https://doi.org/10.1515/9781400881420
- Edward Fadell and Lee Neuwirth, *Configuration spaces*, Mathematica Scandinavica 10 (1962),
111–118. https://doi.org/10.7146/math.scand.a-10517
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Braids, Links and Mapping Class Groups I: Artin's Presentation of the Braid GroupTextbook
## Motivation
The **braid group** is one of the places where group theory, low-dimensional topology and
knot theory meet. Artin introduced it in 1925 (E. Artin, *Theorie der Zöpfe*, Abh. Math. Sem.
Univ. Hamburg 4 (1925), 47–72) and returned to it in 1947; since then it has become standard
equipment in the study of links (closed braids and Markov's theorem), of mapping class groups
of punctured surfaces, and of configuration spaces. Birman's *Braids, Links, and Mapping Class
Groups* (Annals of Mathematics Studies 82, Princeton University Press, 1974) is the classical
reference that develops all three subjects from the braid group outwards, and its Chapter 1
is the foundation on which the rest of the book rests.
The chapter's structure is itself the reason to formalize it first: everything later in the
book — the closed-braid picture of links, Markov's theorem, the Magnus representations, the
mapping class group of the punctured sphere — is phrased in terms of the group
$\pi_1 B_{0,n}E^2$ and of the presentation established here. A mission that fixes faithful Lean
definitions of the configuration spaces and of the abstract braid group therefore fixes the
vocabulary for the whole series.
Timeline of the results collected here: Artin (1925) gave the presentation and the
characterization of braid automorphisms of a free group; Chow (1948) determined the centre;
Fadell–Neuwirth (1962) introduced the configuration-space fibrations, and Fadell–Van Buskirk
(1962) used them to give the proof of the presentation reproduced by Birman.
## Setting
Write $E^2$ for the Euclidean plane, identified throughout with the complex numbers $\mathbb{C}$.
For $n \ge 0$ let
$$F_{0,n}E^2 = \{\,(z_1,\dots,z_n) \in \mathbb{C}^n : z_i \neq z_j \text{ for } i \neq j\,\}$$
be the **ordered configuration space** of $n$ points in the plane, topologized as a subspace of
$\mathbb{C}^n$. The symmetric group $\Sigma_n$ acts on it by permuting coordinates; the quotient
$$B_{0,n}E^2 = F_{0,n}E^2 / \Sigma_n,$$
with the quotient topology, is the **unordered configuration space**. A point of $B_{0,n}E^2$ is
an unordered set of $n$ distinct points of the plane. The base configuration is
$\bar z^{\,0} = (1,2,\dots,n)$, and all fundamental groups below are taken at $\bar z^{\,0}$ or at
its image.
The **braid group of the plane** is $\pi_1 B_{0,n}E^2$: a loop is a motion of $n$ points of the
plane returning to the same set of points, and homotopy classes of such motions compose as
braids. The **pure braid group** is $P_n = \pi_1 F_{0,n}E^2$, the subgroup of motions returning
each point to its own starting position.
Separately, let $B_n$ denote the abstract group given by generators $\sigma_1,\dots,\sigma_{n-1}$
subject to
$$\sigma_i\sigma_j = \sigma_j\sigma_i \quad (|i-j| \ge 2), \qquad
\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1} \quad (1 \le i \le n-2).$$
These are equations (1-1) and (1-2) of the book (p. 11). Geometrically $\sigma_i$ interchanges the
$i$-th and $(i+1)$-st points along a semicircle.
## Formalization targets
### Goal — Theorem 1.8 (Artin, 1925; Birman p. 18)
$$B_n \;\cong\; \pi_1 B_{0,n} E^2 .$$
The group of motions of $n$ points of the plane is the group with generators
$\sigma_1,\dots,\sigma_{n-1}$ and the two families of relations above: the relations are not only
valid but *defining*.
### Milestones
The milestone list follows the chapter: the covering-space description of the projection
$F_{0,n}E^2 \to B_{0,n}E^2$ (Proposition 1.1, p. 11), the Fadell–Neuwirth exact sequence
(Theorem 1.4, p. 14), the semidirect-product decomposition of the pure braid group
(Corollary 1.8.1, p. 24), the faithful representation of $B_n$ by automorphisms of a free group
(Corollary 1.8.3, p. 25), the centre of $B_n$ (Corollary 1.8.4, p. 28, due to Chow), and Artin's
algebraic characterization of the braid automorphisms (Theorem 1.9, p. 30).
## Significance
Theorem 1.8 is what makes the braid group *computable*: with defining relations in hand one can
combine braids into the normal form of Corollary 1.8.2 and solve the word problem, represent
braids by automorphisms of a free group, and pass to the link-theoretic material of Chapters 2
and 5 where braid words, not motions, are the objects manipulated. Corollary 1.8.3 turns braids
into concrete data — a braid is determined by what it does to the generators of a free group —
and Theorem 1.9 says exactly which endomorphisms arise this way; both are the algebraic engine
behind the conjugacy-problem and Magnus-representation chapters.
For formalization the state of play is that Mathlib has free groups, presented groups, the
fundamental groupoid and fundamental group, covering maps and fibre bundles, but no braid
groups and no configuration spaces: nothing here can be assembled from existing declarations.
The mission therefore produces reusable infrastructure — configuration spaces of the plane, the
symmetric-group quotient, the Artin presentation, the Artin action on a free group — as well as
machine-checked proofs of results that are classical but, as far as the mission's search of the
library showed, not yet formalized in Mathlib.
## Difficulty
The generators and relations are easy to write down and easy to verify *in* $\pi_1 B_{0,n}E^2$;
what is hard is completeness, i.e. that no further relations are needed. The naive route —
draw the braid, push it into a normal form by hand — is exactly what a formal proof cannot do.
The Fadell–Van Buskirk argument reproduced by Birman instead runs an induction on $n$ driven by
the fibration $F_{0,n}E^2 \to F_{0,n-1}E^2$: its homotopy exact sequence gives a split extension
of $P_{n-1}$ by a free group, presentations are assembled along the extension, and finally the
covering $F_{0,n}E^2 \to B_{0,n}E^2$ with deck group $\Sigma_n$ transfers the answer from the pure
braid group to the full braid group. Each of those steps needs genuine algebraic topology —
local triviality of the projection, exactness of the homotopy sequence, freeness of
$\pi_1$ of a punctured plane — which is where the formalization work actually lies.
## Formalization scope
The plane is $\mathbb{C}$. $F_{0,n}E^2$ is the subtype of injective functions
$\mathrm{Fin}\,n \to \mathbb{C}$; $B_{0,n}E^2$ is its quotient by the equivalence "differ by
precomposition with a permutation", with the quotient topology. Base point: the configuration
$i \mapsto i+1$, i.e. $(1,2,\dots,n)$, and its image. Fundamental groups are Mathlib's
`FundamentalGroup` at those base points. Braid generators are indexed by $\mathrm{Fin}(n-1)$
with $0$-based indices ($i$ stands for $\sigma_{i+1}$), and free-group generators by
$\mathrm{Fin}\,n$; the abstract braid group is a `PresentedGroup` on that index set. Truncated
subtraction makes the generator set empty for $n \le 1$, so $B_0$ and $B_1$ are trivial, as
intended. Two milestones are stated with the shift $n \mapsto n+1$ (i.e. for the projection
$F_{0,n+1}E^2 \to F_{0,n}E^2$) to avoid truncated subtraction in the maps.
Two conventions are worth flagging because they weaken what the Lean text asserts relative to
the prose. First, the goal asserts the existence of *some* isomorphism $B_n \cong \pi_1B_{0,n}E^2$;
it does not pin the isomorphism down on the geometric generators of Figure 2, since those loops
are not part of the formal development. Second, Artin's representation is formalized as the
existence of a homomorphism $\xi$ from $B_n$ to the automorphism group of the free group whose
value on each $\sigma_i$ is the explicit endomorphism of equation (1-14), together with its
injectivity; Theorem 1.9 is then stated for an arbitrary such $\xi$, given as a hypothesis, and
is non-vacuous precisely because Corollary 1.8.3 supplies one.
No trivializing formalization is available: the goal is an isomorphism statement between two
groups that are both defined independently of it, and the degenerate cases $n \le 1$ (both sides
trivial) are genuine special cases of it, not the content.
Infrastructure a complete development needs, all reusable: freeness of $\pi_1$ of a punctured
plane, local triviality of the Fadell–Neuwirth projection, the homotopy exact sequence of a
fibration in the range needed, presentations of split extensions, and the transfer of a
presentation along a regular covering. Contributions of any of these as standalone lemmas are
welcome, as is a formalization of the geometric generators (1-9) that would let the goal be
strengthened to pin the isomorphism on $\sigma_i$.
## Selected references
- E. Artin, *Theorie der Zöpfe*, Abhandlungen aus dem Mathematischen Seminar der Universität
Hamburg 4 (1925), 47–72. https://doi.org/10.1007/BF02950718
- E. Artin, *Theory of braids*, Annals of Mathematics 48 (1947), 101–126.
https://doi.org/10.2307/1969218
- W.-L. Chow, *On the algebraical braid group*, Annals of Mathematics 49 (1948), 654–658.
https://doi.org/10.2307/1969333
- E. Fadell, L. Neuwirth, *Configuration spaces*, Mathematica Scandinavica 10 (1962), 111–118.
https://doi.org/10.7146/math.scand.a-10517
- E. Fadell, J. Van Buskirk, *The braid groups of $E^2$ and $S^2$*, Duke Mathematical Journal 29
(1962), 243–257. https://doi.org/10.1215/S0012-7094-62-02925-3
- J. S. Birman, *Braids, Links, and Mapping Class Groups*, Annals of Mathematics Studies 82,
Princeton University Press, 1974. https://doi.org/10.1515/9781400881420
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Captain: Lucas
Gribov Ambiguity: no continuous gauge fixing (Singer 1978)Research Paper
## Motivation
In the Feynman path-integral approach to a non-abelian gauge theory one wants to integrate a
gauge-invariant weight over the space $\mathfrak{A}$ of vector potentials (connections) of a
principal bundle. The integrand is constant on the orbits of the group $\mathcal{G}$ of gauge
transformations, so the integral over $\mathfrak{A}$ diverges and one is supposed to integrate
instead over the orbit space $\mathfrak{R} = \mathfrak{A}/\mathcal{G}$. The Faddeev–Popov
procedure realizes this by *fixing a gauge*: choosing, continuously in the orbit, exactly one
vector potential on each orbit, and correcting by a Jacobian determinant.
V. N. Gribov (SLAC Translation 176, 1977) observed that for $SU(2)$ potentials on
$\mathbb{R}^3$ (or $\mathbb{R}^4$) with suitable conditions at infinity, the Coulomb gauge
condition does not do this: the Coulomb slice through the zero potential meets the orbit of the
zero potential again, far from the origin. These extra intersections are the **Gribov copies**;
R. Jackiw, I. Muzinich and C. Rebbi (Phys. Rev. D 17 (1978) 1576) analyzed them in detail.
I. M. Singer, *Some Remarks on the Gribov Ambiguity* (Commun. Math. Phys. **60** (1978) 7–12),
showed that the phenomenon is not a defect of the Coulomb gauge. If the conditions at infinity
are those of Gribov — gauge transformations extending to the one-point compactification with
value $I$ at infinity, so that the base manifold is $M = S^3$ or $M = S^4$ — then **no**
continuous gauge fixing exists at all, in any gauge. The obstruction is topological: the space
of irreducible connections is weakly contractible, while the gauge group is not, and a
weakly contractible principal bundle admits no global continuous section.
## Setting
Fix $N \ge 2$ and take the structure group $SU(N)$, the group of $N \times N$ complex matrices
$U$ with $U^\ast U = I$ and $\det U = 1$, topologized as a subspace of matrices. Let
$S^r$ denote the unit sphere of $\mathbb{R}^{r+1}$, with base point $m$ the north pole.
For the trivial $SU(N)$-bundle over a space $M$, a gauge transformation is a map
$\varphi : M \to SU(N)$, and the **gauge group** is
$$\mathcal{G}(M,N) \;=\; C\bigl(M, SU(N)\bigr),$$
continuous maps with pointwise multiplication and the compact-open topology. Two subobjects
matter. The **based gauge group** $\mathcal{G}_m = \{\varphi : \varphi(m) = I\}$ is the subgroup
of transformations that are the identity at the base point. The constant transformations with
value in the centre $Z_N = \{e^{2\pi i k/N} I\}$ of $SU(N)$ form a normal subgroup, and the
**reduced gauge group** is the quotient
$$\overline{\mathcal{G}}(M,N) \;=\; \mathcal{G}(M,N)/Z_N$$
with the quotient topology. The centre acts trivially on vector potentials, so
$\overline{\mathcal{G}}$ is the group that acts effectively.
A group $G$ acting continuously on a space $\mathfrak{A}$ has orbit space
$\mathfrak{A}/G$ with the quotient topology, and a **gauge fixing** is a continuous map
$s : \mathfrak{A}/G \to \mathfrak{A}$ with $p \circ s = \mathrm{id}$, where
$p : \mathfrak{A} \to \mathfrak{A}/G$ is the projection: a continuous choice of exactly one point
on each orbit. The action is **principal** when it is free and the division map, which sends a
pair of points on one orbit to a group element carrying the second to the first, can be chosen
continuously; this is the topological content of "$p$ is a principal $G$-bundle". The space
$\mathfrak{A}$ is **weakly contractible** when it is nonempty and all its homotopy groups vanish.
In the paper, $\mathfrak{A}$ is the affine space of connections, $\mathfrak{R}$ its set of
irreducible members, and Theorems 1 and 2 say exactly that $\mathfrak{R}$ is a weakly
contractible principal $\overline{\mathcal{G}}$-space.
## Formalization targets
### Goal — Corollary 4 (no gauge fixing)
For $r \in \{3,4\}$, $N \ge 2$, and every weakly contractible principal
$\overline{\mathcal{G}}(S^r,N)$-space $A$:
$$\nexists\, s : A/\overline{\mathcal{G}}(S^r,N) \longrightarrow A \quad\text{continuous with}\quad p \circ s = \mathrm{id}.$$
By Theorems 1 and 2 of the paper the space of irreducible connections over $S^3$ or $S^4$ is such
an $A$, so the goal contains Singer's Corollary 4 for that space; it leaves the analytic
construction of the space of connections unfixed, which is what makes it statable today.
### Milestone level — Theorem 3
$$\exists\, j \ge 1: \quad \pi_j\bigl(\overline{\mathcal{G}}(S^r,N)\bigr) \neq 0, \qquad r \in \{3,4\},\ N \ge 2 .$$
### Milestone level — Theorem 5 and its homotopy inputs
$$\pi_j\bigl(\mathcal{G}_m(S^r,N)\bigr) \;\cong\; \pi_{j+r}\bigl(SU(N)\bigr), \qquad
\pi_3(SU(N)) \cong \mathbb{Z}, \qquad \pi_4(SU(N)) = 0 \ (N\ge 3), \qquad \pi_4(SU(2)) \cong \mathbb{Z}/2 .$$
## Significance
The result rules out the existence of a global gauge in the topological sense: every gauge
condition used in practice is at best a local slice, and the Faddeev–Popov construction has to be
read as a local statement, patched with a partition of unity over the orbit space (as the last
section of the paper proposes). It is the mathematical reason why the Gribov ambiguity cannot be
repaired by a cleverer gauge condition, and it is the origin of the Gribov–Zwanziger restriction
of the functional integral to a fundamental domain.
Formalizing it adds a machine-checked version of an argument that is quoted far more often than
it is checked, and it forces into Lean a piece of infrastructure that Mathlib currently lacks:
homotopy groups of mapping spaces, the long exact sequence of a fibration in the form needed for
$0 \to \mathcal{G}_m \to \mathcal{G} \to SU(N) \to 0$, and the classical computations
$\pi_3(SU(N)) \cong \mathbb{Z}$, $\pi_4(SU(N)) = 0$ for $N \ge 3$, $\pi_4(SU(2)) \cong \mathbb{Z}/2$.
Singer's results are proved mathematics; none of them is formalized, and Mathlib as of the pinned
revision contains homotopy groups as a definition together with their group structure, but
essentially no computation of them.
## Difficulty
The naive approach to the goal — build a section by hand, or average over the group — fails
because $\overline{\mathcal{G}}$ is neither compact nor contractible and the obstruction is
global: locally, slices do exist (that is the content of the generalized Coulomb gauge), so no
local argument can produce a contradiction. The proof has to convert a section into a
homotopy-theoretic statement: a section of a principal bundle trivializes it, exhibiting the
group as a retract of the total space, so all homotopy groups of the group would vanish; the work
is then to show that some homotopy group of the reduced gauge group does not vanish, which needs
the identification of the based gauge group with a mapping space, the exact sequences relating
$\mathcal{G}_m$, $\mathcal{G}$ and $\overline{\mathcal{G}}$, and non-trivial homotopy groups of
$SU(N)$ — including $\pi_6(S^3) \cong \mathbb{Z}/12$ for the $SU(2)$ case of Theorem 3.
## Formalization scope
The formalization commits to the following conventions, all of them visible in the definitions of
this mission.
- The bundle is the **trivial** $SU(N)$-bundle, so gauge transformations are literally maps
$M \to SU(N)$. This is the case of Gribov's original setting over $S^3$; over $S^4$ the paper
also treats bundles of nonzero Pontrjagin index, which are out of scope here.
- Gauge transformations are **continuous**, not smooth, with the compact-open topology; Singer's
Theorem 5 uses smoothing homotopies to pass between the two, and the homotopy-theoretic content
is the same.
- $SU(N)$ is the special unitary group of complex $N \times N$ matrices, with its subspace
topology; $S^r$ is the unit sphere of $\mathbb{R}^{r+1}$ with its subspace topology.
- Homotopy groups are Mathlib's `HomotopyGroup`, based at the identity element.
- The **space of connections is not constructed**: Mathlib has no space of connections on a
principal bundle, and building one is a mission of its own. The goal therefore quantifies over
an arbitrary topological space carrying a weakly contractible principal action of the reduced
gauge group — exactly the properties Theorems 1 and 2 establish for the irreducible
connections.
- This quantification is not vacuous: such spaces exist (the total space of a universal
$\overline{\mathcal{G}}$-bundle is one), so the goal is a genuine non-existence statement and
not a statement about an empty class. Conversely it is not trivially true: the hypotheses do
not mention any homotopy invariant of the gauge group, and refuting a section requires
Theorem 3.
- The paper's analytic statements — Theorem 1 (openness and density of the irreducible
connections, principal bundle structure), Theorem 2 (weak contractibility), Theorem 6
($\pi_1$ of the irreducible orbit space), Theorem 7 (no flat connection), Theorem 8 (tangency
of orbits to the Coulomb slice) and Theorem 9 (the canonical connection and its curvature) —
are out of scope until a space of connections exists in Lean. Contributions that build one, in
reusable form, are welcome and would let this mission be extended to them.
## Selected references
- V. N. Gribov, *Instability of non-abelian gauge theories and impossibility of choice of Coulomb
gauge*, SLAC Translation 176 (1977); Nucl. Phys. B **139** (1978) 1–19,
[doi:10.1016/0550-3213(78)90175-X](https://doi.org/10.1016/0550-3213(78)90175-X).
- I. M. Singer, *Some Remarks on the Gribov Ambiguity*, Commun. Math. Phys. **60** (1978) 7–12,
[doi:10.1007/BF01609471](https://doi.org/10.1007/BF01609471).
- R. Jackiw, I. Muzinich, C. Rebbi, *Coulomb gauge description of large Yang-Mills fields*,
Phys. Rev. D **17** (1978) 1576, [doi:10.1103/PhysRevD.17.1576](https://doi.org/10.1103/PhysRevD.17.1576).
- H. Toda, *Composition methods in homotopy groups of spheres*, Annals of Mathematics Studies 49,
Princeton University Press (1962).
8 thms1 active userReviewed
Captain: ryanshin
Smooth 4-dimensional Poincaré conjecture: foundations and reductionsOpen Problem
## Motivation
The **smooth four-dimensional Poincaré conjecture** asks whether a smooth manifold with the topology of the four-sphere must also have its standard smooth structure, up to diffeomorphism. The distinction is between the existence of continuous coordinates and the compatibility of differentiable coordinates. The mission concerns this precise sphere question, listed as open in Problem 4.1 of *K3 — A New Problem List in Low-Dimensional Topology*. It does not treat a collection of algebraic obstructions as an existing proof of the conjecture. [Baykur–Kirby–Ruberman, Problem 4.1](https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf)
### Historical landmarks
- **1961:** Smale proved that a closed smooth manifold homotopy equivalent to a sphere of dimension at least five is homeomorphic to that sphere. This is not a theorem that all such smooth manifolds are diffeomorphic to the standard sphere. [Smale, Theorem A](https://www.math.uchicago.edu/~shmuel/tom-readings/Smale,%20PC.pdf)
- **1982:** Freedman established the topological four-dimensional Poincaré theorem: a topological four-manifold homotopy equivalent to the four-sphere is homeomorphic to it. [Freedman, Theorem 1.6](https://www.maths.gla.ac.uk/~mpowell/1982_The%20topology%20of%20four-dimensional%20manifolds.pdf)
- **2026:** The *K3* problem list continues to distinguish this established topological result from the open smooth sphere problem. [Problem 4.1, pp. 191–192](https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf)
## Setting
Let $S^4$ be the unit sphere in $ℝ^5$, with its standard stereographic smooth structure. A **homeomorphism** is a continuous bijection with continuous inverse; a **diffeomorphism** is a smooth bijection with smooth inverse. A **smooth atlas** is a collection of local Euclidean coordinates whose transition maps are smooth.
The manifold $M$ is compact and Hausdorff, has no boundary, and is equipped with a specified smooth atlas modeled on $ℝ^4$. The given atlas is arbitrary: it is not defined by transporting the standard structure from $S^4$.
For a homeomorphism $e:N\to S^4$, let $\mathcal A_e$ denote the atlas transported from the standard sphere along $e$. A **structomorphism** for the smooth structure groupoid is a homeomorphism whose coordinate expressions belong to that groupoid. The predicate $\mathsf{SPC4Pullback}$ requires, for every given smooth atlas $\mathcal A$ on such an $N$ and every such $e$, a structomorphism between $(N,\mathcal A)$ and $(N,\mathcal A_e)$. It does not require that structomorphism to be the identity. These are the conventions of the source definitions, not additional uniqueness assumptions. [Shin, *SPC4.lean*, lines 53–81 and 211–221]
## Formalization targets
### Main open goal
For every manifold $M$ with the preceding hypotheses, the goal is
$$
M\cong_{\mathrm{Top}}S^4
\quad\Longrightarrow\quad
M\cong_{\mathrm{Diff}}S^4.
$$
This is the source predicate $\mathsf{SPC4}$. Its conclusion asserts the existence of a diffeomorphism; it does not assert that a particular supplied homeomorphism is smooth.
### Structural and literature milestones
The atlas formulation has the exact equivalence
$$
\mathsf{SPC4}\quad\Longleftrightarrow\quad\mathsf{SPC4Pullback}.
$$
The source supplies a proof of this equivalence without invoking Freedman's theorem or assuming the conjecture as an unconditional fact. It is a reformulation, not a solution. Its foundations include the correspondence
$$
\operatorname{Structomorph}(\mathcal G^{\infty},M,N)
\simeq \operatorname{Diff}^{\infty}(M,N),
$$
where $\mathcal G^{\infty}$ is the smooth coordinate-change groupoid for the common model. [Shin, *SPC4.lean*, lines 334–365; *Bridge.lean*]
Write $F_4$ for the following compact Hausdorff, boundaryless instance of Freedman's topological theorem:
$$
M\simeq S^4\quad\Longrightarrow\quad M\cong_{\mathrm{Top}}S^4,
$$
where $\simeq$ denotes homotopy equivalence and only topological manifold charts are assumed. This is established mathematics, but a proof in the present formal development remains a target. If $\mathsf{SPC4Homotopy}$ denotes the analogous smooth conclusion from a homotopy equivalence, the relation to the main goal is recorded with its hypothesis visible:
$$
F_4\quad\Longrightarrow\quad
(\mathsf{SPC4}\Longleftrightarrow\mathsf{SPC4Homotopy}).
$$
Explicit standard-disk foundations form another track. For every $m\geq0$, they concern the manifold-with-boundary structure on $\overline B^{m+1}$, its boundary set $S^m$, and the smooth collar
$$
c:S^m\times[0,1]\longrightarrow\overline B^{m+1},
\qquad c(u,t)=(1-t/2)u.
$$
The collar is a closed embedding, has image
$$
\{z\in\overline B^{m+1}:\|z\|\geq1/2\},
$$
and satisfies $c(u,0)=u$, using the boundary inclusion. Its image is a neighborhood of every boundary point in the disk. A companion interface characterizes a $C^k$ map from a $C^k$ manifold with corners into the disk as precisely a continuous map whose inclusion into Euclidean space is $C^k$. These targets concern the actual disk smooth structure. [Shin, *Disk.lean*, lines 1076–1141 and 1263–1318]
### Topological two-disk gluing
For each integer $m\geq0$, let $D^{m+1}=\overline B^{m+1}$ be the closed unit disk in $\mathbb R^{m+1}$ and let $\varphi:S^m\to S^m$ be any homeomorphism of its boundary. The **twisted double** identifies the boundary point $u$ in a left copy of the disk with $\varphi(u)$ in a right copy. With the quotient topology, the target is
$$
X_\varphi:=\bigl(D^{m+1}_L\sqcup D^{m+1}_R\bigr)/(u_L\sim\varphi(u)_R)
\quad\cong_{\mathrm{Top}}\quad S^{m+1}.
$$
This statement is published as [SP4Gluing.twistedSphere_homeomorphic](https://prove2.me/theorems/fe8e71c7-85fd-4392-8327-453dda13f24c). The theorem and its supporting [continuity](https://prove2.me/theorems/e09b0118-b3a4-44e3-9ba2-e70fb31a2faa) and [injectivity](https://prove2.me/theorems/12c0403c-fdcf-4202-8b53-9f12893b568f) lemmas have accepted Lean proofs contributed by [carlok](https://prove2.me/users/fca9fd8a-84f4-46ca-8845-a4a2b665381d). All three accepted proofs have also been checked locally with their proved dependencies. It concerns these explicit topological quotients, not arbitrary homotopy spheres or a prescribed smooth structure.
### Seam–interior smooth compatibility
For every regional chart base point, the open-bicollar and left-interior transitions are smooth in both directions. Right-interior-to-seam smoothness requires smooth $\varphi^{-1}$; the reverse requires smooth $\varphi$. The [single compatibility target](https://prove2.me/theorems/f1e93fb9-c414-46aa-8b6c-fc6978243ee7) concerns exact overlap sources, combining four source results internally. It provides neither a global smooth-manifold instance nor smooth standardness. [Shin, *Hemisphere.lean*, lines 2439–3577]
## Significance
A proof of the main goal would identify every smooth structure in its stated sphere class with the standard one, up to diffeomorphism. A proof of the transported-atlas equivalence instead locates the same unresolved comparison in a different formal language. The distinction matters: constructing a smooth structure by transport is not the same as identifying an arbitrary pre-existing one.
The bridge, explicit disk atlas, and stated collar properties have accepted kernel-checked Lean proofs. The clean atlas equivalence also has a proof with no admitted theorem among its axioms. The conjecture remains open, and Freedman's topological theorem remains unproved in this formal development despite its published mathematical proof.
The [topological two-disk gluing result](https://prove2.me/theorems/fe8e71c7-85fd-4392-8327-453dda13f24c) identifies the homeomorphism type of these quotients for every boundary homeomorphism and every disk dimension at least one. The accepted formalization supplies a global topological comparison for this explicit quotient. It does not resolve the comparison with a prescribed smooth structure or recognition of general smooth four-manifolds.
Four supporting algebraic tracks concern orbit coinvariants, homology dimension budgets, finite-support shift rigidity, and Laurent-polynomial positivity. Their source results arose in route-specific obstruction studies. As of 6 September 2026, all eleven theorem targets in these algebraic tracks have accepted Lean proofs. The five additional formal proofs were contributed by [wamlart](https://prove2.me/users/70d2064f-6b47-4a3f-bb90-74e71a55cbb7): [orbit augmentation](https://prove2.me/theorems/85478003-b7db-44ef-be81-1c0fc5d7a8b4), [region homology budgets](https://prove2.me/theorems/e3d78c98-ef71-4b2a-a04a-7fd93fe9246f), [two-corner homology budgets](https://prove2.me/theorems/f6cc0221-29eb-4e8a-9649-8856b2954f2a), [the Laurent mass threshold](https://prove2.me/theorems/2d22450e-c4ce-4a0c-8829-b51d65e80eb5), and [mass-two positivity](https://prove2.me/theorems/988bc029-7a8a-43d7-9413-9e9cc3f86c24). No theorem currently connects their completion to a proof or disproof of $\mathsf{SPC4}$. They are exploratory tools, not established milestones in a proof of the main goal.
## Difficulty
A homeomorphism can transport the standard atlas, but that observation does not compare the transported atlas with the one already specified on the manifold. Treating those two atlases as equal would remove the central mathematical question by changing its hypotheses.
Likewise, topological recognition does not supply a smooth recognition theorem. Standard disk and collar constructions establish local models; they do not establish a smooth gluing or recognition theorem for an arbitrary prescribed smooth structure, a recognition theorem for arbitrary smooth balls, or a smooth Schoenflies theorem. The missing global comparison cannot be replaced by successful finite algebraic tests or by constructing a standard local chart.
## Formalization scope
The sphere goal quantifies over `Type` in universe zero, exactly as in the source. It uses real four-dimensional Euclidean chart models, compactness, the Hausdorff condition, and smoothness of order $\infty$. Boundaryless manifolds are built into that model. No orientation, fixed parametrization, or identity-map uniqueness is imposed.
The geometric foundations use charted spaces, structure groupoids, models with corners, homotopy equivalences and diffeomorphisms. Disk results include every $m\geq0$, so their dimensions are $m+1\geq1$. The boundary-set identification does not by itself construct a general induced smooth boundary structure. Nor is smoothness asserted for a radial clamp across its nonsmooth locus.
The separate source assertion `SPC4Ball` is not treated as equivalent to the sphere goal: the required formal boundary, capping and gluing bridge is absent. The transported-annulus product diffeomorphism is not a current target; its chart instances serve only as constructor support. No unconditional implication is taken through the source's admitted Freedman declaration. Gaussian coupling, transport defects, partition incidence and merge-score results remain outside this mission because no mathematical dependency on them has been established.
## Selected references
- R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., *K3 — A New Problem List in Low-Dimensional Topology*, Mathematical Surveys and Monographs 295, American Mathematical Society, 2026, Problem 4.1, pp. 191–192. [Author PDF](https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf).
- Michael Hartley Freedman, *The topology of four-dimensional manifolds*, Journal of Differential Geometry 17 (1982), 357–453, Theorem 1.6, p. 371. [DOI](https://doi.org/10.4310/jdg/1214437136); [primary-article scan](https://www.maths.gla.ac.uk/~mpowell/1982_The%20topology%20of%20four-dimensional%20manifolds.pdf).
- Stephen Smale, *Generalized Poincaré's Conjecture in Dimensions Greater Than Four*, Annals of Mathematics 74 (1961), 391–406, Theorem A. [DOI](https://doi.org/10.2307/1970239); [primary-article scan](https://www.math.uchicago.edu/~shmuel/tom-readings/Smale,%20PC.pdf).
- Ryan Shin, *SPC4.lean*, *Bridge.lean* and *Disk.lean*, unpublished source files, 2026; no public manuscript URL available. SHA-256, respectively: `b17fdb932034e5211d0db8171c08e2b3a182016bceaecdd2deb49c39d6bfd5cc`, `e8ea6b66f6bd675ca272e862e0825ab2db1f8bb792eaffe1b9e8f5d89024d302`, `889a9eccf9d2350aee7051ab7b6895e565f9f1a0c84e7120fb45c15acae0097e`.
- Ryan Shin, *Hemisphere.lean*, unpublished Lean source file, 2026, declaration `twistedSphereHomeoSphere`; source SHA-256 `c48843d2c4ec6987acfd7f7ab3a92bfed990376206142e74712795b4e9399828`. [Published topological two-disk gluing target](https://prove2.me/theorems/fe8e71c7-85fd-4392-8327-453dda13f24c); the recovered local construction is checked; the accepted proof and its two supporting lemmas were contributed by carlok.