折り図Seam Theory, folded
What is studied?
Seam theory, the study of seams: whether incarnations in different theories can be joined, in how many ways, whether the joins are consistent around a cycle of theories, how they depend on the identification of symmetry groups, what the maps that are not joins forget, and where a join is impossible. The objects are classical; the joins, not the pieces, are at the centre.
The last question gives the subject its shape: a table of objects against theories in which some cells are empty by necessity, a negative space bounded by the seams that do exist.
Existence
two sets are joined exactly when their stabilizer classes agree.
Number
the seams between two incarnations form a torsor under .
Consistency
seams are unique and coherent exactly when the object is rigid, and otherwise natural seams can carry monodromy.
Dependence on markings
an inner change of marking changes nothing up to isomorphism, while an outer automorphism moves the stabilizer class, as it exchanges the points and the lines of the Fano plane, and a bridge refuted for one marking is built over the outer automorphism.
Forgetting
a map between theories that is not a seam is a description, and what it forgets at a point is its kernel, a stabilizer.
Impossibility
the negative space of absences, with their windows and imprints.
Each concept is one fold of a single square, made through the points of the concepts it uses, so no fold comes before what it needs. A seam is drawn as a fold too: two faces, two incarnations of one object, laid one onto the other along a crease.
Fold the square into eighths, valley and mountain in turn: one band for each floor.
0
What does each theory supply before anything is compared?
Groups acting on sets, stabilizers, characters, and the classical groups with their geometries. The objects of the book are classical, and so is the group theory it uses: orbits and stabilizers, normalizers, automorphisms of permutation groups.
Several classical tools are adopted as they are: orbital graphs, which carry structure across seams; permutation isomorphisms; Gassmann equivalence; Burnside’s marks; power maps; the Frobenius–Schur indicator; equivariant bundles over a finite -set; Hurwitz groups; the Bruhat–Tits building; and the triangle presentations of Cartwright, Mantero, Steger and Zappa. What the book isolates is the matchings themselves.
Precrease. Fold the square into eighths, valley and mountain in turn: one band for each floor.
1
When do two theories name one object?
An object of a group is a transitive -set. A theory supplies a set and a group acting on it, both defined without reference to ; a marking identifies with a subgroup of that group, and the marked set is an incarnation of an object when it is -isomorphic to it.
The floor rests on the stabilizer principle: an object is determined by its stabilizer class, so an entry of an atlas of objects is a conjugacy class of subgroups. The group of order 168 has exactly fifteen objects, and the Fano plane, the projective line over and the Klein quartic each carry an incarnation of every one of them.
Its double cover adds objects on which acts without fixed points, sets that come from no set of the group of order 168. There are exactly four of these new objects, one over each class of subgroups of odd order.
Floor 1 folds onto floor 0. The stabilizer principle joins incarnation to the classical floor.
Seams 1, chapter 1 · Un objet, plusieurs noms
What single datum decides which object a set is?
The conjugacy class formed by the stabilizers of an object’s points; it determines the object up to isomorphism.
Valley-fold and unfold through point 1.
Seams 1, chapter 1 · Un objet, plusieurs noms
How are the symmetry groups of two theories compared?
An injective homomorphism from the reference group into the group a theory supplies; it makes the theory’s set a set acted on by the reference group.
Valley-fold and unfold through points 1 and 2.
Seams 1, chapter 1 · Un objet, plusieurs noms
When is a set in some theory a form of a given object?
A set acted on by the group, usually a theory’s marked set, that admits an equivariant bijection from the object.
Valley-fold and unfold through points 1, 2 and 3.
Seams 1, chapter 1 · Un objet, plusieurs noms
How is an incarnation laid over its object, point by point?
An isomorphism from the object onto one of its incarnations; there are as many as the object has automorphisms.
Valley-fold and unfold through points 1 and 4.
Seams 3, chapter 11 · Le revêtement double et le miroir
Which objects does a double cover add to those of the group below it?
A transitive set of the double cover on which acts without fixed points, so that it comes from no set of the group of order 168; there are exactly four, one over each class of subgroups of odd order.
Valley-fold and unfold through points 1, 2 and 4.
2
In how many ways are two incarnations one, and do the ways agree?
A seam is a -isomorphism between two incarnations of one object. The seams between two incarnations form a torsor under the automorphism group of the object, so they are unique, and consistent around every cycle, exactly when the stabilizer is self-normalizing. Six of the fifteen objects of the group of order 168 are rigid in this sense.
The other nine carry freedom. When theories supply their seams by their own constructions, a cycle of natural seams can return a nontrivial automorphism, its monodromy. On the object of size 24 the flex-tangent map of the Klein quartic closes a cycle of length two with monodromy of order 3, and the power of an automorphism makes such monodromies comparable across theories.
Read in a choice of alignments, a family of seams over a graph is a lattice gauge connection with gauge group , and monodromy is its holonomy. A seam over an automorphism of , a seam after twisting the action by it, joins incarnations that a fixed marking keeps apart, as the polarity joins the lines of the Fano plane to its points.
Floor 2 folds onto floor 1. Seams join incarnations, unique when the stabilizer is self-normalizing and otherwise carrying monodromy.
Seams 1, chapter 1 · Un objet, plusieurs noms
How are two incarnations of one object matched?
An equivariant bijection between two incarnations of one object; the seams between two incarnations form a torsor under the object’s automorphisms.
Valley-fold and unfold through points 4 and 5.
Seams 1, chapter 1 · Un objet, plusieurs noms
How are all the seams of a family recorded at once?
The groupoid whose vertices are a family of incarnations and whose arrows are their seams; consistency means it is the pair groupoid.
Valley-fold and unfold through point 7.
Seams 1, chapter 1 · Un objet, plusieurs noms
When is the seam between two incarnations forced?
An object with no automorphism but the identity; equivalently its stabilizers are self-normalizing, and then every seam is unique.
Valley-fold and unfold through points 2, 5 and 7.
Seams 1, chapter 1 · Un objet, plusieurs noms
Do seams chosen one at a time agree around every route?
A family of seams is coherent when every route between two incarnations gives the same map; automatic for rigid objects, and otherwise the same as coming from one choice of alignments.
Valley-fold and unfold through points 5, 8 and 9.
Seams 1, chapter 4 · La monodromie des sutures
Which seams do the theories themselves supply?
A family of incarnations with a chosen set of seams among them, loops allowed: typically the natural identifications that the theories provide.
Valley-fold and unfold through points 4 and 7.
Seams 1, chapter 4 · La monodromie des sutures
What does going around a loop of natural identifications do?
The composite of seams around a closed walk, an automorphism of the incarnation; it measures how far a family of seams is from one choice of alignments.
Valley-fold and unfold through points 5, 10 and 11.
Seams 1, chapter 4 · La monodromie des sutures
What does a seam system become once an alignment is chosen at every incarnation?
A choice of alignments, one for each incarnation of a seam system over a graph; it turns the seams into link variables in , so that a seam system is a lattice gauge connection and its monodromy is holonomy.
Valley-fold and unfold through points 5, 11 and 12.
Seams 1, chapter 4 · La monodromie des sutures
How can automorphisms of incarnations in different theories be compared?
For a self-centralizing cyclic stabilizer, the residue k such that every seam to a conjugacy class turns the automorphism into the k-th power map; it depends on no seam, class or marking.
Valley-fold and unfold through points 4 and 7.
Seams 1, chapter 4 · La monodromie des sutures
How are automorphisms named when no power is available?
The stabilizer class of the quotient of an incarnation by an automorphism; seams preserve it, and it names the three involutions of the object of size 84.
Valley-fold and unfold through points 1, 2 and 7.
Seams 3, chapter 10 · La table en deux, en sept et à l’infini
What does a symmetry that normalizes the group, rather than commuting with it, give?
The element c by which a symmetry conjugates the marking; correcting the symmetry by c gives an automorphism, its equivariant twist, whose power is inverse to that of c.
Valley-fold and unfold through points 3 and 14.
Seams 1, chapter 1 · Un objet, plusieurs noms
What is a seam after twisting by an automorphism of the group?
A bijection that carries the action of each element to the action of its image under an automorphism of the group; a seam is a seam over the identity, and a bridge refuted for one marking can be built over an outer automorphism.
Valley-fold and unfold through points 3, 7 and 9.
3
What has been proved about a bridge, and what is proved not to exist?
A bridge asserts that two sets, given in two theories, are incarnations of one object, and its status records what is known: built, type, name or refuted. Only built bridges are theorems. For two marked sets of one group the stabilizer principle decides every type bridge, which is either built or refuted. Type recurrence is not identification.
Beside the statuses stands the negative space: absences, theorems that something is not there, with their windows, the parameter values where the excluded thing can still happen, and their imprints, the structures an absence forces to exist. The absences reduce to one another in three clusters: the group of order 168, the octonions and Hilbert space.
A map between theories that is not a seam is a description: it goes one way and forgets something, and what it forgets at a point is its kernel, a stabilizer. Read so, each concept of the floor is a statement about a description and what it forgets: a built bridge is a description that forgets nothing, an absence is an empty fibre, a carrier imprint is induced from what an orbit description forgets, and monodromy is what remains of the loops once the kernel of the holonomy is divided out.
Floor 3 folds onto floor 2. A bridge has a status, which the stabilizer principle decides for two marked sets of one group, and beside the statuses stand the absences.
Seams 1, chapter 1 · Un objet, plusieurs noms
What exactly is claimed when two theories are said to name the same thing?
The assertion that two sets, given in two theories, are incarnations of one object; only a built bridge is a theorem.
Valley-fold and unfold through points 4 and 7.
Seams 1, chapter 1 · Un objet, plusieurs noms
What is a map between theories that is not a seam?
A surjective equivariant map between sets on which one group acts: it goes one way and may forget something, and it forgets nothing exactly when it is a seam.
Valley-fold and unfold through points 1 and 7.
Seams 1, chapter 1 · Un objet, plusieurs noms
What does a description forget?
The stabilizer of the image of a point under a description: what the description cannot tell apart there. It covers the kernel of a homomorphism, the stabilizer of an orbit and the congruence kernel of a reduction.
Valley-fold and unfold through point 21.
Seams 1, chapter 2 · L’espace en creux
What is a theorem that something does not exist, taken as an object of study?
A theorem that a collection of structures, specified by explicit axioms, has no member with a stated property; it marks where the atlas cannot be stitched.
Mountain-fold and unfold through point 20.
Seams 1, chapter 3 · La table des sutures du groupe d’ordre 168
Which objects can the simplest figures of a theory not reach?
A class of subgroups that no basic figure of a theory has as its stabilizer class; in the seam table every forced gap is filled by a composite figure, and only the Coxeter graph reaches every class with its simplest figures.
Mountain-fold and unfold through points 2, 4 and 23.
Seams 1, chapter 2 · L’espace en creux
What structure does an absence force to exist?
A structure that exists, with a theorem characterizing it by an absence: terminal (the survivors at the edge of a window), carrier (what carries local data that do not globalize) or separating (a finer invariant).
Mountain-fold and unfold through points 23 and 25.
Seams 1, chapter 2 · L’espace en creux
Which absences are the same fact seen twice?
An absence reduces to another when the book proves it from the other without reproving it; the reductions sort the absences into three clusters that meet only through bridges.
Mountain-fold and unfold through points 18, 23 and 26.
4
When does one group carry the geometries of two families?
A life of a group is an isomorphism onto a member of the families , acting on its projective space, or , acting on letters. Isomorphisms between members of different families are rare: by Artin’s absence exactly four groups have a double life, , , and . The floor consists of those survivors, read as seams.
Each double life comes with a dictionary of which natural sets of the two lives are one object, computed by matching stabilizers. For the group of order 168 the dictionary is the seam table itself. In three of the four double lives an outer automorphism exchanges two dual objects of one life and is unremarkable in the other.
Floor 4 folds onto floor 3. Artin’s absence leaves exactly four groups with a double life.
Seams 2, chapter 6 · Quatre groupes à double vie
Which groups carry the geometries of two families at once?
Two lives of one group in different members of the families; by Artin’s absence exactly four groups have one: , , and .
Valley-fold and unfold through points 3, 23 and 28.
Seams 2, chapter 6 · Quatre groupes à double vie
Which natural sets of one life are which natural sets of the other?
For each object of a group with a double life, the natural sets of each life that are incarnations of it, computed by matching stabilizers.
Valley-fold and unfold through points 2, 4 and 29.
5
Where do the finite geometries sit inside buildings over local fields?
Each life of a double life is a geometry over a finite field , the residue field of . The geometry is the link of a vertex of the Bruhat–Tits building over , and the stabilizer of the vertex acts on it through the finite group. A completion of a finite projective geometry is such a building with such a vertex.
So a double life sits at two vertices: the group of order 168 acts on the Heawood graph, the link of a vertex of the building of , and on , the link of a vertex of the tree of . The octonion multiplication table glues Fano links into the building of over , and no subgroup of finite index of its group is isomorphic to one of Mumford’s lattice.
Kato’s hermitian form glues the same links, without symmetry, into the building over , as Mumford’s lattice; a gluing that a Frobenius group of order 21 respects is the octonion one, and it lives in characteristic 2. At 7 Mumford’s form has its own tree, whose base link is the sky, and over the group of order 168 is the stabilizer of a vertex, Klein’s lattice, at which it carries both lives.
At Klein’s lattice the finite geometry is found among short vectors and neighbours: the stabilizer of a point acts on the neighbour through it as the rotations of a cube, a flag is a pair of vectors of norm 2 whose reflection is a half-turn of that cube, and an antiflag is one of its diagonals, of norm 3. One step beyond the link the two trees at 7 carry a doublet and its symmetric square; the object of the points of the Fano plane is carried along the one tree in exactly one way and along the other in none; and the two parents carry independent flips, the sign changes of and , of which only the first is seen by the oriented cells of the link complement.
The two parents are joined only by fiber products: across scales none keeps the finite line attached, and at one scale the attachment is forced. Around the loops of the scale tree a single relabelling carries the signed octonion table without reversals exactly on the Iwahori subgroup, and on seven loops in eight it must reverse two units; carried observer by observer, every loop returns each fiber changed only by colour, consistently with the meetings.
Floor 5 folds onto floor 4. Residue fields make each life the link of a vertex of a building over a local field: at 2 the octonion table glues such links into a building over and Kato’s hermitian form glues them, without symmetry, into the building over ; a gluing with the Frobenius symmetry of order 21 is the octonion one; and over the group of order 168 is the stabilizer of the vertex of Klein’s lattice, carrying both lives, where the finite geometry is the geometry of short vectors and neighbours; one step beyond the link the two trees at 7 carry a doublet and its symmetric square, the object of the points is carried along one tree and not the other, and the two parents carry independent flips, of which only the first is seen by the oriented cells of the link complement. The two parents are joined only by fiber products, which across scales keep no attachment of the finite line and at one scale force it; around the loops of the scale tree a single relabelling carries the signed table without reversals exactly on the Iwahori subgroup, and on seven loops in eight must reverse two units, while carried observer by observer every loop returns each fiber changed only by colour.
Seams 3, chapter 9 · Immeubles et réseaux
Where does a finite geometry sit inside a building over a local field?
A building over a local field with a vertex whose link is the flag complex of a finite projective geometry; the finite geometry lives over the residue field.
Valley-fold and unfold through points 4, 28 and 29.
Seams 3, chapter 13 · Orientation et charge
What does each parent’s flip change, and what can see it?
Each arithmetic parent of the group of order 168 carries an orientation: the sign change of turns the congruence link complement into its mirror image, and that of exchanges the octonion table with its Weil mirror. The two flips are independent, and once the signs of the units at the cusps are treated as a convention, only the first is seen by the structures of the link complement.
Valley-fold and unfold through points 13, 17 and 31.
6
How do the finite objects reappear in real and complex geometry?
A finite object can appear in a continuous geometry in two ways: as a configuration of points fixed in place by a finite group of symmetries, an embedded continuum, or as a set of classes of an arithmetic configuration modulo a congruence subgroup, an arithmetic one. Which kind an object can have is decided by absences.
The archimedean place joins this floor to the completions: the congruence that gives a residue field at a prime gives, over , Thurston’s congruence link complement, whose eight cusps are the points of and whose cells are objects of the group of order 168. The projective line has no embedded continuum in or in Klein’s plane, only this arithmetic one.
Every row of the seam table is a configuration of cells of that manifold, and the Fano incidence among them is the absence of a shared face. In the Cayley plane one point and one imaginary unit carry the intersection of two maximal subgroups of found by Todorov and Dubois-Violette; in its complexification the same point carries the of .
On the link complement the spinor system, the local system of the defining representation of , carries the first of the two parents’ flips at the cusps, and the operators that move between pairs of cusps generate a Clifford algebra whose centre is a single sign.
Floor 6 folds onto floor 5. The archimedean place gives, over , Thurston’s link complement, whose cells are objects of the group, with the Fano incidence among them the absence of a shared face; the group has two arithmetic parents at 7, which share the sky but not its completion; absences decide which continua exist; and in the Cayley plane one point and one imaginary unit carry the intersection of Todorov and Dubois-Violette.
Seams 3, chapter 14 · Les continus
How does a finite object reappear inside a continuous geometry?
A homogeneous space of a Lie group that carries the object, either as an equivariant configuration (embedded) or as classes modulo a congruence subgroup (arithmetic).
Valley-fold and unfold through points 1, 23 and 31.
Seams 3, chapter 13 · Orientation et charge
What does the defining representation of the Bianchi group carry at the cusps of the link complement?
The local system on the congruence link complement given by the defining representation of . Its boundary scattering is one constant times the Paley matrix; its cusp lines transform as and not as its mirror ; and the moves between pairs of cusps keep that class.
Valley-fold and unfold through points 32 and 33.
Seams 3, chapter 13 · Orientation et charge
Which operators commute with every move at a point of the Fano plane?
At a unit of the octonions, the algebra generated by the six left multiplications by the other units, each tensored with a flip of a two-state counter, and by the counter’s sign. It is the complex Clifford algebra , a sum of two matrix algebras, and its centre is spanned by the identity and one sign, the chirality read with the parity of the number of moves.
Valley-fold and unfold through points 32 and 34.
R
What is studied?
The roof folds onto floor 6.
Seams 1, chapter 1 · Un objet, plusieurs noms
What is the subject?
The study of seams: when they exist, how many there are, whether they are consistent, how they depend on markings, what the maps that are not seams forget, and where seams are impossible.
Valley-fold and unfold through points 2, 3, 7, 9, 12, 22, 23 and 26.
Unfold all 36 steps and the section is one square: a point for each concept, a crease from each concept to each concept it uses, and the floors as pleats. Each floor is joined to the one below it by a theorem.
No point chosen.
The stabilizer principle joins incarnation to the classical floor.
Seams join incarnations, unique when the stabilizer is self-normalizing and otherwise carrying monodromy.
A bridge has a status, which the stabilizer principle decides for two marked sets of one group, and beside the statuses stand the absences.
Artin’s absence leaves exactly four groups with a double life.
Residue fields make each life the link of a vertex of a building over a local field: at 2 the octonion table glues such links into a building over and Kato’s hermitian form glues them, without symmetry, into the building over ; a gluing with the Frobenius symmetry of order 21 is the octonion one; and over the group of order 168 is the stabilizer of the vertex of Klein’s lattice, carrying both lives, where the finite geometry is the geometry of short vectors and neighbours; one step beyond the link the two trees at 7 carry a doublet and its symmetric square, the object of the points is carried along one tree and not the other, and the two parents carry independent flips, of which only the first is seen by the oriented cells of the link complement. The two parents are joined only by fiber products, which across scales keep no attachment of the finite line and at one scale force it; around the loops of the scale tree a single relabelling carries the signed table without reversals exactly on the Iwahori subgroup, and on seven loops in eight must reverse two units, while carried observer by observer every loop returns each fiber changed only by colour.
The archimedean place gives, over , Thurston’s link complement, whose cells are objects of the group, with the Fano incidence among them the absence of a shared face; the group has two arithmetic parents at 7, which share the sky but not its completion; absences decide which continua exist; and in the Cayley plane one point and one imaginary unit carry the intersection of Todorov and Dubois-Violette.
An object of a group is a transitive -set: a set with a left action of such that for some, equivalently every, . For a subgroup the cosets form an object, and the stabilizer of the coset is .
The group of order 168 has exactly fifteen objects, and the Fano plane, the projective line over and the Klein quartic each carry an incarnation of every one of them.
Exactly six of the fifteen objects of are rigid: those with stabilizers , , , , and , of sizes 28, 21, 8, 7, 7 and 1. For each of them all seams between incarnations are unique and consistent. The other nine, of sizes 168, 84, 56, 42, 42, 42, 24, 14 and 14, have automorphism groups , , , , , , , and .
Each unit is creased once from its centre for every seam between any two of its incarnations; a rigid object’s sheet has only its pinch.
1681
seams between any two incarnations
The group acting on itself: trivial stabilizer, and the whole group of order 168 as its automorphisms.
84
seams between any two incarnations
Stabilizer and automorphism group : three involutions, each named by its quotient class.
56
seams between any two incarnations
Stabilizer and automorphism group : the points of contact of the bitangents, the triples of the projective line, the faces of Thurston’s link complement.
42
seams between any two incarnations
Stabilizer and automorphism group : the edges of the Coxeter graph, the directed 4-cycles on the quadrangles of the Fano plane, and the imaginary points of , whose automorphism is the Frobenius at 7.
42
seams between any two incarnations
Stabilizer and automorphism group : ordered pairs of lines of the Fano plane; it shares its permutation character with the class-b object and is not that object.
Exchanged by the outer automorphism. One permutation character; for one marking every bridge between them is refuted.
42
seams between any two incarnations
Stabilizer and automorphism group : ordered pairs of points of the Fano plane, where natural seams match swaps with swaps and rotations with rotations.
28
in the program, anchored observers
rigid: one seam between any two incarnations
Stabilizer , rigid: the antiflags, pairs, Sylow 3-subgroups, bitangents and Coxeter vertices are one object, with exactly one seam between any two.
24
seams between any two incarnations
Stabilizer and automorphism group : the flexes of the Klein quartic, where seam monodromy first appears.
21
rigid: one seam between any two incarnations
Stabilizer , rigid: the flags of the Fano plane, the involutions of the group, and the centres of involutions in Klein’s plane.
14
seams between any two incarnations
Stabilizer and automorphism group : the oriented quadrilaterals of the Fano plane, and half of the ideal tetrahedra of Thurston’s link complement.
Exchanged by the outer automorphism. One permutation character; for one marking every bridge between them is refuted.
14
seams between any two incarnations
Stabilizer and automorphism group : the oriented quadrangles of the Fano plane, fourteen of the lattices in the octonions, and the other half of the ideal tetrahedra.
8
in the program, the sky
rigid: one seam between any two incarnations
Stabilizer 7:3, rigid: the points of the projective line over , the Sylow 7-subgroups, the flex triangles and the cyclic orientations of the Fano plane.
7
in the program, clocks
rigid: one seam between any two incarnations
Stabilizer , rigid: the points of the Fano plane, the unit lines of the octonions and Coxeter’s seven octavian orders.
Exchanged by the outer automorphism. One permutation character; for one marking every bridge between them is refuted.
7
in the program, vantage lines
rigid: one seam between any two incarnations
Stabilizer , rigid: the lines of the Fano plane, its complete quadrangles and the quaternion subalgebras of the octonions.
1
rigid: one seam between any two incarnations
The one-point object: what the whole group fixes, the plane, the line, the curve, the graph and Kirmse’s lattice.
One fold in depth · fold 12, floor 2 · les sutures
What does going around a loop of natural identifications do?
The composite of seams around a closed walk, an automorphism of the incarnation; it measures how far a family of seams is from one choice of alignments.
Three flexes of the Klein quartic, (1:0:0), (0:1:0) and (0:0:1), on the washi face of a triangle; its vermilion face is their tangents. A pin marks (1:0:0).
A flex goes to its tangent line: (1:0:0) to y = 0, (0:0:1) to x = 0, (0:1:0) to z = 0.
A flex tangent meets the quartic at its flex with multiplicity 3 and at exactly one other point, again a flex, and the tangent goes to that point: y = 0 to (0:0:1). Two natural seams, and the pin is not where it began.
The cycle flexes → flex tangents → flexes has monodromy τ: (1:0:0) ↦ (0:0:1) ↦ (0:1:0) ↦ (1:0:0), of order 3 and power 4. It permutes every flex triangle cyclically.
DefinitionSeam system, monodromy
The monodromy of a cycle of a seam system, starting and ending at , is
The word is used as for coverings: going around a loop of identifications returns a permutation of the fibre. Here the fibre is an incarnation and the permutation is an automorphism of the object; through an alignment it is an element of , well defined up to conjugation.
Proposition
If is abelian, then for each incarnation the isomorphism given by an alignment does not depend on the alignment, and monodromy is a homomorphism from the fundamental group of the graph of the system to .
Proof
Two alignments differ by an automorphism of , and the two isomorphisms differ by conjugation by , which is trivial in an abelian group. Concatenating cycles composes monodromies.
TheoremMonodromy of the object of size 24
Let be the composite of the tangent and residual-point seams: a flex of the Klein quartic goes to the other flex on its tangent. Then is an automorphism of the flexes of power 4, so the cycle flexes flex tangents flexes, along the two natural seams, has monodromy of order 3. It permutes each flex triangle cyclically:
Proof
The rotation of is , and the rotation of is the element acting there by , which is , since acts there by and so by . As , the rotation seam carries to a map sending to , which is the fourth-power map.
Remark
So the answer for non-rigid objects is mixed. Seams fixed by the conventions of their theories are consistent wherever they meet. But a single theory may supply two natural seams between the same two incarnations, and then a cycle of length two already has nontrivial monodromy: the contact point and the residual point of a flex tangent, or the roles of a point in its line, whose three seams have relative powers 1, 2 and 4. The monodromy is then an invariant of the theory; here it is the cyclic order that the tangents put on each flex triangle, a fact of the projective geometry of the quartic.
TheoremThe Coxeter edges and the marking
Let the Coxeter graph be in its antiflag model, with acting through a marking . Each edge has the form , with and meeting in the third point of the line .
(a) The point rule, which goes from each point of off to the third point of its line with , and from each point of off to the third point of its line with , traces a directed 4-cycle on the quadrangle complementary to . The line rule traces, dually, a directed 4-cycle on the four lines missing . Each rule, followed by the element of order 4 that advances its cycle one step, is a seam from the edges to , and the two rules give mutually inverse elements.
(b) The vertex seam sends an antiflag to the pair of points of with the same stabilizer, and an edge to a harmonic pair of disjoint pairs . Of the two directed 4-cycles and , exactly one has a nonzero square, and the bracket rule sends the edge to the element of order 4 advancing that cycle one step.
(c) If differs from by an inner automorphism, the bracket rule agrees with the point rule on every edge; if by an outer one, it agrees with the line rule.
Consequently the seam system for formed by the Coxeter edges, the harmonic pairs of pairs and the class , with the vertex seam, the bracket rule and the point rule, is coherent when the marking is in the class of , and its monodromy is the nontrivial automorphism otherwise.
Proof
(a) The rules use only incidence and treat the two antiflags of an edge alike, so they are -maps; that they give inverse elements was checked by machine. (b) In each point occurs twice, so its square class does not depend on the coordinate vectors, and it is invariant under . With , , harmonicity gives , and the products for the cycle are all in the square class of , while the reverse cycle gives that of ; as is not a square modulo 7, exactly one cycle has a square product. (c) For the agreement was checked on all 42 edges. An inner change of marking is induced by a collineation, which commutes with all the constructions. An outer change, by conjugation with a Möbius map of non-square determinant, multiplies every bracket by a non-square, so it reverses the bracket rule.
PropositionMonodromy as the kernel of holonomy
Let a seam system over a connected graph have holonomy , read as a description; its kernel is the group of loops around which the seams close up. The system is coherent if and only if , and the group of monodromies is . So monodromy is what remains of the loops once the kernel of the holonomy is divided out.
Proof
Holonomy is a homomorphism on the fundamental group, and the system is coherent exactly when every holonomy is trivial.
Examplecomputed
Monodromy can be the spinor sign. In the lattice preserved by for one class of tetrahedra of Thurston’s manifold, the 224 half-roots form two copies and of the new object of size 112, whose automorphism group is . The reflection seam, changing the sign of a half-root at the point of its support fixed by its stabilizer, is a seam from to and back, and the cycle it forms has trivial monodromy, since its square is the identity. The sign seam, the sign pattern of on the support of , with the conference matrix of the Weil representation, equals the reflection seam on and its negative on : the cycle it forms has monodromy . Half of off the support is an automorphism of of order 4 with square , the integral shadow of multiplication by , and it generates the automorphism group.
Open questionopen
On the object the automorphism group is , not abelian, so monodromy is defined only up to conjugation. A natural seam system with non-abelian monodromy is not known: the natural seams found there, the elation and centre seams, carry swaps to swaps and rotations to rotations, and are coherent.