La tour assemblée

Du fonds classique à ce qui est suFrom the classical stock to what is known

The first three joins are the language of the first part. The stabilizer principle joins the classical floor to incarnation: an object is determined by its stabilizer class, and its incarnations are the sets in which theories meet it. Seams join incarnations; they are unique when the stabilizer is self-normalizing, and otherwise they carry monodromy, the holonomy of a lattice gauge connection.

A bridge between two theories has a status, built, type, name or refuted. For two marked sets of one group the stabilizer principle decides it, built or refuted. Beside the statuses stand the absences, theorems that something is not there, with their windows, where the excluded thing can still happen, and their imprints, the structures an absence forces to exist.

L’absence d’ArtinArtin’s absence

From floor three to floor four the join is an absence. Isomorphisms between members of the families $\PSL(n,q)$ and $A_m$ are rare: by Artin’s absence exactly four groups have a double life, $A_5$, $\PSL(2,7)$, $A_6$ and $A_8$. Each double life comes with a dictionary of which natural sets of the two lives are one object, and for the group of order 168 that dictionary is the seam table itself.

Les corps résiduelsResidue fields

From floor four to floor five: each life is the link of a vertex of a building over a local field. At 2 the octonion table glues such links into a building over $\F_2(\!(t)\!)$, and Kato’s Hermitian form glues them, without symmetry, into the building over $\Q_2$; a gluing with the symmetry of order 21 is the octonion one, in characteristic 2. At 7 Mumford’s form has its own tree, whose base link is $\Proj^1(\F_7)$, and over $\Z[1/14]$ the group of order 168 is a vertex stabilizer carrying both lives. That vertex is Klein’s lattice, whose vectors of norms 2 and 3 reduce at 2, at 7 and at the complex place to the objects of sizes 21 and 28, and where the first part’s 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, and 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. The two parents carry independent flips, the sign changes of $\sqrt{-3}$ and $\sqrt{-7}$, of which only the first is seen by the oriented cells of the link complement. They 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 table without reversals exactly on the Iwahori subgroup and must reverse two units on seven loops in eight; carried observer by observer, every loop returns each copy changed only by a relabelling that fixes its clock.

La place archimédienneThe archimedean place

From floor five to floor six: the congruence that gives a residue field at a prime gives, over $\C$, Thurston’s link complement, whose cells are objects of the group. Every row of the seam table is a configuration of its cells, and the Fano incidence among them is the absence of a shared face. The group of order 168 has two arithmetic parents at 7, of opposite real type, which share $\Proj^1(\F_7)$ but not its completion.

The sphere at infinity of hyperbolic space is the celestial and the Bloch sphere. Absences decide which continua exist. In the Cayley plane one point and one imaginary unit carry the intersection of Todorov and Dubois-Violette; in the complexified plane the same point carries the $\mathbf{16}$ of $\mathfrak{so}(10)$, and the copy that is not self-conjugate appears only outside $\mathfrak f_4$.

Sept objets en montant la tourSeven objects up the tower

Seven objects of the group of order 168 go up the tower. In the plane life the object of size 7 is the points or the lines; in the line life the bisections, in two orbits; it is rigid, the two classes exchanged by the outer automorphism and refuted for one marking; at floor five it is the vertices of the 2-adic link; at floor six the complementary pairs of tetrahedra of $M$, or the invariant conics in Klein’s plane. The object of size 8 is the cyclic orientations of the plane and the points of the line, rigid, the neighbours in the 7-adic tree and the cusps of $M$, with no orbit in $\Proj^1(\C)$ or Klein’s plane. The object of size 21 is the flags and a bisection orbit, rigid, the edges of the 2-adic link and the vectors of norm 2 of Klein’s lattice, the partitions of the cusps of $M$ and the centres of involutions in Klein’s plane.

The object of size 28, the antiflags and the pairs, rigid with a refuted bridge to the tetrahedra of $M$, reaches every floor: pairs at distance 3 in the 2-adic link, pairs of neighbours at 7, vectors of norm 3 of Klein’s lattice, the edges of $M$ and the bitangents. The objects of sizes 14, 24 and 56, not rigid, with automorphism groups $C_2$, $C_3$ and $C_2$, reach floor six, as the tetrahedra of $M$, the cusps of $M$ with a parallel class or the flexes of the Klein quartic, and the faces of $M$ or the points of contact of the bitangents. At floor five they are unexplored.

Le toitThe roof

The roof is seam theory itself, 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. The tower’s table is of the other kind, with cells empty only because no one has looked.