status

Floor 3, Ce qui est su · introduced in Chapter 1, Un objet, plusieurs noms

How much is known about a bridge?

Built, type, name or refuted: a record of what is known about a bridge, not a property of the objects.

Plate 3.2The four statuses of a bridge and the moves between them, with examples around the object of size 28. A status records knowledge, not the objects.

Definition(Bridge, status)

The status of a bridge is one of the following.

Built: the symmetry groups are marked by $G$ , and a seam has been written down and proved to be a $G$ -map.

Type: the two sides are known to share some invariant of objects, such as the group, the number of elements, a stabilizer up to abstract isomorphism or the permutation character, but no seam has been written down.

Name: the two sides share a name, and nothing more is known.

Refuted: it is proved that the two sides are not incarnations of one object, that is, that no seam exists between them for the markings in question.

A status describes knowledge, not the objects. A bridge of status type or name may later be built, or may be shown to be false.

Remark

The strongest invariant short of the stabilizer class that a type bridge usually records is the permutation character, and it does not suffice: two sets can have isomorphic permutation representations over $\C$ and still admit no seam. Type recurrence is not identification.

An empty cell is not the same as a cell empty by necessity. In the table that follows objects up the tower, a cell is filled when an incarnation there is proved, and an empty cell is unexplored. In the dictionary of a double life, a missing entry means unnamed in that life, not refuted.

Example

Around the object of size 28: the pairs, the Sylow 3-subgroups, the antiflags, the bitangents, the Coxeter vertices and the 28 ideal edges of Thurston’s congruence link complement are joined by built bridges. The 28 ideal tetrahedra of the same manifold have the same size and the same group, a type bridge, and it is refuted. The number three that is the threshold of Gleason’s theorem and the number three in $\mathrm{PG}(2,2)=\Proj(\F_2^3)$ share only a name: one counts the dimension at which contexts first overlap, the other the coordinates of a plane, and no map relates them.

Proposition(Statuses)

Let $X$ and $Y$ be transitive $G$ -sets, incarnations of objects in two theories with fixed markings. (a) A description $X\to Y$ exists if and only if a stabilizer of $X$ is contained in a stabilizer of $Y$ ; it forgets nothing, its kernel at each point being the stabilizer of that point, exactly when it is a seam. (b) The bridge between $X$ and $Y$ is built if and only if there is a description $X\to Y$ that forgets nothing. If $|X|=|Y|$ , every description $X\to Y$ is a seam, so the bridge is refuted if and only if there is no description from $X$ to $Y$ at all.

Proof

(a) A $G$ -map $G/H\to G/L$ sending $H$ to $gL$ exists exactly when $H\le gLg^{-1}$ , and it is surjective; it forgets a copy of $gLg^{-1}/H$ at the base point, which is trivial exactly when the map is a bijection. (b) A surjection between finite sets of equal size is a bijection; then apply the stabilizer principle.

Remark(Statuses inside the table)

Inside the closed seam table a bridge between two entries is built when they lie in one row and refuted when they lie in different rows, since objects with different stabilizer classes are not isomorphic for a fixed marking. So no bridge inside the table has status type or name. The refuted bridges met by name are the points and the lines of the Fano plane, the other two Gassmann pairs, the two classes of tetrahedra of Thurston’s manifold, and the tetrahedra against the object of size 28; for the pairs exchanged by the outer automorphism the refutation holds for a fixed marking, and the bridge is built over the outer automorphism.

Example(Two completions of the sky)

The group of order 168 is the reduction at a prime over 7 of two arithmetic groups, the Bianchi group $\PSL(2,\Z[\zeta])$ , $\zeta=(1+\sqrt{-3})/2$ , and Mumford’s group, and each acts on a tree at 7 whose base link is the sky. The bridge between the two parents’ completions of the sky is built on the links, which are incarnations of the rigid object of size 8 joined by exactly one seam; a type on the bare trees, both regular of valence 8 with no distinguished isomorphism; and refuted on the trees with their symmetry, where two separating invariants tell them apart.