Why G31 is K(pi,1)

the K(π, 1) conjecture for complex reflection arrangementsscissors and glue

well-generated complex reflection arrangements are K(π, 1)abstract nonsense

Why G31 is K (π, 1)

David Bessis

Bochum, 22/2/2013

David Bessis Why G31 is K(π, 1)

Conjecture (Brieskorn and ???, early 1970s)

Let W ⊆ GL(V ) be a finite group generated by complexreflections. Let A be the reflection arrangement of W .Then V −⋃H∈AH is a K (π, 1) space.

Brieskorn stated the conjecture in 1971 for complexified realreflection groups.

Deligne solved this case in 1972.

It’s not clear who stated it first in the general complex case.

Candidates for ???: Arnold, Brieskorn, Orlik, Saito...

The conjecture is mentioned in Orlik-Terao’s book (1992).

As of 1992, all but 6 exceptional cases had been solved:Deligne (1972), Nakamura (1983), Orlik-Solomon (1988).

The six missing cases were G24, G27, G29, G31, G33 and G34.

Remark:

K (π, 1)⇔ connected and universal cover is contractible

Remark:

Let’s begin with an interesting case of the conjecture:

Theorem (K (π, 1) conjecture for the A1 arrangement)

The punctured complex line C∗ is a K (π, 1) space.

To prove this, we need:

scissors

imagination

A recipe for constructing the universal cover of C∗:1 slice open C∗ along the negative imaginary half-line below 0

2 call this basic piece U0

3 make copies (Uk)k∈Z4 glue the copies together in a daisy-chain: each copy Uk get

glued along its “left-of-the-cut” side onto the“right-of-the-cut” side of Uk+1.

Disclaimer:

don’t try this at home!

it will fall apart

you cannot glue along a half-line

basic safety principle for a robust glueing:always glue along an open set

Disclaimer:

it will fall apart

Disclaimer:

it will fall apart

Disclaimer:

it will fall apart

Definition (fat basepoint trick)

Let X be a topological space. Let U be a contractible subspace ofX . Let π1(X ) be the fundamental groupoid of X .The fundamental group of X with respect to the fat basepoint U isdefined as the transitive limit

π1(X ,U) := lim−→

u,v∈UHomπ1(X )(u, v)

for the transitive system of isomorphisms given by homotopyclasses of paths within U .

If you don’t like transitive limits, just remember this:

any path starting in u ∈ U and ending in v ∈ U represents anelement of π1(X ,U)

if your intuition requires you to really see a loop, draw a pathwithin U connecting u and v

the product of an element represented by a path withendpoints u, v ∈ U with an element represented by a pathwith endpoints u′, v ′ ∈ U is well-defined

if your intuition requires you to see this product asconcatenation, draw a path within U connecting v and u′

because U is contractible, all the paths you can draw within Uare homotopic

Let X be (path-connected, locally simply-connected) space and letx0 ∈ X be a “basepoint”.To construct the (or rather a) universal cover of X , onetraditionally proceeds as follows:

let π1(X ) be the fundamental groupoid of X

consider Xx0 = x g→ y ∈ Ω(X )|x = x0this set can be equipped with a natural topology(neighborhoods around y)

the group π1(X , x0) naturally acts on it (by concatenation onthe left)

the map (xg→ y) 7→ y is a natural covering whose fibers are

transitively permuted by π1(X , x0)

The same construction works if we replace the basepoint x0 by afat basepoint U :

XU = x g→ y ∈ Ω(X )|x ∈ Ucan naturally be viewed as a universal cover of X .

Let’s go back to the first theorem we want to prove:

Theorem (K (π, 1) conjecture for the A1 arrangement)

We choose as fat basepoint for C∗ the complement of the negativeimaginary half-line:

U = a +√−1b ∈ C∗|a = 0⇒ b > 0.

This fat basepoint is used to define “the” universal group and“the” universal cover.

Definition (tunnels and simples of type A1)

Let x ∈ U and L ∈ R≥0. Assume that the whole affine segment[x , x + L] is within C∗. Then we say that the path [0, 1]→ C∗,t 7→ x + tL is a semitunnel of length L.If in addition x + L ∈ U , we say that the semitunnel is a tunnel.A simple element in π1(C∗,U) is an element that can berepresented by a tunnel.The set of simple elements is denoted by S .

Clearly,S = 0, 1.

Semitunnels represent a certain subspace

within the universal cover:

C∗U = x g→ y ∈ Ω(C∗)|x ∈ U.

Intuitively: U0 is an open patch of five-quarters of a turn. Theadditional quarter can be used to glue copies of it.The fundamental group

π1(C∗,U) ' Z

acts on C∗U by left concatenation. For k ∈ Z, we set

Uk := kU0.

If k ∈ Z is simple, then U0 ∩ Uk 6= ∅

Proof: k can be represented by a tunnel from x to x + L (= anelement of U0) can be viewed as the concatenation of a tunnelfrom x to x + L (= a simple) and a tunnnel from x + L to x + L(= an element of U0).The concatenation of a simple and an element of U0, byconstruction, an element of Uk .

If k ∈ Z is simple, then U0 ∩ Uk 6= ∅

Proof: k can be represented by a tunnel from x to x + L (= anelement of U0) can be viewed as the concatenation of a tunnelfrom x to x + L (= a simple) and a tunnnel from x + L to x + L(= an element of U0).The concatenation of a simple and an element of U0, byconstruction, an element of Uk .

Lemma (“the glueing goes well”)

1 the subspaces(Uk)k∈Z

are open sets in C∗U

2 we have C∗U =⋃

k∈Z Uk3 for all non-empty A ⊆ Z, the intersection

⋂k∈A Uk is either

empty or contractible:

Uk is contractible (“five quarters of a turn”)

Uk ∩ Uk+1 is contractible (“one quarter of a turn”)other intersections are empty.

In other words: S controls the combinatorics of the glueing rule.

k∈Z Uk

3 for all non-empty A ⊆ Z, the intersection⋂

k∈A Uk is eitherempty or contractible:

Uk ∩ Uk+1 is contractible (“one quarter of a turn”)

other intersections are empty.

Theorem (“Topology is a myth, only combinatorics matter.”)

Let C be a collection of open subsets of X , such that:

X =⋃

U∈C U

for all A ⊆ C , the intersection⋂

U∈A U is either empty ofcontractible.

Then X is homotopy equivalent to the geometric realization of thenerve of C (the simplicial complex NC whose k-simplices are setsA ⊆ C such that

⋂U∈A U 6= ∅ and |A| = k + 1).

Disclaimer:

actually, for this to hold, X ought to be paracompact

the saying is: “Unless things are locally horrible, topology is amyth, and only combinatorics matter.”

Theorem (“Topology is a myth, only combinatorics matter.”)

Let C be a collection of open subsets of X , such that:

X =⋃

U∈C U

for all A ⊆ C , the intersection⋂

U∈A U is either empty ofcontractible.

Then X is homotopy equivalent to the geometric realization of thenerve of C (the simplicial complex NC whose k-simplices are setsA ⊆ C such that

⋂U∈A U 6= ∅ and |A| = k + 1).

Disclaimer:

actually, for this to hold, X ought to be paracompact

the saying is: “Unless things are locally horrible, topology is amyth, and only combinatorics matter.”

Corollary (“great, now it’s group theory”)

The universal cover of C∗ is homotopy-equivalent to the Cayleygraph of Z with respect to the generating set 1.

Definition

Let G be a group. A Garside structure on G consists of a finite setS of simple elements satisfying the following axioms:

1 S generates G ,

2 the partial ordering of S defined by