of random 2-complexes Michael Farber University of Warwick, UK - - PowerPoint PPT Presentation

Geometry and Topology

• f random 2-complexes

Michael Farber University of Warwick, UK July 2013

Geometry and topology of random 2-complexes 1

Let be a 2-dimensional finite simplicial complex. is called if ( ) = 0. Equivalently, is if the universal cover is contractible. Examples of aspherical 2-complex aspheric es: wi al

2 g

aspheric X X X X al X   th > 0; N with > 1. Non-aspherical are and (the real projective plane).

2 2 g

g g S P

The Whitehead Conjecture

Geometry and topology of random 2-complexes 2

Is every subcomplex of an aspherical 2-complex also asph In 1941, J.H.C. Whitehead suggested the following question: ? This question is known as the Whitehead conjec e t rical ure.

Geometry and topology of random 2-complexes 3

Geometry and topology of random 2-complexes 4

Equivalently one may ask: suppose that is a connected 2-complex with and let be obtained by attaching a 2-cell. Is (L) 0?

2 2 1 2

    

f

K K L K D f S K ( ) , :  

Geometry and topology of random 2-complexes 5

(J.F. Adams, 1955): If and (K) while (L) then the kernel of the homomorphism contains a nontrivial perfect subgroup. This implies some (earlier) results of W.H. Coc

2 2 2 1 1

    

f

L K D K L Theorem , , ( ) ( )     kcroft who considered the cases when is finite, free, or free abelian.

1 K

( ) 

Geometry and topology of random 2-complexes 6

Can one test the Whitehead Conjecture probabilistically?

• 1. Produce aspherical 2-complexes randomly;
• 2. Estimate the probability that the Whitehead Conjecture

is satisfied Question : Tasks :

Geometry and topology of random 2-complexes 7

 

Consider the complete graph

• n vertices

A random 2-complex X is obtained from by adding each potential 2-simplex at random, with probabiltiy 1 2 0 1 

n n

K n n K ijk p The Linial - Meshulam model , , , . ( ) ( , ), independently of each other. The finite probability space contains simplicial complexes satisfying and the probability function is given by

3 1 2 3

2 1

          

    

n n n n f Y

Y n p Y P Y n p R P Y p p

( ) ( ) ( )

( , ) : ( , ) ( ) ( )  

  f Y ( )

.

Geometry and topology of random 2-complexes 8

Case n=4:

p4

 

p p 

3

4 1  

p p 

2 2

6 1

 

p p 

3

4 1

 

p 

4

1

Geometry and topology of random 2-complexes 9

For simplicity I will assume that where If then for any fixed finite group of coefficients

• ne has

H a.a.s. ( ). asymptotically almost sur in ely (L Topology of random 2 - complexes , . ( ; ) , G Y G p n       

1

1 ial-Meshulam). If then Y simplicially collapses to a graph, a.a.s. (Kozlov, Costa-Cohen-Farber-Kappeler, Aronshtam-Linial-Luczak-Meshulam). If >-1 then and (Kozlov) ( ) ( ) , . . . H Y Y a a s       

2 2

1

Geometry and topology of random 2-complexes 10

If then is simply connected, a.a.s. If < then is nontrivial and is hyperbolic in the sense of Gromov, a.a.s. Babson, Hoffman, Kahle, 2011.

1

1 2 1 2  Y Y ( )   

Geometry and topology of random 2-complexes 11

• 1/2
• 1

H H H H

1 1 1 1

=0 =0 =0

2 2 0

Phase transitions

Geometry and topology of random 2-complexes 12

 



if extends to an embedding

• therwise

If then contains a subcomplex isomorphic to the tetrahedron T.

, ,

: ( ) , ,..., : ( , ) ( ) ( )

g g T Y g g

Y g V T n J Y n p J Y E J p 



     

1 4

1 1 2

T

Geometry and topology of random 2-complexes 13

X counts the number of tetrahedra in a random 2-complex. if

( )

, : ( , ) ( ) , .

g g

X J X Y n p n E X p n p n







             



4 4 4 4 1

4 1

Geometry and topology of random 2-complexes 14

The results stated below were obtained jointly with Armindo Costa.

Geometry and topology of random 2-complexes 15

If < -3/5 then the fundamental group

• f a random

2-complex has cohomological dimension 2, a.a.s. In particular, is torsion free, a.a.s. Moreover, if then cd( Theorem : ( ) ( , ) ( ) / ( A Y Y Y n p Y Y           

1 1 1

1 3 5 )) , . . . a a s  2

Geometry and topology of random 2-complexes 16

 

Y       

1

3 5 1 2 If the probability parameter satisfies then the fundamental group contains elements of order 2, a.a.s. Th Theorem B : / / / /

Geometry and topology of random 2-complexes 17

1 2 3 1 2 3

Triangulation of projective plane with 6 vertices and 10 faces. (Note: 3/5=6/10)

Geometry and topology of random 2-complexes 18

 

m n Y Y n p         3 1 2 Let be an odd prime. If the proability parameter satisfies then, with probability tending to one as a random 2-complex has the following property: the fundamental gro Th Theorem C : / , , , , Y Y m    up of any subcomplex has no torsion '

Geometry and topology of random 2-complexes 19



• 1
• 3/5
• 1/2

Free No torsion Has 2-torsion Trivial Torsion in the fundamental group of a random 2-complex

Geometry and topology of random 2-complexes 20



• 1
• 3/5
• 1/2

CD=1 CD=∞ CD=2 Cohomological dimension of fundamental group of a random 2-complex

Geometry and topology of random 2-complexes 21

 

aspherical Assume that the probability parameter satisfies <-1/2. Then a random 2-complex , , with probability tending to one has the following property: any subcomplex is i Th Theorem D : , ' Y Y n p p n Y Y



    

 

f and only if it contains no subcomplexes with at most faces which are homeomorphic to the sphere the real projective plane

• r to the complexes

shown below. , , , , , S P Z Z 



 

1 2 2 2 3

4 1 2

Geometry and topology of random 2-complexes 22

Z Z

2 3

Complexes (left) and (right)

Geometry and topology of random 2-complexes 23

 

Y Y n p Y Y      1 2 Assume that Then a random 2-complex with probability tending to one has the following property: any aspherical subcomplex Whitehe satisfies the a , i.e. d Conje e c ve t ry su ure Co Corollary : / . / . , ' Y Y  bcomplex is also aspherical. '' '' '

Geometry and topology of random 2-complexes 24

 

     

X X

X S X A X I X A        

1 1

Let be a simplicial 2-complex. For a simplicial null-homotopic loop

• ne defines the

and the The length area isoperimetric

• f

is constant defined as Is Isoperimetr tric consta tants ts : . in inf ;

   

S X I X X             

1 1

0 iff the fundamental group hyperbol is . ic : . : .

Geometry and topology of random 2-complexes 25

   

The inequality means that an is satisfied for any null-homotopic loop It is known that in the class of hype isoperimetric inequ rbolic groups the word problem, the conj a ugacy lity : . : .

X

I X a A a S X   



    

1 1

problem as well as the isomorphism problem are algorithmocally solvable.

Geometry and topology of random 2-complexes 26

   

Y Y n p p n       2011 1 2 : If the probability parameter satisfies then the fundamental group of a random 2-complex is hyperbolic, a.a.s. Th Theorem Babson, Ho Hoffman,Kahle, / , , , , ,

Geometry and topology of random 2-complexes 27

 

C Y Y n p p n

 

     If the probability parameter satisfies <-1/2 then there exists a constant such that, with probability tending to one, a random 2-complex , has the following property: any sub Th Theorem : , , , , ,

 

Y Y I Y C   complex satisfies ' ' ' ' .

Geometry and topology of random 2-complexes 28

T 

2

For <-1/2 a random 2-complex contains no subcomplexes homeomorphic to the torus , a.a.s. Co Corollary :

Geometry and topology of random 2-complexes 29

   

Y Y M Y   

2

Let be a simplicial complex with Define as the minimal number of faces in a 2-complex homeomorphic to 2-sphere such that there exists a homotopically nontrivial simplicial m Mi Minimal sp sphere res .

   

Y M Y Y     

2

ap We also define if . .

Geometry and topology of random 2-complexes 30

   

Y I Y c M Y c         

2

16 If is a 2-complex satisfying then The proof of this deterministic statement uses an inequality

• f Papasoglu for Cheeger constants of triangulations of the sphere.

Th Theorem : .

Geometry and topology of random 2-complexes 31

           

S

Y S A M Y S h A S A A S S



     



              

2

2 Consider a homotopically nontrivial simplicial map where is homeomorphic to and Cons Cheeger con ider the

• f ,

Here is a subcomplex homeomo sta rph nt Pr Proof : . ; / ; / .

mi min

ic to the disc.

Geometry and topology of random 2-complexes 32

S

Geometry and topology of random 2-complexes 33

             

I Y c h c h A M Y A h c                          

2 2

16 16 16 One can show that implies . On the other hand, proved an inequality Combining we Papasogl

• btain

u . .

Geometry and topology of random 2-complexes 34

 

C Y Y n p p n Y Y

 

        1 2 If the probability parameter satisfies then for some constant a random 2-complex with probability tending to

• ne has the following property: for any subcomplex
• Th

Theorem : / , , , , , '

 

M Y C  ne has ' . ' .

Geometry and topology of random 2-complexes 35

   

Let be a finite 2-complex and let be a constant such that any pure subcomplex having at most faces satisfies Then Gr Gromov's local to global Principle Th Theorem : . . X C S X C I S C I X C



       

3 1

44 44

Geometry and topology of random 2-complexes 36

         

A finite 2-complex is said to be a if and for any proper subcomplex

• ne has

For a minimal cycle we denote We are interest minimal c ed ycle i Cl Classification of minimal cycles ' ' ' ' . . Z b Z Z Z b Z Z v Z Z f Z      

2 2

1

 

n describing all minimal cycles satisfying / . / . Z   1 2

Geometry and topology of random 2-complexes 37

 

Any minimal cycle satisfying is homeomorphic to one of four complexes where and are shown on the following slide.

,

Th Theorem MinCycle : / , , , , , , Z Z Z S Z Z Z P P P Z Z         

2 2 2 1 2 3 4 2 2 1 2 3

1 2

Geometry and topology of random 2-complexes 38

Z Z

2 3

Complexes (left) and (right)

Geometry and topology of random 2-complexes 39

     

A random 2-complex with probability tending to one has the following property: for a subcomplex the following properties are equivalent: is aspherical; contains no Th Theorem : , , , , , / ' ' ' Y Y n p p n Y Y A Y B Y

 

     1 2

 

subcomplexes with at most faces which are homeomorphic to , , , , , . S P Z Z 



 

1 2 2 2 3

4 1 2

Geometry and topology of random 2-complexes 40

 

Y Y n p Y Y      1 2 Assume that Then a random 2-complex with probability tending to one has the following property: any aspherical subcomplex Whitehe satisfies the a , i.e. d Conje e c ve t ry su ure Co Corollary : / . / . , ' Y Y  bcomplex is also aspherical. '' '' '

Geometry and topology of random 2-complexes 41

       

 

Let Then a.a.s. There are finitely many isomorphism types of triangulations

• f the 2-sphere with at most

faces. There are also finitely many simplicial quotients Pr Proof

• f

' , ' , , , . ' , ' ,

j

B A Y Y Y Y n p p n M Y C S C

  

    

 

 

 

 

 

 

• f such triangulations.

The quotients satisfying cannot be embedded into Thus we shall only consider the quotients satisfying . / . / .

j j j j j j

S S Y S             1 2

Geometry and topology of random 2-complexes 42

 

 

     

   

Case when Then the image contains a minimal cycle which by Theorem MinCycle is homeomorphic to one of implies which contradicts our assumption . , , , , , , . .

j j j j j j

b S S Z Z Z Z Z f Z A     



     

2 1 2 3 4 1

4 1 2

Geometry and topology of random 2-complexes 43

 

 

 

   

Case Then one shows that the image contains a projective plane with at most faces which contradicts our assumption . .

j j j j

b S S A   



  

2 1

4 1 2

Geometry and topology of random 2-complexes 44

Wha What does all thi his mean for th the dete terministi tic White tehead Conjectu ture?

Geometry and topology of random 2-complexes 45

 

m n Y Y n p         3 1 2 Let be an odd prime. If the proability parameter satisfies then, with probability tending to one as a random 2-complex has the following property: the fundamental gro Th Theorem C : / , , , , Y Y m    up of any subcomplex has no torsion '

Geometry and topology of random 2-complexes 46

Geometry and topology of random 2-complexes 47

   

 

   

Consider the Moore surface Maps inducing mono on describe torison in

.

Sk Sketch ch of the he proof , , , , , , .

m m m m m

M S D M M Y m Y        

1 2 1 1 1

1 1 1

Geometry and topology of random 2-complexes 48

• triangulation of the Moore surface.

We shall consider simplicial maps such that:

• they induce mono on
• have shortest possible length of the singular curve
• have smallest possible area (th

Y C      

1

 

e number of faces) One defines the number as the number of faces in above.

m

N Y 

Geometry and topology of random 2-complexes 49

       

If then The proof uses systollic inequality (Gromov, Katz, Rudyak,...)

/

Le Lemma :

m

I Y c m N Y c sys A            

2 1 2

6 6