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

Geometry and Topology of random 2-complexes Michael Farber University of Warwick, UK July 2013 Geometry and topology of random 2-complexes 1

The Whitehead Conjecture Let be a 2-dimensional finite simplicial complex. X  is called if ( ) = 0. X aspheric al X 2 Equivalently, is aspheric al if the universal cover is contractible. X X  Examples of aspherical 2-complex es: wi th > 0; g g N with > 1. g g Non-aspherical are and (the real projective plane). 2 2 S P Geometry and topology of random 2-complexes 2

In 1941, J.H.C. Whitehead suggested the following question: Is every subcomplex of an aspherical 2-complex also asph e rical ? This question is known as the Whitehead conjec t 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 K 2-complex with   ( K ) 0 2 and let    2 1 L K D , f : S K f be obtained by attaching a 2-cell. Is   (L) 0? 2 Geometry and topology of random 2-complexes 5

    (J.F. Adams, 1955): If 2 and (K) Theorem L K D 0 , f 2   while (L) then the kernel of the homomorphism 0 , 2    ( K ) ( L ) 1 1 contains a nontrivial perfect subgroup. This implies some (earlier) results of W.H. Coc kcroft who  considered the cases when is finite, free, or free abelian. 1 K ( ) Geometry and topology of random 2-complexes 6

Question : Can one test the Whitehead Conjecture probabilistically? Tasks : 1. Produce aspherical 2-complexes randomly; 2. Estimate the probability that the Whitehead Conjecture is satisfied Geometry and topology of random 2-complexes 7

The Linial - Meshulam model   Consider the complete graph on vertices K n 1 2 , , , n . n A random 2-complex X is obtained from by adding each K n  potential 2-simplex at random, with probabiltiy ( ijk ) p ( , ), 0 1 independently of each other. The finite probability space Y n p ( , )   n     contains 3 simplicial complexes satisfying 2     ( 1 ) ( 2 ) Y n n  and the probability function is given by P Y n p : ( , ) R   n    f ( Y )     f ( Y ) 3 P Y ( ) p ( 1 p ) . Geometry and topology of random 2-complexes 8

Case n=4: p 4   3  4 p 1 p    3 4 p 1 p   4  1 p   2  2 6 p 1 p Geometry and topology of random 2-complexes 9

Topology of random 2 - complexes  n    For simplicity I will assume that where p , 0 .    If then for any fixed finite group of coefficients one has 1 G  H a.a.s. ( asymptotically almost sur ely ). (L in ial-Meshulam). ( Y G ; ) 0 , 1    If then Y simplicially collapses to a graph, a.a.s. 1 (Kozlov, Costa-Cohen-Farber-Kappeler, Aronshtam-Linial-Luczak-Meshulam).     If >-1 then and (Kozlov) H ( Y ) 0 ( Y ) 0 , . . . a a s 2 2 Geometry and topology of random 2-complexes 10

1   If then is simply connected, a.a.s. Y 2 1   If < then is nontrivial and is hyperbolic ( Y ) 1 2 in the sense of Gromov, a.a.s. Babson, Hoffman, Kahle, 2011. Geometry and topology of random 2-complexes 11

1 1 H =0 H =0 1 1 H 2 0 H =0 2 -1 0 -1/2 Phase transitions Geometry and topology of random 2-complexes 12

   If then contains a subcomplex isomorphic 1 Y to the tetrahedron T.    g V T : ( ) 1 2 , ,..., n  J : Y n p ( , ) g    if extends to an embedding 1 , g T Y J ( Y ) otherwise g 0 ,  4 E J ( ) p g T Geometry and topology of random 2-complexes 13

   X J , X : Y n p ( , ) g g X counts the number of tetrahedra in a random 2-complex.   n       4 4 4 4 1 ( ) E X ( )   p n p n ,   4    if 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

Theorem A :   If < -3/5 then the fundamental group of a random ( Y ) 1   2-complex has cohomological dimension 2, a.a.s. Y Y n p ( , )  In particular, is torsion free, a.a.s. ( Y ) 1        2 Moreover, if then cd( 1 3 / 5 ( Y )) , . . . a a s 1 Geometry and topology of random 2-complexes 16

Theorem B : Th If the probability parameter satisfies  / / / /      3 5 1 2   then the fundamental group  Y 1 contains elements of order 2, a.a.s. Geometry and topology of random 2-complexes 17

3 2 1 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

Theorem C : Th Let be an odd prime.  m 3 If the proability parameter satisfies  / then, with probability tending to one as    1 2   , , a random 2-complex , , has the following    n Y Y n p property: the fundamental gro up of any subcomplex ' has no torsion    Y Y m Geometry and topology of random 2-complexes 19

No torsion Has 2-torsion Free Trivial  -1 -1/2 -3/5 Torsion in the fundamental group of a random 2-complex Geometry and topology of random 2-complexes 20

CD=2 CD=∞ CD=1  -1 -1/2 -3/5 Cohomological dimension of fundamental group of a random 2-complex Geometry and topology of random 2-complexes 21

Theorem D : Th Assume that the probability parameter satisfies <-1/2.     Then a random 2-complex , , , with probability    Y Y n p p n tending to one has the following property: any subcomplex ' is aspherical i f and only if it  Y Y    contains no subcomplexes with at most 1    4 1 2 faces which are homeomorphic to the sphere , the 2 S real projective plane , , or to the complexes , , 2 P Z Z 2 3 shown below. Geometry and topology of random 2-complexes 22

Complexes (left) and (right) Z Z 2 3 Geometry and topology of random 2-complexes 23

Corollary : Co Assume that / / . .    1 2   Then a random 2-complex , with probability  Y Y n p tending to one has the following property: any aspherical subcomplex ' satisfies the Whitehe a d Conje c t ure ,  Y Y i.e. e ve ry su bcomplex '' '' ' is also aspherical.  Y Y Geometry and topology of random 2-complexes 24

Isoperimetr Is tric consta tants ts Let be a simplicial 2-complex. For a simplicial null-homotopic X   loop : one defines the length and    1 1 S X   the area . The isoperimetric constant of is defined as  A X X        inf in ; : : . .    1   I X S X     A    X     0 iff the fundamental group is hyperbol ic .   I X X 1 Geometry and topology of random 2-complexes 25

  The inequality means that an isoperimetric   I X a 0   inequ a lity is satisfied for any null-homotopic      1 A a X loop : : . .   1 S X It is known that in the class of hype rbolic groups the word problem, the conj ugacy problem as well as the isomorphism problem are algorithmocally solvable. Geometry and topology of random 2-complexes 26

  Theorem Babson, Ho Th Hoffman,Kahle, : 2011 If the probability parameter satisfies /     1 2 then the fundamental group of a random 2-complex   , , , , , is hyperbolic, a.a.s. n    Y Y n p p Geometry and topology of random 2-complexes 27

Th Theorem : If the probability parameter satisfies <-1/2 then   there exists a constant , such that,  C 0  with probability tending to one, a random 2-complex   , , , , , has the following property:    Y Y n p p n   any sub complex ' ' satisfies ' ' .   Y Y I Y C  Geometry and topology of random 2-complexes 28

Co Corollary : For <-1/2 a random 2-complex contains no subcomplexes  homeomorphic to the torus , a.a.s. 2 T Geometry and topology of random 2-complexes 29

Minimal sp Mi sphere res   Let be a simplicial complex with .   Y Y 0 2   Define as the minimal number of faces in a M Y 2-complex homeomorphic to 2-sphere such that  there exists a homotopically nontrivial simplicial m ap .   Y     We also define if .    M Y 0 Y 0 2 Geometry and topology of random 2-complexes 30

Th Theorem :   If is a 2-complex satisfying then   Y I Y c 0 2   16   .   M Y    c The proof of this deterministic statement uses an inequality of Papasoglu for Cheeger constants of triangulations of the sphere. Geometry and topology of random 2-complexes 31