Outstanding Problems in Low-Dimensional Topology and Group Theory

Tim Riley

Fields Workshop in Asymptotic Group Theory and Cryptography at Carleton University, Ottawa

December 14, 2007 The Poincaré Conjecture

Every closed simply connected 3-manifold is homeomorphic to the 3-sphere.

J.H. Poincaré

W.P. Thurston R. Hamilton G. Perelman “How not to prove the Poincaré Conjecture” M a closed 3-manifold

A Heegaard decomposition of M : U1 U2 M = ∪

→

! Σ = U U = ∂U = ∂U g 1 ∩ 2 1 2

Let Γ g := π 1 Σ g and F g = π 1 U 1 = π 1 U 2 . φ U1 Fg Σg induces Γ g ! and φ ψ : Γg Fg Fg ψ × → × that is surjective on each factor — a splitting homomorphism.

φ ψ is surjective iff π 1 M = 1 . × φ ψ 1 × 1 Splitting homomorphisms φ 1 ψ 1 Γg Fg Fg and φ ψ are equivalent when× −→ × 2 2 θ α β there exist× automorphisms × ! ↓ ◦ θ : Γ g and α , β : F g such that: φ ψ 2 × 2

The Poincaré Conjecture is true iff every epimorphism Γ g ! F g F g is equivalent to the standard one: × a , b , . . . , a , b g [a , b ] ! 1 1 g g | i=1 i i " ! a1, . . . , a!g b1, . . . , bg . J.R. Stallings ! " × ! "

Stallings invoked results of Jaco, Papakyriakopoulus, and Waldhausen. In fact, this formulation is due to Hempel. The Andrews–Curtis Conjecture Every balanced presentation a1, . . . , am r1, . . . , rm of the trivial group can! be converted |to " a , . . . , a a , . . . , a ! 1 m | 1 m" using the moves

ri, rj rirj, rj, i = j !→1 # ri ri− !→ 1 1 r a ∓ r a ± . i !→ k i k

Stable Version. Also allow the move

a1, . . . , am r1, . . . , rm ! a , . . . , a| , a r" ,↔. . . , r , a ! 1 m m+1 | 1 m m+1" Some Candidate Counterexamples — all are known to present the trivial group

Akbulut–Kirby (open for n 3 ) ≥ a, b aba = bab, an+1 = bn ! | "

Miller–Schupp a, b a2b = ba3, w = 1 ! | " where the exponent sum of the b ‘s in w is 1 . ±

B.H. Neumann (proposed as a counterexample by Rapaport) 1 2 1 2 1 2 a, b, c c− bc = b , a− ca = c , b− ab = a ! | " (A.D. Miasnikov, A.G. Miasnikov and V. Shpilrain Example trivialization (A.J. Casson) were the first to find some trivialization) a, b aba = bab, a3 = b2 a, b a, b ! | " ! ! | "

1 1 1 3 2 abab− a− b− , a b−

3 2 2 1 1 1 a b− , a b− aba− b−

2 1 1 1 1 1 1 a b− aba− b− , ab− ab− a− b

1 1 1 2 1 1 ab− ab− a− , a b− ab−

2 1 1 2 1 a b− ab− , a b−

2 1 1 a b− , ab−

1 ab− , a a a, b b The Grigorchuk–Kurchanov Conjecture

Let = a , . . . , a . Every 2 n -tuple A { 1 n} ( r 1 , . . . , r n , q1, . . . , qn) 1 of words on ± such that 1 A t− r t i = 1, . . . , n; t F (q , . . . , q ) { i | ∈ 1 n } generates F ( ) , can be converted to the -tuple A R. Grigorchuk (a1, . . . , an, 1, . . . , 1) using the moves:

ri, rj rirj, rj, (i = j) !→1 # ri ri− !→ 1 1 ? r q ∓ r q ± i !→ k i k qi qirj !→ 1 q q − P.F. Kurchanov i !→ i q , q q q , q , (i = j) i j !→ i j j # . The GK-Conjecture implies the AC-Conjecture. — take ( a 1 , . . . , a n ) = ( q 1 , . . . , q n ) .

The GK-Conjecture is true iff every epimorphism F F F is equivalent to the standard one: 2g ! g × g a1, b1, . . . , ag, bg a1, . . . , ag b1, . . . , bg . ! " ! ! " × ! "

(This theorem is established via results of Grigorchuk, Kurchanov, Lysenok. Cf. work of Craggs.) From presentations to 2-complexes

Γ = a , . . . , a r , . . . , r ! 1 m | 1 n" !

a3 r1 r2 r3 a2 a4 K = a1 r r4 n am

π1(K) = Γ by the Seifert–van Kampen Theorem. If m = n and Γ is trivial, then χ ( K ) = 1 and K is contractible. The Smooth 4-Dimensional Poincaré Conjecture

Every closed C ∞ 4-manifold M homotopy equivalent to the 4-sphere is diffeomorphic to the 4-sphere.

K = the pres. 2-complex of a balanced pres. of the trivial group. P 5 Embed K in R . W = a regular nbhd of in B2 B3 = B5 (1-handles) (2-handles) × ∪ B1 B4‘s ∪ B2 B3‘s × × ‘s

M = ∂W is a closed 4-manifold B1 B4 with × χ(M) = χ(S4) = 2 B5

π1M = π1(W ! K) = π1W = π1K = 1. So M S 4 . !h.e. Unstable AC-moves handle slides on W ←→ Stabilization adding a cancelling 1-handle, 2-handle pair to

So if is AC-trivializable, then M is diffeomorphic to S 4 . P

So potential counter-examples to the AC-Conjecture generate potential counterexamples to the C ∞ 4-diml Poincaré Conjecture.

But — (Gompf) The Akbulut–Kirby presentations all give standard 4-spheres. Simple Homotopy and Collapsibility

K , L CW-complexes

Elementary Elementary expansions collapses L K are the K e L : inv!erses of elementary ! collapses. n Write K ! L (or K ! L ) when K and L are related by a sequence of elementar!y collapses and !expansions (involving cells of dim n ). ≤

Spaces X , Y are simple-homotopy equivalent when they are homeomorphic to , such that .

is collapsible ( X pt ) when it is homeomorphic to a complex that can be reduced to a point! via a sequence of elementary collapses. Examples 1. The Dunce Hat: K = — contractible, but not collapsible

K I = × ∪

pt !

n n 1 2. a triangulated manifold with boundary. Then M N − ! for some ( n 1) -complex N n 1 (a spine of ). − − 3. M n a triangulated manifold. M n pt iff M n = B n . ! ∼ Folk-theorem (P. Wright, J.R. Stallings). The Stable AC-Conjecture 3 is true iff K ! pt for every finite contractible 2-complex K . !

Moot Corollary. Were a spine of a closed 3-manifold a counterexample to the Stable AC-Conjecture (as above), would be a counterexample to the Poincaré Conjecture. The Zeeman Conjecture

If K 2 is a finite contractible 2-complex

K2 I pt. × "

E.C. Zeeman

⇒ The= Zeeman Conjecture

The Stable The Poincaré Conjecture AC-Conjecture M a closed simply connected 3-manifold

K a finite Assume is simplicially triangulated. Let N be contractible with the interior of a 3-simplex removed and K be 2-complex a spine of . K K I χ(K) = χ(N) = χ(M) + 1 = 1 ! × = is π1(K) = π1(N) = π1(M) = 1 ⇒ contractible Then by Zeeman, { }! So N I K I pt by Zeeman. K I pt × " × " × " . 4 3 So N I ∼= B and N 0 ∂ ( N I ) ∼= S . So 3 × × { } ⊆ × 2 2 3 K ! pt. But ∂ N ∼= S and in the PL-category an S in an S ! bounds a B 3 by the Schönflies Theorem. 3 3 So N ∼= B and M ∼= S . Whitehead’s Asphericity Question

Is every subcomplex of an aspherical 2-complex, itself aspherical?

J.H.C.Whitehead and friend An Eilenberg-Maclane Space (a K ( Γ , 1) ) for a group Γ is an aspherical CW-complex with π 1 K ( Γ , 1) = Γ .

The geometric dimension, gd( Γ ) , of is the minimal n such that there is a for of dimension .

The cohomological dimension, cd( Γ ) , of is the minimal such that Z admits a resolution 0 Pn P1 P0 Z 0 → → · · · → → → → by projective Z Γ -modules.

Note. cd( Γ ) gd( Γ ) — the cellular chain complex of a yields a ≤projective resolution. The Eilenberg–Ganea Conjecture

Theorem (Stallings–Swan) cd(Γ) = 1 gd(Γ) = 1 Γ is free. ⇐⇒ ⇐⇒

Eilenberg–Ganea Theorem cd(Γ) 3 = cd(Γ) = gd(Γ) ≥ ⇒ ? Eilenberg–Ganea Conjecture cd(Γ) = 2 = gd(Γ) = 2 ⇒ S. Eilenberg T. Ganea

(I am unaware of anywhere Eilenberg and Ganea actually stated this as conjecture.) Γ is of type F n if it admits a K ( Γ , 1) with finitely many n -cells.

Note. Type F 2 is finite presentability.

is of type if Z admits a partial projective resolution Pn P1 P0 Z 0 → · · · → → → → by finitely generated projective Z Γ -modules.

Note. Type implies type FP n .

is of type FP if admits a projective resolution

0 Pn P1 P0 Z 0 → → · · · → → → → by finitely generated projective -modules. Bestvina–Brady Groups A finite graph G with vertices and edges determines a right-angled Artin grAoup E A = [a , a ] = 1, (a , a ) !A | i j ∀ i j ∈ E$. Define Γ to be the kernel of the homomophism A in which a 1 , a . Z !→ ∀ ∈ A → M. Bestvina N is the flag complex with 1-skeleton .

Theorem is

of type FP n iff H ( N ) = 0 , i < n . i ∀ of type FP iff is acyclic.

finitely presentable (of type F 2 ) iff is simply connected. N. Brady Corollary Suppose N is a spine of the Poincaré Homology Sphere. Then Γ is FP but not finitely presentable. (In fact, cd( Γ ) = 2 .) Moreover, either Γ is a counterexample to the Eilenberg–Ganea Conjecture (i.e. has gd( Γ ) = 3 ), or Whitehead’s Asphericity question has a negative answer. (The universal cover of a 2-dimensional K ( Γ , 1) would have a non- contractible subcomplex.)

Main tools – Morse Theory CAT(0) geometry The Relation Gap Problem Γ = a , . . . , a r , . . . , r = F/R ! 1 m | 1 n" where F = F ( a , . . . , a ) and R = r , . . . , r . 1 m !! 1 n"" R F acts on R by conjugation, so induces an action of on R ab = . [R, R]

ab s1, . . . , sk F, Rank of R as a Z Γ -module min k ∃ ∈ . ≤ R = s1, . . . sk ! " $$ %% # " The difference is the relation gap. " " Open question. Is there a presentation with (finite) non-zero relation gap?

If Γ is of type FP 2 then is finitely generated as a -module. So the Bestvina–Brady example has infinite relation gap. Bridson–Tweedale example

m s s m+1 x = 1, x xx− = x , Γ = x, y, s, t n t t n+1 y = 1, y yy− = y ! " # " where m, n > " 1 , and ( m + 1) m 1 and ( n + 1) n 1 "are coprime. − − M. Bridson ab The Z Γ -module R is generated by s s m 1 t t n 1 m n x xx− x− − , y yy− y− − , x y . Express Γ as F ( x , y , s, t ) /R .

Conjecture. R is not the normal closure of 3 elements. M. Tweedale (Cf. examples of K. Gruenberg & P. Linnell, and also some Bestvina–Brady- style examples of Brisdon & Tweedale.) The D (2) Conjecture X a space with universal cover X . enjoys the D ( n ) property when !

Hi(X) = 0 for all i > n , and H n +1 ( X , ) = 0 for all local coefficient systems! M on X . M C.T.C. Wall

Theorem. For n = 2 , a finite CW complex is homotopic to a finite n -dimensional CW! complex iff enjoys the property.

Folk Conjecture. The same is true when n = 2 . 3 Theorem (M. Dyer). If there is a group Γ with H ( Γ , Z Γ) = 0 and a presentation that has a relation gap, and realises the deficiency of , then the D (2) -Conjecture is false. J. Harlander wrote up a proof. See also M. Tweedale’s thesis.

The Bridson–Tweedale examples would satisfy these conditions if they have a relation gap. Magnus Problem

Given a balanced presentation a , . . . , a r , . . . , r ! 1 m | 1 m" of the trivial group, can some r i always be replaced by a primitive element of F ( a 1 , . . . , a m ) whilst triviality of the group is preserved?

W. Magnus

S.V. Ivanov gave an explicit negative example (with m = 3 ). Kervaire–Laudenbach Conjecture

If Γ is non-trivial and r Γ Z , ∈ ∗ Γ Z then ∗ is non-trivial. r ? "" ##

M. Kervaire F. Laudenbach

(Klyachko.) The conjecture is true ? for all torsion free .

There is a good account by R. Fenn. A. Klyachko Remark (S.V. Ivanov). Special case of the Kervaire–Laudenbach Conjecture –

If a 1 , . . . , a m r 1 , . . . , r m is a balanced presentation of the ! | 1 " trivial group and a m ± does not occur in r 1 , . . . , r m 1 , then − Γ = a 1 , . . . , a m 1 r 1 , . . . , r m 1 is also trivial. ! − | − "

By Klyachko, in a counterexample, Γ would contain torsion elements and so its presentation 2-complex would not be aspherical, answering Whitehead’s problem negatively. “Toy” Problems

1.) Can a, b aba = bab, a 4 = b 3 be converted to ! | " a, b a, b using Andrews–Curtis moves? ! | "

2.) Do there exist r 1 , r 2 , r 3 such that 2 s s 3 x = 1, x xx− = x , x, y, s, t 3 t t 4 y = 1, y yy− = y ! " # " " = x, y, s, t r1, r2, r3 ? " ! | "

3.) Does the Bestvina–Brady example have gd( Γ ) = 3 ? Triumphs of Geometry

Poincaré Conjecture — Geometrization + Ricci Flow

Bestvina–Brady — CAT(0) geometry and Morse Theory

Magnus Problem (Ivanov) — small cancellation A (far from complete) list of references / sources of further reading – •S. Akbulut and R. Kirby, A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schonflies conjecture, and the Andrew-Curtis conjecture, Topology 24, no. 4 (1985), 375-390. •J.J. Andrews and M.L. Curtis, Free Groups and Handlebodies, Proc. AMS, Vol. 16, No. 2., pp. 192-195, 1965 •M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math., Vol.129, pp. 445-470, 1997 •K. Brown,Cohomology of Groups •M.M. Cohen, A course in simple-homotopy theory, Springer, Graduate texts in mathematics, 1973 •D.J.Collins, R.I.Grigorchuk, P.F.Kurchanov and H.Zieschang, Combinatorial Group Theory and Applications to Geometry, Springer •R. Craggs, Free Heegaard diagrams and extended Nielsen Transformations I, Michigan Math. J., 26, 1979 •R. Craggs, Free Heegaard diagrams and extended Nielsen Transformations II, Ill. J. Math., 23, 1979 •S. Eilenberg, T. Ganea, On the Lusternik-Schnirelmann category of abstract groups, Annals of Mathematics, 2nd Ser., 65 (1957), no. 3, 517 – 518 •J. Harlander, C. Hog-Angeloni, W. Metzler, S. Rosebrock, Problems in Low-dimensional topology, Encyclopaedia of Mathematics, Springer •S.V. Ivanov, On balanced presentations of the trivial group, Inventiones, Volume 165, Number 3 / September, 2006 •C. Hog-Angeloni, W.Metzler, A.J. Sieradski (eds.), Two-dimensional Homotopy and Combinatorial Group Theory, LMS Lecture Note Series No. 197, C.U.P., 1993 • A.D.Myasnikov, A.G.Myasnikov and V.Shpilrain, On the Andrews-Curtis equivalence, Contemp. Math., Amer. Math. Soc. 296 (2002), 183-198. •C.F. Miller III and P.E. Schupp, Some presentations of the trivial group •E.S. Rapaport, Remarks on groups of order 1, Amer. Math. Monthly, 75, (1968), 714–720 •D. Rolfsen, Cousins of the Poincaré Conjecture (lecture notes), http://www.math.ubc.ca/~rolfsen/ •J.R. Stallings, How not to prove the Poincaré Conjecture, http://math.berkeley.edu/~stall/ •The World of Groups - open problems list, http://www.grouptheory.info/ •P. Wright, Group presentations and formal deformations, Trans. Amer. Math. Soc. 208, (1975), 161–169 •E.C. Zeeman, On the dunce hat, Topology 2, 1964, 341–358.