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Outstanding Problems in Low-Dimensional Topology and Group Theory

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Outstanding Problems in Low-Dimensional Topology and Group Theory 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) 1 using the moves: r r − i !→ i r , r r r , r , (i = j) i j !→ i j j # 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-handleP pair to So if is AC-trivializable, then M is diffeomorphic to S 4 . 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 and are related by a sequence Kof elementarL !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.) is of type F n if it admits a with finitely many n -cells. K(Γ, 1) Γ Note. Type F 2 is finite presentability. is of type if admits a partial projective resolution Pn P1 P0 Z 0 → · · · → → → → by finitely generated projective -modules. Z Note. Type implies type FP n . ZΓ 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 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 –FP 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 = .
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