Tight triangulations: a link between combinatorics and topology

Transcription

1 Tight tringultions: link between ombintoris nd topology Jonthn Spreer Melbourne, August 15, 2016

2 Topologil mnifolds (Geometri) Topology is study of mnifolds (surfes) up to ontinuous deformtion Complited deformtions mke reognition of mnifold diult (if not impossible) Reognition might still be esy for suiently nie deformtions

3 Convexity Topologil blls / spheres onvexity Limited use in geometry nd topology: mny objets re not blls / nnot be onvex Intuitive notion: mny objets look more onvex thn others

4 Tightness An embedding of topologil spe D into some Euliden spe E d is sid to be tight, if it is s onvex s possible given its topologil onstrints D is tight if nd only if interseting D with hlf spe h does not introdue new topologil fetures (eg. onneted omponents or holes) h H 0 (D) = F H 0 (D h) = F 2 D D Tightness generlises onvexity, nd minimises totl bsolute urvture. 1 1 Developed by Alexndrov, 1938; Milnor, Chern nd Lshof, Kuiper, 1950's.

5 Tightness

6 Morse theory Möbius. Theorie der elementren Verwndtshft, 1863

7 Morse theory Given mnifold M, look t funtion f M R f hs nitely mny ritil levels f 1 (α) f height funtion, f 1 (M) ontour lines Denes hndle deomposition of M nite desription of M Powerful: used to prove h-obordism theorem

8 Morse theory nd tightness Some height funtions re better thn others: fewer ritil levels mens more eient hndle deomposition How good n Morse funtion be? Theorem (Morse reltions) Let M be mnifold, nd let f M R be Morse funtion. Then µ(f ) d i=0 β i (M, Z 2 ) where β i (M, Z 2 ) is the i-th Betti number of M with Z 2 -oeients. If f ttins equlity, it is lled perfet Diretion long whih n objet ppers tight Embedding is tight if nd only if ll projetions re perfet Morse funtions. Deiding whether suh funtion exist is NP-omplete. 2 Is deiding whether ll Morse funtions re perfet onp-omplete? 2 Joswig nd Pfetsh 2006.

9 Finite desription of mnifolds Represent mnifolds (surfes) s simpliil omplexes ,2,4, 1,2,6, 1,3,4, 1,3,7, 1,5,6, 1,5,7, 2,3,5, 2,3,7, 2,4,5, 2,6,7, 3,4,6, 3,5,6, 4,5,7, 4,6, A PL tringultion of mnifold M is simpliil omplex C suh tht every vertex link is PL homeomorphi to the stndrd sphere stc(6) lkc(6) C 3 Question: n we link ombintoril properties of C to topologil properties of M? 2 b d

10 Finite desription of mnifolds Represent mnifolds (surfes) s simpliil omplexes ,2,4, 1,2,6, 1,3,4, 1,3,7, 1,5,6, 1,5,7, 2,3,5, 2,3,7, 2,4,5, 2,6,7, 3,4,6, 3,5,6, 4,5,7, 4,6, A PL tringultion of mnifold M is simpliil omplex C suh tht every vertex link is PL homeomorphi to the stndrd sphere stc(6) lkc(6) C 3 Question: n we link ombintoril properties of C to topologil properties of M? 2 b d Euler Chrteristi

11 Finite desription of mnifolds Represent mnifolds (surfes) s simpliil omplexes ,2,4, 1,2,6, 1,3,4, 1,3,7, 1,5,6, 1,5,7, 2,3,5, 2,3,7, 2,4,5, 2,6,7, 3,4,6, 3,5,6, 4,5,7, 4,6, A PL tringultion of mnifold M is simpliil omplex C suh tht every vertex link is PL homeomorphi to the stndrd sphere stc(6) lkc(6) C 3 Question: n we link ombintoril properties of C to topologil properties of M? Euler Chrteristi, rst Z 2 -Betti number(?) 2 b d

12 Abstrt disrete version of tightness Gol: intrinsi intertion between ombintoris nd topology. C onneted bstrt simpliil omplex with vertex set V (C). W V (C), C[W ] subomplex of C indued by W. C is tight if nd only if W V (C), C[W ] does not introdue ny new topologil fetures. 3 Tight tringultions re onjetured to be strongly miniml 4 3 Bnho 1970, Kühnel Kühnel Lutz

13 Tightness ounting topologil fetures New top. fetures non-perfet PL Morse funtion Compute verge no. of ritil levels per PL Morse funtion For eh subset W V (C), think of PL Morse funtions f C R, s.t. f () < f (b) for ll pirs W, b V (C) W Count ll suh Morse funtions Count topologil fetures of lkc (v) W, for ll v V (C) Morse reltion: Weighted sum (verge) must equl sum of Betti numbers

14 Seprtion nd µ index Denition C simpliil omplex with n verties, the µ index of C is given by where µ(c) = 1 n σ(lk C (v)) = σ(lk C (v)), v V (C) is lled the seprtion index of lk C (v). #(lk C (v)[w ]) 1 W V (C) ( n W ) Theorem (Combintoril Morse reltions) C tringultion of (onn.) mnifold M. Then µ(c) β 1 (M, Z 2 ).

15 Known tight tringultions dim(c) 2: C tight every pir of verties of C spns edge Surfe types dmitting tight tringultions: most orientble nd non-orientble surfes S with Euler hrteristi k Z, k > 3. χ(s) = 1 k(k 7), 6 Mnifolds known to dmit tight tringultions in dim > 2: K 3 surfe, CP 2, SU(3)/SO(3), innitely mny sphere bundles Fous of this tlk is dim(c) = 3: Equlity in ombintoril Morse reltion if nd only if C is tight. 5 whih we will show is purely ombintoril ondition

16 Known tight tringultions dim(c) 2: C tight every pir of verties of C spns edge Surfe types dmitting tight tringultions: most orientble nd non-orientble surfes S with Euler hrteristi k Z, k > 3. χ(s) = 1 k(k 7), 6 Mnifolds known to dmit tight tringultions in dim > 2: K 3 surfe, CP 2, SU(3)/SO(3), innitely mny sphere bundles Fous of this tlk is dim(c) = 3: Equlity in ombintoril Morse reltion if nd only if C is tight. Result: Tight tringultions of 3-mnifolds re strongly miniml, nd must hve stked 2-spheres s vertex links 5 5 whih we will show is purely ombintoril ondition

17 Stked spheres A stked (d + 1)-bll is dened reursively: A (d + 1)-simplex is stked (d + 1)-bll. A simpliil omplex obtined from stked (d + 1)-bll B by gluing (d + 1)-simplex long d-boundry fe of B is stked (d + 1)-bll. Stked (d 1)-sphere: boundry omplex of stked d-bll.

18 Stked spheres A stked (d + 1)-bll is dened reursively: A (d + 1)-simplex is stked (d + 1)-bll. A simpliil omplex obtined from stked (d + 1)-bll B by gluing (d + 1)-simplex long d-boundry fe of B is stked (d + 1)-bll. tringulted (d 1)-spheres boundries of simpliil d-polytopes stked (d 1)-spheres Stked (d 1)-sphere: boundry omplex of stked d-bll.

19 Stked spheres We know: µ is n upper bound for the rst Betti number Question: given simpliil omplex C with n verties, how lrge n µ(c) be? Mximise onneted omponents of subsets of links (i.e., mximise σ(lk C (v)) Intuition: mximum is ttined when number of edges is smll Theorem (Kli 1987) Let S be tringulted d-sphere, d 3. Then S hs t lest s mny edges s stked d-sphere with equlity i S is stked.

20 Stked spheres Vertex links lk C (v) re tringultions of the 2-sphere with n verties, e edges nd t tringles. 2e = 3t (every edge is ontined in extly two tringles) n e + t = 2 (Euler hrteristi) f (S) = (n, 3n 6, 2n 4) number of edges is lwys the sme

21 Stked spheres Vertex links lk C (v) re tringultions of the 2-sphere with n verties, e edges nd t tringles. 2e = 3t (every edge is ontined in extly two tringles) n e + t = 2 (Euler hrteristi) f (S) = (n, 3n 6, 2n 4) number of edges is lwys the sme

22 Stked spheres Vertex links lk C (v) re tringultions of the 2-sphere with n verties, e edges nd t tringles. 2e = 3t (every edge is ontined in extly two tringles) n e + t = 2 (Euler hrteristi) f (S) = (n, 3n 6, 2n 4) number of edges is lwys the sme Tringultion S σ(s) Tringultion S σ(s) 0 1/5 b b 27/35 5/7 b b 11/9 71/63 b b 23/21 22/21 b b

23 Stked spheres Theorem (Burton, Dtt, Singh, S. 2014) Let S be n n-vertex tringulted 2-sphere. Then (n 8)(n + 1) σ(s), 20 where equlity ours if nd only if S is stked sphere. Bound on σ

24 Stked spheres Theorem (Burton, Dtt, Singh, S. 2014) Let S be n n-vertex tringulted 2-sphere. Then (n 8)(n + 1) σ(s), 20 where equlity ours if nd only if S is stked sphere. Bound on σ Bound on µ

25 Stked spheres Theorem (Burton, Dtt, Singh, S. 2014) Let S be n n-vertex tringulted 2-sphere. Then (n 8)(n + 1) σ(s), 20 where equlity ours if nd only if S is stked sphere. Bound on σ Bound on µ Bound on β 1 (M, Z 2 )

26 Results Corollry (See lso Lutz, Sulnke, Swrtz, 2008) Let C be tringultion of 3-mnifold M with n verties. Then n 1 2 ( β 1 (M, Z 2 )) nd C is tight if equlity is ttined.

27 Results Corollry (See lso Lutz, Sulnke, Swrtz, 2008) Let C be tringultion of 3-mnifold M with n verties. Then n 1 2 ( β 1 (M, Z 2 )) nd C is tight if equlity is ttined. Theorem (Bghi, Dtt, S. 2016) Let C be tight tringultion of 3-mnifold with n verties. Then ll of its vertex links re (n 1)-vertex stked 2-spheres.

28 Results Corollry (See lso Lutz, Sulnke, Swrtz, 2008) Let C be tringultion of 3-mnifold M with n verties. Then n 1 2 ( β 1 (M, Z 2 )) nd C is tight if nd only if equlity is ttined. Theorem (Bghi, Dtt, S. 2016) Let C be tight tringultion of 3-mnifold with n verties. Then ll of its vertex links re (n 1)-vertex stked 2-spheres.

29 Results Corollry (See lso Lutz, Sulnke, Swrtz, 2008) Let C be tringultion of 3-mnifold M with n verties. Then n 1 2 ( β 1 (M, Z 2 )) nd C is tight if nd only if equlity is ttined. Theorem (Bghi, Dtt, S. 2016) Let C be tight tringultion of 3-mnifold with n verties. Then ll of its vertex links re (n 1)-vertex stked 2-spheres. Tightness for 3-mnifold is purely ombintoril

30 Results Corollry (See lso Lutz, Sulnke, Swrtz, 2008) Let C be tringultion of 3-mnifold M with n verties. Then n 1 2 ( β 1 (M, Z 2 )) nd C is tight if nd only if equlity is ttined. Theorem (Bghi, Dtt, S. 2016) Let C be tight tringultion of 3-mnifold with n verties. Then ll of its vertex links re (n 1)-vertex stked 2-spheres. Tightness for 3-mnifold is purely ombintoril Tight tringultions of 3-mnifolds re miniml, nd their rst Betti number is given by their number of verties

31 Thnk you B. Burton, B. Dtt, N. Singh, J. Spreer. Seprtion index of grphs nd stked 2-spheres. J. Combin. Theory (A), 136: , B. Bghi, B. Dtt, J. Spreer. A hrteriztion of tightly tringulted 3-mnifolds, rxiv: