Formulas for the Reidemeister, Lefschetz and Nielsen Coincidenc. Infra-nilmanifolds

Transcription

1 Formulas for the Reidemeister, Lefschetz and Nielsen Coincidence Number of Maps between Infra-nilmanifolds Jong Bum Lee 1 (joint work with Ku Yong Ha and Pieter Penninckx) 1 Sogang University, Seoul, KOREA International Conference on Nielsen Fixed Point Theory and Related Topics Capital Normal University, Beijing, China June 20-24, 2011

2 Motivation For a self map f : M M on a torus, the Nielsen number N(f ) and Lefschetz number L(f ) are equal up to a sign, i.e., N(f ) = L(f ) = det(i f ), where f : π 1 (M) π 1 (M) is the homomorphism on π 1 (M) induced by f. R. B. S. Brooks, R. F. Brown, J. Pak and D. H. Taylor, Nielsen numbers of maps of tori, Proc. Amer. Math. Soc., 52 (1975),

3 Let L be a connected, simply connected nilpotent Lie group, Γ a lattice of it, and M = Γ\L a nilmanifold. Any f : M M is homotopic to a map obtained from an endomorphism F : L L for which F(Γ) Γ. Let F be the corresponding endomorphism of the Lie algebra of L. Then N(f ) = L(f ) = det(i F ). D. V. Anosov, The Nielsen numbers of maps on nil-manifolds, Uspekhi. Mat. Nauk, 40 (1985),

4 They obtained two results: Anosov relation N(f ) = L(f ) Computation Formula L(f ) = det(i F )

5 Generalizations: Relation (1) The Anosov relation N(f, g) = L(f, g) holds for nilmanifolds. C. K. McCord, Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds, II, Topology Appl., 75 (1997), (2) The relation N(f, g) L(f, g) holds for orientable solvmanifolds. P. Wong, Reidemeister number, Hirsch rank, coincidences on polycyclic groups and solvmanifolds, J. reine angew. Math., 524 (2000),

6 Generalizations:Relation and Computation Formula (3) Let M be an infra-nilmanifold with the holonomy group Ψ and f : M M be any self map. Then L(f ) = 1 Ψ N(f ) = 1 Ψ A Ψ det(a f ) = 1 det(i A f ) det A Ψ A Ψ det(a f ). A Ψ K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), S. W. Kim, J. B. Lee and K. B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J., 178 (2005), J. B. Lee and K. B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys., 56 (2006),

7 Generalizations (4) (Computation Formula) L(f, g) = det(g F ) holds for special solvmanifolds of type (R). S. W. Kim and J. B. Lee, Anosov theorem for coincidences on nilmanifolds, Fund. Math., 185 (2005), K. Y. Ha, J. B. Lee and P. Penninckx, Anosov theorem for coincidences on special solvmanifolds of type (R), P. Amer. Math. Soc., 139 (2011), Definition A connected solvable Lie group S is called of type (R) (or completely solvable) if ad(x) : S S has only real eigenvalues for each X S.

8 Goal Our Goal is to obtain averaging formulas for the Reidemeister/Lefschetz/Nilesen coincidence numbers on orientable infra-nilmanifolds, generalizing (3) from fixed point theory to coincidence theory and generalizing (4) from nilmanifolds to infra-nilmanifolds.

9 Algebraic Reidemeister Coincidence Number Suppose we have a commutative diagram of groups: i 1 Γ 1 u 1 1 Π1 Π1 /Γ 1 1 ψ ϕ ϕ ϕ ψ ψ i 1 Γ 2 u 2 2 Π2 Π2 /Γ 2 1 where the top and bottom sequences are exact and the quotient groups Π 1 /Γ 1 and Π 2 /Γ 2 are finite.

10 Algebraic Reidemeister Coincidence Number For any α Π 2, have a commutative diagram i 1 Γ 1 u 1 1 Π1 Π1 /Γ 1 1 ψ τ αϕ τ αϕ τᾱ ϕ ψ ψ i 1 Γ 2 u 2 2 Π2 Π2 /Γ 2 1 have an exact sequence of groups 1 coin(τ α ϕ, ψ ) iα 1 coin(τ α ϕ, ψ) uα 1 coin(τᾱ ϕ, ψ) and have an exact sequence of sets R[τ α ϕ, ψ ] î2 α û2 R[τ α ϕ, ψ] α R[τᾱ ϕ, ψ] 1

11 Algebraic Reidemeister Coincidence Number Lemma For γ Γ 2 and α Π 2, R[ϕ, ψ] = im (îα 2 ), R[τ α ϕ, ψ ] = Theorem [ᾱ] R[ ϕ, ψ] With the diagram before, we have R(ϕ, ψ) [γ] im (îα 2 ) [coin(τᾱ ϕ, ψ) : u γα 1 (coin(τ γαϕ, ψ))] 1 [Π 1 : Γ 1 ] ᾱ Π 2 /Γ 2 R(τ α ϕ, ψ ). When either side of the inequality is finite, then equality occurs if and only if coin(τ α ϕ, ψ) ) Γ 1 for each α Π 2.

12 Topological Reidemeister Coincidence Number Theorem Let f, g : M 1 M 2 be a pair of maps between closed manifolds M i inducing a commutative diagram before where ϕ, ψ : Π 1 Π 2 on π 1. Then: 1 (P. Wong) If coin(τᾱ ϕ, ψ) = { 1} for all α Π 2, then R(f, g) = [ᾱ] R[ ϕ, ψ] R(ᾱ f, ḡ). 2 R(f, g) is finite iff R(ᾱ f, ḡ) is finite for every α Π 2. 3 We have R(f, g) 1 [Π 1 : Γ 1 ] ᾱ Π 2 /Γ 2 R(ᾱ f, ḡ). When either side of the inequality is finite, then equality occurs if and only if coin(τ α ϕ, ψ) ) Γ 1 for each α Π 2.

13 Topological Reidemeister Coincidence Number To translate algebraic results to topological results, we need to know the existence of a commutative diagram before. The following lemma guarantees the existence of such a diagram for infra-nilmanifolds. Lemma Let Π 1 and Π 2 be almost-crystallographic groups and let Γ i be the maximal normal nilpotent subgroup of Π i. Then there exist fully invariant subgroups Λ i Γ i of Π i, which are of finite index, so that any homomorphism Π 1 Π 2 maps Λ 1 into Λ 2. J. B. Lee and K. B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys., 56 (2006),

14 Topological Reidemeister Coincidence Number Theorem Let f, g : M 1 M 2 be a pair of maps between orientable infra-nilmanifolds M i inducing a commutative diagram of groups of the previous slide where ϕ, ψ : Π 1 Π 2 on π 1. Then R(f, g) = 1 [Π 1 : Γ 1 ] ᾱ Π 2 /Γ 2 R(ᾱ f, ḡ). In fact, if f and g have an inessential coincidence class, then both sides are. When all coincidence classes are essential, we can show that coin(τ α ϕ, ψ) is a trivial group.

15 Averaging formula for Lefschetz number:known Averaging formula for Lefschetz fixed point number L(f ) = 1 L( f ) [Π : K ] B. Jiang, Lectures on Nielsen fixed point theory, Contemp. Math., 14, Amer. Math. Soc., Averaging formula for Lefschetz coincidence number C. K. McCord, Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds, II, Topology Appl., 75 (1997),

16 Averaging formula for Lefschetz number:new Proof Averaging formula for Lefschetz coincidence number All spaces are orientable of equal dimension, and covering projections are orientation-preserving. Then L(f, g) = 1 [Π 1 : Γ 1 ] ᾱ Π 2 /Γ 2 L(ᾱ f, ḡ) Use Decomposition of the Coincidence Set p ( Coin(ᾱ f, ḡ) ) = ( ) Coin(γα f, g) Coin(f, g) = [γ] im (îα 2 ) p p [ᾱ] R[ ϕ, ψ] [γ] im (îα 2 ) ( ) Coin(γα f, g).

17 Averaging formula for Nielsen number:known All spaces are orientable of equal dimension. Then N(f, g) 1 [Π 1 : Γ 1 ] ᾱ Π 2 /Γ 2 N(ᾱ f, ḡ) and ( equality occurs ) iff coin(τ α ϕ, ψ) Γ 1 for each α Π 2 with p Coin(α f, g) essential. If the spaces M 1 and M 2 are orientable infra-nilmanifolds, then we( can show that ) coin(τ α ϕ, ψ) = {1} for each α Π 2 with p Coin(α f, g) essential. Hence the equality occurs. S. W. Kim and J. B. Lee, Averaging formula for Nielsen coincidence numbers, Nagoya Math. J., 186 (2007),

18 Practical formulas for Ridemeister/Lefschetz/Nielsen number Let M 1 and M 2 be orientable infra-nilmanifolds of equal dimension with holonomy groups Ψ 1 and Ψ 2 respectively. Recall that the averaging formulas depend on the choice of fully invariant subgroups of almost-bieberbach groups which induce a commutative diagram. For example, N(f, g) = This depends on Γ 1 and Γ 2. 1 [Π 1 : Γ 1 ] ᾱ Π 2 /Γ 2 N(ᾱ f, ḡ) Our goal is to obtain such formulas depending only on Ψ i.

19 Necessary Facts Let M 1 and M 2 be infra-nilmanifolds of equal dimension. Then M i = Π i \G i where G i is a connected simply connected nilpotent Lie group and Π i is an AB-group modeled on G i, and Γ i = Π i G i is the unique maximal nilpotent subgroup of Π i of finite index. Let f, g : M 1 M 2 be maps. Then there exist Lie group homomorphisms D, D : G 1 G 2 and d, d G 2 so that λ d D, λ d D : G 1 G 2 are homotopy lifts of f, g respectively. K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995),

20 Theorem Let M 1 and M 2 be orientable infra-nilmanifolds of equal dimension. Let f, g : M 1 M 2 be maps. Then L(f, g) = 1 Ψ 1 N(f, g) = 1 Ψ 1 R(f, g) = 1 Ψ 1 det(d A D ) A Ψ 2 det(d A D ) A Ψ 2 σ(det(d A D )) A Ψ 2 where σ(0) = and σ(x) = x for x 0; the differential D : G 1 G 2 of D is expressed with respect to preferred bases of Γ 1 and Γ 2.

21 Corollary Let M 1 and M 2 be orientable infra-nilmanifolds of equal dimension. Let f, g : M 1 M 2 be maps. Then N(f, g) = L(f, g) iff det(d A D ) 0 for every A Ψ 2 ; N(f, g) = L(f, g) iff det(d A D ) 0 for every A Ψ 2. This generalizes Theorem 2.2 in: K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), from fixed point theory to coincidence theory.

22 Necessary Fact Theorem Let G i be a connected simply connected nilpotent Lie group and D, D : G 1 G 2 be Lie group homomorphisms. Then for any g G 2, det(d D ) = det(d Ad(g)D ). PROOF. By complexifying G i we may assume G i is a complex Lie group. The right hand side is a polynomial f (Y ) = det(d Ad(g)D ) = det(d Ad(exp(Y ))D ) = det(d exp(ad(y ))D )

23 Necessary Fact Need a fact that for any g G 2, D D is surjective iff D Ad(g)D is surjective. K. Dekimpe and P. Penninckx, The finiteness of the Reidemeister number of morphisms between almost-crystallographic groups, to appear in J. Fixed Point Theory and Appl. Then either the polynomial f is trivial or it has no zeros. If f has no zeros then by FTA, f is a constant polynomial. Hence f (g) = det(d D ).

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