Formulas for the Reidemeister, Lefschetz and Nielsen ...maths.sogang.ac.kr/jlee/Symp/CNU2011.pdf ·...

Formulas for the Reidemeister, Lefschetz andNielsen Coincidence Number of Maps between

Infra-nilmanifolds

Jong Bum Lee1

(joint work with Ku Yong Ha and Pieter Penninckx)

1Sogang University, Seoul, KOREA

International Conferenceon Nielsen Fixed Point Theory and

Related TopicsCapital Normal University, Beijing, China

June 20-24, 2011

Jong Bum Lee (joint work with Ku Yong Ha and Pieter Penninckx) Formulas for the Reidemeister, Lefschetz and Nielsen Coincidence Number of Maps between Infra-nilmanifolds

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∗ : π1(M) → π1(M) is thehomomorphism on π1(M) induced by f .

R. B. S. Brooks, R. F. Brown, J. Pak and D. H. Taylor, Nielsennumbers of maps of tori, Proc. Amer. Math. Soc., 52 (1975),398–400.


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 anendomorphism F : L → L for which F (Γ) ⊂ Γ.Let F∗ be the corresponding endomorphism of the Lie algebraof 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), 133–134.


They obtained two results:

Anosov relation N(f ) = |L(f )|Computation Formula L(f ) = det(I − F∗)


Generalizations: Relation

(1) The Anosov relation N(f ,g) = |L(f ,g)| holds fornilmanifolds.

C. K. McCord, Lefschetz and Nielsen coincidence numberson nilmanifolds and solvmanifolds, II, Topology Appl., 75(1997), 81–92.

(2) The relation N(f ,g) ≥ |L(f ,g)| holds for orientablesolvmanifolds.

P. Wong, Reidemeister number, Hirsch rank, coincidenceson polycyclic groups and solvmanifolds, J. reine angew.Math., 524 (2000), 185–204.


Generalizations:Relation and Computation Formula

(3) Let M be an infra-nilmanifold with the holonomy group Ψand f : M → M be any self map. Then

L(f ) =1|Ψ|

∑A∈Ψ

det(A∗ − f∗)det A∗

=1|Ψ|

∑A∈Ψ

det(I − A∗f∗)

N(f ) =1|Ψ|

∑A∈Ψ

|det(A∗ − f∗)|.

K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168(1995), 157–166.

S. W. Kim, J. B. Lee and K. B. Lee, Averaging formula forNielsen numbers, Nagoya Math. J., 178 (2005), 37–53.

J. B. Lee and K. B. Lee, Lefschetz numbers for continuousmaps, and periods for expanding maps oninfra-nilmanifolds, J. Geom. Phys., 56 (2006), 2011-2023.


Generalizations

(4) (Computation Formula) L(f ,g) = det(G∗ − F∗) holds forspecial solvmanifolds of type (R).

S. W. Kim and J. B. Lee, Anosov theorem for coincidenceson nilmanifolds, Fund. Math., 185 (2005), 247–259.

K. Y. Ha, J. B. Lee and P. Penninckx, Anosov theorem forcoincidences on special solvmanifolds of type (R), P. Amer.Math. Soc., 139 (2011), 2239–2248.

Definition

A connected solvable Lie group S is called of type (R) (orcompletely solvable) if ad(X ) : S → S has only realeigenvalues for each X ∈ S.


Goal

Our Goal is to obtain

averaging formulas for the Reidemeister/Lefschetz/Nilesencoincidence numbers on orientable infra-nilmanifolds,

generalizing (3) from fixed point theory to coincidence theoryand generalizing (4) from nilmanifolds to infra-nilmanifolds.


Algebraic Reidemeister Coincidence Number

Suppose we have a commutative diagram of groups:

1 −−−−→ Γ1i1−−−−→ Π1

u1−−−−→ Π1/Γ1 −−−−→ 1

ϕ′yψ′ ϕ

yψ ϕ

yψ1 −−−−→ Γ2

i2−−−−→ Π2u2−−−−→ Π2/Γ2 −−−−→ 1

where the top and bottom sequences are exact and thequotient groups Π1/Γ1 and Π2/Γ2 are finite.



For any α ∈ Π2, have a commutative diagram

1 −−−−→ Γ1i1−−−−→ Π1

u1−−−−→ Π1/Γ1 −−−−→ 1

ταϕ′yψ′ ταϕ

yψ ταϕ

yψ1 −−−−→ Γ2

i2−−−−→ Π2u2−−−−→ Π2/Γ2 −−−−→ 1

have an exact sequence of groups

1 −→ coin(ταϕ′, ψ′)

iα1−→ coin(ταϕ,ψ)uα

1−→ coin(ταϕ, ψ)

and have an exact sequence of sets

R[ταϕ′, ψ′]

iα2−→ R[ταϕ,ψ]uα

2−→ R[ταϕ, ψ] −→ 1



Lemma

For γ ∈ Γ2 and α ∈ Π2,

|R[ϕ,ψ]| =∑

[α]∈R[ϕ,ψ]

|im (iα2 )|,

|R[ταϕ′, ψ′]| =

∑[γ]∈im (iα2 )

[coin(ταϕ, ψ) : uγα1 (coin(τγαϕ,ψ))]

Theorem

With the diagram before, we have

R(ϕ,ψ) ≥ 1[Π1 : Γ1]

∑α∈Π2/Γ2

R(ταϕ′, ψ′).

When either side of the inequality is finite, then equality occursif and only if coin(ταϕ,ψ)

)⊂ Γ1 for each α ∈ Π2.


Topological Reidemeister Coincidence Number

Theorem

Let f ,g : M1 → M2 be a pair of maps between closed manifoldsMi inducing a commutative diagram before whereϕ,ψ : Π1 → Π2 on π1. Then:

1 (P. Wong) If coin(ταϕ, ψ) = {1} for all α ∈ Π2, then

R(f ,g) =∑

[α]∈R[ϕ,ψ]

R(αf , g).

2 R(f ,g) is finite iff R(αf , g) is finite for every α ∈ Π2.3 We have

R(f ,g) ≥ 1[Π1 : Γ1]

∑α∈Π2/Γ2

R(αf , g).

When either side of the inequality is finite, then equalityoccurs if and only if coin(ταϕ,ψ)

)⊂ Γ1 for each α ∈ Π2.



To translate algebraic results to topological results, we need toknow the existence of a commutative diagram before.

The following lemma guarantees the existence of such adiagram for infra-nilmanifolds.

Lemma

Let Π1 and Π2 be almost-crystallographic groups and let Γi bethe maximal normal nilpotent subgroup of Πi . Then there existfully invariant subgroups Λi ⊂ Γi of Πi , which are of finite index,so that any homomorphism Π1 → Π2 maps Λ1 into Λ2.

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), 2011-2023.



Theorem

Let f ,g : M1 → M2 be a pair of maps between orientableinfra-nilmanifolds Mi inducing a commutative diagram of groupsof the previous slide where ϕ,ψ : Π1 → Π2 on π1. Then

R(f ,g) =1

[Π1 : Γ1]

∑α∈Π2/Γ2

R(αf , g).

In fact, if f and g have an inessential coincidence class, thenboth sides are ∞.When all coincidence classes are essential, we can show thatcoin(ταϕ,ψ) is a trivial group.


Averaging formula for Lefschetz number:Known

Averaging formula for Lefschetz fixed point number

L(f ) =1

[Π : K ]

∑L(f )

B. Jiang, Lectures on Nielsen fixed point theory, Contemp. Math.,14, Amer. Math. Soc., 1983.

Averaging formula for Lefschetz coincidence number

C. K. McCord, Lefschetz and Nielsen coincidence numbers onnilmanifolds and solvmanifolds, II, Topology Appl., 75 (1997),81–92.


Averaging formula for Lefschetz number:New Proof

Averaging formula for Lefschetz coincidence numberAll spaces are orientable of equal dimension, and coveringprojections are orientation-preserving. Then

L(f ,g) =1

[Π1 : Γ1]

∑α∈Π2/Γ2

L(αf , g)

Use “Decomposition of the Coincidence Set”

p(Coin(αf , g)

)=

∐[γ]∈im (iα2 )

p(

Coin(γαf , g))

Coin(f ,g) =∐

[α]∈R[ϕ,ψ]

∐[γ]∈im (iα2 )

p(

Coin(γαf , g)).


Averaging formula for Nielsen number:Known

All spaces are orientable of equal dimension. Then

N(f ,g) ≥ 1[Π1 : Γ1]

∑α∈Π2/Γ2

N(αf , g)

and equality occurs iff coin(ταϕ,ψ) ⊂ Γ1 for each α ∈ Π2 with

p(

Coin(αf , g))

essential.

If the spaces M1 and M2 are orientable infra-nilmanifolds, thenwe can show that coin(ταϕ,ψ) = {1} for each α ∈ Π2 with

p(

Coin(αf , g))

essential. Hence the equality occurs.

S. W. Kim and J. B. Lee, Averaging formula for Nielsencoincidence numbers, Nagoya Math. J., 186 (2007), 69–93.


Practical formulas for Ridemeister/Lefschetz/Nielsennumber

Let M1 and M2 be orientable infra-nilmanifolds of equaldimension with holonomy groups Ψ1 and Ψ2 respectively.

Recall that the averaging formulas depend on the choice of fullyinvariant subgroups of almost-Bieberbach groups which inducea commutative diagram. For example,

N(f ,g) =1

[Π1 : Γ1]

∑α∈Π2/Γ2

N(αf , g)

This depends on Γ1 and Γ2.

Our goal is to obtain such formulas depending only on Ψi .


Necessary Facts

Let M1 and M2 be infra-nilmanifolds of equal dimension. Then

Mi = Πi\Gi

where Gi is a connected simply connected nilpotent Lie groupand Πi is an AB-group modeled on Gi , and Γi = Πi ∩Gi is theunique maximal nilpotent subgroup of Πi of finite index.

Let f ,g : M1 → M2 be maps. Then there exist Lie grouphomomorphisms D,D′ : G1 → G2 and d ,d ′ ∈ G2 so thatλd ◦D, λd ′ ◦D′ : G1 → G2 are homotopy lifts of f ,g respectively.

K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168(1995), 157–166.


Theorem

Let M1 and M2 be orientable infra-nilmanifolds of equaldimension. Let f ,g : M1 → M2 be maps. Then

L(f ,g) =1|Ψ1|

∑A∈Ψ2

det(D′∗ − A∗D∗)

N(f ,g) =1|Ψ1|

∑A∈Ψ2

|det(D′∗ − A∗D∗)|

R(f ,g) =1|Ψ1|

∑A∈Ψ2

σ(det(D′∗ − A∗D∗))

where σ(0) = ∞ and σ(x) = |x | for x 6= 0; the differentialD∗ : G1 → G2 of D is expressed with respect to preferred basesof Γ1 and Γ2.


Corollary

Let M1 and M2 be orientable infra-nilmanifolds of equaldimension. Let f ,g : M1 → M2 be maps. Then

N(f ,g) = L(f ,g) iff det(D′∗ − A∗D∗) ≥ 0 for every A ∈ Ψ2;

N(f ,g) = −L(f ,g) iff det(D′∗ − A∗D∗) ≤ 0 for every A ∈ Ψ2.

This generalizes Theorem 2.2 in:

K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168(1995), 157–166

from fixed point theory to coincidence theory.


Necessary Fact

Theorem

Let Gi be a connected simply connected nilpotent Lie groupand D,D′ : G1 → G2 be Lie group homomorphisms. Then forany g ∈ G2,

det(D′∗ − D∗) = det(D′

∗ − Ad(g)D∗).

PROOF. By complexifying Gi we may assume Gi 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∗)


Necessary Fact

Need a fact that for any g ∈ G2,

D′∗ − D∗ is surjective iff D′

∗ − Ad(g)D∗ is surjective.

K. Dekimpe and P. Penninckx, The finiteness of the Reidemeisternumber 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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