THE LEFSCHETZ INVARIANT

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THE LEFSCHETZ INVARIANT

VESTA COUFAL

Received 30 November 2004; Accepted 21 July 2005

In classical Lefschetz-Nielsen theory, one defines the Lefschetz invariantL(f) of an endomorphism f of a manifoldM. The definition depends on the fundamental group ofM, and hence on choosing a base point∗ ∈Mand a base path from∗tof(∗). At times, it is inconvenient or impossible to make these choices. In this paper, we use the fundamental groupoid to define a base-point-free version of the Lefschetz invariant.

Copyright © 2006 Vesta Coufal. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In classical Lefschetz fixed point theory [3], one considers an endomorphism f :M→M of a compact, connected polyhedronM. Lefschetz used an elementary trace construction to define the Lefschetz invariantL(f)∈Z. The Hopf-Lefschetz theorem states that if L(f)=0, then every map homotopic to f has a fixed point. The converse is false. How- ever, a converse can be achieved by strengthening the invariant. To begin, one chooses a base point ∗of M and a base path τ from ∗to f(∗). Then, using the fundamental group and an advanced trace construction one defines a Lefschetz-Nielsen invariant L(f,∗,τ), which is an element of a zero-dimensional Hochschild homology group [4].

Wecken proved that whenMis a compact manifold of dimensionn >2,L(f,∗,τ)=0 if and only if f is homotopic to a map with no fixed points.

We wish to extend Lefschetz-Nielsen theory to a family of manifolds and endomor- phisms, that is, a smooth fiber bundlep:E→Btogether with a map f :E→Esuch that p=p◦f. One problem with extending the definitions comes from choosing base points in the fibers, that is, a sectionsofp, and the fact thatf is not necessarily fiber homotopic to a map which fixes the base points (as is the case for a single path connected space and a single endomorphism.) To avoid this diﬃculty, we reformulate the classical definitions of the Lefschetz-Nielsen invariant by employing a trace construction over the fundamental groupoid, rather than the fundamental group.

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 34143, Pages1–20 DOI10.1155/FPTA/2006/34143

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InSection 2, we describe the classical (strengthened) Lefschetz-Nielsen invariant following the treatment given by Geoghegan [4] (see also Jiang [6], Brown [3] and L¨uck [8]). We also introduce the Hattori-Stallings trace, which will replace the usual trace in the construction of the algebraic invariant.

InSection 3, we develop the background necessary to explain our base-point-free definitions. This includes the general theory of groupoids and modules over ringoids, as well as our version of the Hattori-Stallings trace.

InSection 4, we present our base-point-free definitions of the Lefschetz-Nielsen invariant, and show that they are equivalent to the classical definitions.

2. The classical theory

2.1. The geometric invariant. In this section,Mⁿis a compact, connected manifold of dimensionn, and f :M→Mis a continuous endomorphism.

The concatenation of two pathsα:I→Xandβ:I→Xsuch thatα(1)=β(0) is defined by

α·β(t)=

⎧⎪

⎪⎨

⎪⎪

⎩

α(2t) if 0≤t≤1 2, β(2t−1) if1

2^≤t≤1.

(2.1)

The fixed point set of f is

Fix(f)=

x∈M| f(x)=x. (2.2)

Note that Fix(f) is compact. Define an equivalence relation∼on Fix(f) by lettingx∼y if there is a pathνinM fromxto ysuch thatν·(f ◦ν)⁻¹ is homotopic to a constant path.

Choose a base point∗ ∈Mand a base pathτfrom∗tof(∗). Letπ=π1(M,∗). Given these choices, f induces a homomorphism

φ:π−→π (2.3)

defined by

φ[w] =

τ·(f ◦w)·τ⁻¹, (2.4)

where [w] is the homotopy class of a pathwrel endpoints. Define an equivalence relation onπ by sayingg,h∈π are equivalent if there is somew∈π such thath=wgφ(w)⁻¹. The equivalence classes are called semiconjugacy classes; denote the set of semiconjugacy classes byπφ.

Define a map

Φ: Fix(f)−→π_φ (2.5)

by

x −→

μ·(f◦μ)⁻¹·τ⁻¹, (2.6)

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wherex∈Fix(f) andμis a path inMfrom∗tox. This map is well-defined and induces an injection

Φ: Fix(f)/∼−→πφ. (2.7)

It follows that Fix(f)/∼is compact and discrete, and hence finite. Denote the fixed point classes byF1,. . .,F_s.

Next, assume that the fixed point set of f is finite. Letxbe a fixed point. LetUbe an open neighborhood ofxinMandh:U→Rⁿa chart. LetVbe an openn-ball neighborhood ofxinUsuch that f(V)⊂U. Then the fixed point index of f atx,i(f,x), is the degree of the map of pairs

id−h f h⁻¹ :h(V),h(V)−

h(x) −→

Rⁿ,Rⁿ− {0} . (2.8) For a fixed point classFk, define

i(f,Fk)=

x∈Fk

i(f,x)∈Z. (2.9)

Definition 2.1. The classical geometric Lefschetz invariant of f with respect to the base point∗and the base pathτis

L^geo(f,∗,τ)= s k=1

i(f,Fk)Φ(Fk)∈Zπφ, (2.10) whereZπφis the free abelian group generated by the setπφ.

2.2. The algebraic invariant. To construct the classical algebraic Lefschetz invariant, let M be a finite connected CW complex and f :M→M a cellular map. Again, choose a base point∗ ∈M (a vertex ofM) and a base pathτfrom∗to f(∗). Also, choose an orientation on each cell inM.

Let p:M→M be the universal cover ofM. The CW structure onM lifts to a CW structure onM. Choose a lift of the base point ∗to a base point∗ ∈ M, and lift the base pathτto a pathτsuch thatτ(0) =∗. Then f lifts to a cellular map f:M→Msuch that

f(∗)=τ(1).

The groupπ=π1(M,∗) acts onMon the left by covering transformations. For each cellσ inM, choose a liftσinMand orient it compatibly withσ. Take the cellular chain complexC(M) of M. The action of π onMmakesCk(M) into a finitely generated free leftZπ-module with basis given by the chosen lifts of the orientedk-cells ofM.

As in the geometric construction, f andτinduce a homomorphismφ:π→π. Since fis cellular, it induces a chain map fk:Ck(M) →Ck(M) which is φ-linear, namely ifσ is ak-cell ofMandg∈π then fk(gσ)=φ(g)fk(σ). Classically, one represents fk by a matrix overZπwhose (i,j) entry is the coeﬃcient ofσ_jin the chainf_k(σ_i), whereσ_iand

σjarek-cells. For eachk, one can now take the trace offk, that is, the sum of the diagonal entries of the matrix which represents fk.

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Definition 2.2. The classical algebraic Lefschetz invariant of f with respect to the base point∗and the base pathτis

L^alg(f,∗,τ)=

k≥0

(−1)^kqtrace fk

∈Zπφ, (2.11)

whereq:Zπ→Zπ_φis the map sendingg∈πto its semiconjugacy class.

2.3. Hattori-Stallings trace. In the classical algebraic construction of the Lefschetz in- variant above, Reidemeister viewed f_k as a matrix and took its trace, the sum of the diagonal entries, to defineL^alg(f). In our generalizations, we will need to use a more sophisticated trace map, namely the Hattori-Stallings trace. Since on finitely generated free modules, the Hattori-Stallings trace agrees with the usual trace of a matrix, we could use it in the classical case as well. We introduce the classical Hattori-Stallings trace here.

(For the special case whenM=R, see [1,2,9].)

LetRbe a ring,ManR-bimodule, andPa finitely generated projective leftR-module.

Let P^∗=HomR(P,R) be the dual of P. Let [R,M] denote the abelian subgroup ofM generated by elements of the formrm−mr, forr∈Randm∈M. The Hattori-Stallings trace map, tr is given by the following composition:

HomR

P,M⊗RP

tr

P^∗⊗RM⊗RP

∼=

M/[R,M]

HH0(R;M)

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The mapP^∗⊗RM⊗RP→Hom_R(P,M⊗RP) is given byα⊗m⊗p →(p1 →α(p1)(m⊗ p)). The mapP^∗⊗RM⊗RP→M/[R,M] is given byα⊗m⊗p →α(p)m.

The fact that the first map is an isomorphism is an application of the following lemma.

Lemma 2.3. Let R be a ring, P a finitely generated projective rightR-module, and N a leftR-module. Define f_P:P^∗⊗RN→Hom_R(P,N) by f_P(α,n)(p)=α(p)n. Then f_Pis an isomorphism of groups.

Proof. Note thatfR:R^∗⊗RN→HomR(R,N) is an isomorphism with inverse given by (g: R→N) →idR⊗Rg(1R). The result follows from the fact that f(₋): (−)^∗⊗RN→HomR(−,

N) preserves finite direct sums.

3. Background on groups and ringoids

In this section, we generalize to the “oid” setting the basic algebraic definitions and results which we will need for our constructions. This treatment is based on [7, Section 9], though we have developed additional material as needed. In particular, inSection 3.2, we generalize the Hattori-Stallings trace.

We use the following notation. IfCis a category, denote the collection of objects inC by Ob(C). Ifxandyare objects inC, denote the collection of maps fromxtoyinCby C(x,y). The category of sets will be denoted Sets, the category of abelian groups will be denoted Ab, and the category of leftR-modules will be denotedR-mod.

Throughout, “ring” will mean an associative ring with unit.

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3.1. General definitions and results

3.1.1. Groupoids and ringoids. LetGbe a group. We may viewGas a category, denoted by G, in which there is one object∗, and for which all of the maps are isomorphisms.

Each map corresponds to an element ofGwith composition of maps corresponding to the multiplication in the group. This idea generalizes to define a groupoid.

Definition 3.1. A groupoidG is a small category (the objects form a set) such that all maps are isomorphisms.

The analogous game can be played with rings in order to define a ringoid, also known as a linear category or as a small category enriched in the category of abelian groups.

Definition 3.2. A ringoid᏾is a small category such that for each pair of objectsxandy,

᏾(x,y) is an abelian group and the composition function᏾(y,z)×᏾(x,y)→᏾(x,z) is bilinear.

Example 3.3. Recall that ifHis a group, then the group ringZHis the free abelian group generated byH. This group ring construction can be generalized to a “groupoid ringoid”

(though we will call it the group ring): letGbe a groupoid andRa ring. The group ring ofGwith respect toR, denotedRG, is the category with the same objects asG, but with maps given byRG(x,y)=R(G(x,y)), the freeR-module generated by the setG(x,y).

3.1.2. Modules. For the remainder of this paper, unless otherwise noted, letGbe a groupoid and letRbe a commutative ring. While much of the following can be done in terms of a ringoid᏾, we will restrict our attention to group ringsRG.

Definition 3.4. A leftRG-module is a (covariant) functorM:G→R-mod. A rightRG- modules is a (covariant) functorsG^op→R-mod.

Definition 3.5. LetMandNbeRG-modules. AnRG-module homomorphism fromMto Nis a natural transformation fromMtoN. The set of allRG-module homomorphisms fromMtoNis denoted by HomRG(M,N).

LetRG-mod denote the category of leftRG-modules, and let mod-RGdenote the category of rightRG-modules.

Definition 3.6. LetMandNbeRG-modules. The direct sumM⊕NofMandNis the left RG-module defined on an objectxby (M⊕N)(x)=M(x)⊕N(x) and on a mapg:x→y by (M⊕N)(g)=M(g)⊕N(g).

Definition 3.7. LetNbe a leftRG-module andM a rightRG-module. Define the tensor product overRGofMandNto be the abelian group

M⊗RGN=P/Q, (3.1)

wherePis the abelian group

P=

x∈Ob(G)

M(x)⊗RN(x), (3.2)

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andQis the subgroup ofPgenerated by

M(f)(m)_⊗n₋m_⊗N(f)(n)_|m_∈M(y),n_∈N(x), f _∈RG(x,y). (3.3) Proposition 3.8. LetM,N, andPbeRG-modules. Then

HomRG(M⊕N,P)^∼=HomRG(M,P)⊕HomRG(N,P). (3.4) Proposition 3.9. LetM,N, andPbeRG-modules. Then

(M⊕N)⊗RGP^∼=

M⊗RGP ⊕

N⊗RGP . (3.5)

Definition 3.10. Given anRG-bimoduleM, defineM/[RG,M] to be theR-module

x∈Ob(G)

M(x,x)

/m−Mg,g⁻¹ (m)|g:x−→y,m∈M(x,x). (3.6) Call this the zero dimensional Hochschild homology ofRGwith coeﬃcients inM, denoted by

HH0(RG;M). (3.7)

Next, we define freeRG-modules. First, we need the following notions.

Given a categoryC, we can view Ob(C) as the subcategory ofCwhose objects are the same as the objects ofC, but whose maps are only the identity maps. A covariant (con- travariant) functor Ob(C)→Sets will be called a left (right) Ob(C)-set. A map of Ob(C)- sets is a natural transformation. Let Ob(C)- Sets denote the category of left Ob(C)-sets, and let Sets - Ob(C) denote the category of right Ob(C)-sets.

Given either a left or right Ob(C)-setB, let Ꮾ=

x∈Ob(C)

B(x), (3.8)

wheredenotes disjoint union, and let

β:Ꮾ−→Ob(C) (3.9)

sendbtoxifb_∈B(x). Given Ob(C)-setsBandB, we sayBis an Ob(C)-subset ofBif for everyx∈Ob(C),B(x)⊂B(x).

SupposeCis a small category andDis a category equipped with a “forgetful functor”

D→Sets. For a functorF:C→D, let|F|: Ob(C)→Sets be the composition Ob(C) C→D→Sets, where the functor D→Sets is the forgetful functor. In particular,|−|: RG-mod→Ob(C)- Sets and|−|: mod-RG→Sets - Ob(G).

Definition 3.11. For each x∈Ob(G), define a left RG-module RGx =RG(x,−) by RGx(y)=RG(x,y). For a mapg:y→z inG, letRGx(g)=g◦(−). Define a rightRG- moduleRG^x=RG(−,x) similarly.

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Definition 3.12. Define a functorRG(₋): Ob(G)- Sets→RG-mod by RG_B=

b∈Ꮾ

RG_β(b)=

b∈Ꮾ

RGβ(b),− . (3.10) Similarly, defineRG⁽⁻⁾: Sets - Ob(G)→mod-RGby

RG^B=

b∈Ꮾ

RG^β(b)=

b∈Ꮾ

RG−,β(b) . (3.11)

Proposition 3.13. The functor RG(−) is a left adjoint to the functor |−|:RG-mod_→ Ob(G)-Sets. The functorRG⁽⁻⁾is a left adjoint to|−|: mod-RG→Sets - Ob(G).

Proof. For an Ob(G)-set B and a left RG-module M, define a set map ψ =ψ_B,M : RG-mod(RG_B,M)→Ob(G)- Sets(B,|M|) by ψ(η)_y(b)=η_y(id_y)∈ |M(y)|, where η: RG_B→Mis a natural transformation andb∈B(y). Thenψis a bijection whose inverse

is defined in the most obvious way.

Notice that for each Ob(G)-setB, we get a natural transformationηB=ψ(id_RG_B) :B→

|RGB|which is universal. This leads to the following definition of a freeRG-module with baseB.

Definition 3.14. An RG-moduleM is free with base an Ob(G)-setB⊂ |M|if for each RG-moduleNand natural transformation f :B→ |N|there is a unique natural transfor- mationF:M→Nwith|F| ◦i=f, whereiis the inclusionB→ |M|.

Example 3.15. TheRG-moduleRGxis a free leftRG-module with baseBx: Ob(G)→Sets given by

Bx(y)=

⎧⎨

⎩{x} ify=x,

∅ ify=x. (3.12)

IfBis any Ob(G)-set,RG_B=

b∈ᏮRG_β(b)=

b∈ᏮRG(β(b),−) is a freeRG-module with baseB.

LetM be anRG-module. LetSbe an Ob(G)-subset of|M|and let Span(S) be the smallestRG-submodule ofMcontainingS,

Span(S)= ∩

N|Nis anRG-submodule ofM, S⊂N. (3.13) Definition 3.16. Say thatMis generated bySifM=Span(S), andMis finitely generated ifSis finite.

Proposition 3.17. If M is a left RG-module, and B is an Ob(G)-subset of |M|, then Span(B) is the image of the unique natural transformationτ:RGB→Mextending id :B→ B⊂ |M|. Furthermore,Mis generated byBifτis surjective.

Proposition 3.18. LetBbe an Ob(G)-set. IfMis a free leftRG-module with baseB, then Mis generated byB. In particular, there is a natural equivalenceτ:RGB→M.

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Proof. Defineτ:RGB→M. Forx∈Ob(G), let τ_x:RG_B(x)=

b∈Ꮾ

RGβ(b),x −→M(x) (3.14) be given by (g :β(b)→x) →M(g)(b). To construct an inverse natural transformation, defineη:B→ |RGB|by settingηx(b)=idx. SinceM is free with baseB,ηextends to a

unique natural transformationM→RG_B.

Definition 3.19. AnRG-modulePis projective if it is the direct summand of a freeRG- module.

3.1.3. Bimodules.

Definition 3.20. AnRG-bimodule is a (covariant) functor

M:G×G^op−→R-mod. (3.15) Denote the category ofRG-bimodules byRG-bimod.

Example 3.21. Let RGbe RGwith the followingRG-bimodule structure. For (x,y)∈ G×G^op, setRG(x,y)=RG(y,x). Notice the change in the order ofxand y. For maps g:x→xinGandh:y→yinG^op, setRG(g,h)=g◦(−)◦h:RG(y,x)→RG(y,x).

We would like to be able to view anRG-bimoduleN as either a right or a leftRG- module. However, there is no canonical way to do so as each choice of object inGpro- duces a diﬀerent left and a rightRG-module structure onN. Instead, we define two functors: (−) ad and ad(−). In essence,Nad encapsulates all of the rightRG-module struc- tures onN induced by objects ofG, and adN encapsulates all of the left RG-module structure onN.

Definition 3.22. Define a covariant functor

(−) ad :RG-bimod−→(mod-RG)^G (3.16) as follows. LetNbe anRG-bimodule. Forx∈Ob(G), let

Nad(x)=N(x,−). (3.17)

Forga map inG, let

Nad(g)₌N(g,₋). (3.18)

Explicitly, Nad(x) :G^op→R-mod is given byNad(x)(y)=N(x,y) and Nad(x)(h)= N(id_x,h) forh:y→za map inG^op.

Definition 3.23. Define a covariant functor

ad(−) :RG-bimod−→(RG-mod)^G^op (3.19)

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as follows. LetNbe anRG-bimodule. Forx∈Ob(G^op), let

adN(x)₌N(₋,x). (3.20)

Forga map inG^op, let

adN(g)=N(−,g). (3.21)

Explicitly, adN(x) :G→R-mod is given by adN(x)(y)=N(y,x) and adN(x)(h)=N(h, idx) forh:y→za map inG.

Example 3.24. Apply the ad functors to theRG-bimoduleRG. For instance, ifx∈Ob(G), then adRG(x)=RG(x,−)=RGx. Hence, adRG(x) :G→R-mod, with adRG(x)(y)= RG(x,y) and adRG(x)(h)=h◦(−) forh:y→za map inG. Also, forg:x→xa map inG^op, adRG(g)=RG(−,g) :RG(x,−)→RG(x,−) is the natural transformation of left RG-modules given by adRG(g)y=(−)◦g:RG(x,y)→RG(x,y).

Next, if N is an RG-bimodule and M is anRG-module, we define HomRG(N,M), HomRG(M,N),N⊗RGM_l andM_r⊗RGN in such a way that they are alsoRG-modules, as one would expect. LetM_l(resp.,M_r) denote a left (resp., right)RG-module.

Definition 3.25. LetNbe anRG-bimodule. Hom_RG(M_l,N) is defined to be the rightRG- module given by the composition

G^{op ad}^N RG-mod ^Hom^RG^(M^l^,⁻⁾ R-mod. (3.22)

HomRG(N,Ml) is defined to be the leftRG-module given by the composition

G^{op ad}^N RG-mod ^Hom^RG⁽⁻^,M^l⁾ R-mod. (3.23)

HomRG(M_r,N) is defined to be the leftRG-module given by the composition

G ^Nad mod-RG ^Hom^RG^(M^r^,⁻⁾ R-mod. (3.24) HomRG(N,Mr) is defined to be the rightRG-module given by the composition

G ^Nad mod-RG ^Hom^RG⁽⁻^,M^r⁾ R-mod. (3.25) Definition 3.26. LetN be anRG-bimodule. DefineN⊗RGMlto be the leftRG-module given by the composition

G ^Nad mod-RG ⁽⁻⁾^⊗^RG^M^l R-mod. (3.26) DefineMr⊗RGNto be the rightRG-module given by the composition

G^{op ad}^N RG-mod ^M^r^⊗^RG⁽⁻⁾ R-mod. (3.27)

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Applying the above definitions to theRG-bimoduleRG, we get the results for Hom and tensor product which we would expect from algebra. These next three propositions justify viewingRGas “the free rank-one”RG-module. Notice that it is not, however, a freeRG-module. The proofs are straightforward and left to the reader.

Proposition 3.27. Given anRG-moduleM, HomRG(RG,M)^∼₌MasRG-modules.

Proposition 3.28. Given a leftRG-moduleM,RG⊗RGM^∼₌Mas leftRG-modules.

Proposition 3.29. Given rightRG-moduleM,M⊗RGRG^∼₌Mas rightRG-modules.

In particular, we can now define the dual of anRG-module.

Definition 3.30. LetMbe a left (right)RG-module. The dual ofMis the right (left)RG- moduleM^∗=HomRG(M,RG).

Proposition 3.31. LetMandNbeRG-modules. Then there is a natural equivalence (M⊕ N)^∗^∼₌M^∗⊕N^∗.

3.1.4. Chain complexes.

Definition 3.32. AnRG-chain complex is a (covariant) functor C:G_→Ch(R), where Ch(R) is the category of chain complexes over the ringR.

Lemma 3.33. The following are equivalent:

(i)Cis anRG-chain complex;

(ii) there exist a family{Cn}ofRG-modules together with a family of natural transfor- mations{d_n:C_n→C_n₋1}, called diﬀerentials, such thatd_n₋1◦d_n=0.

Using the second characterization ofRG-chain complexes, we can now define finitely generated projective chain complexes, chain maps and chain homotopies in the usual manner.

Definition 3.34. AnRG-chain complexPis said to be a finitely generated projective if eachP_nis a finitely generated projectiveRG-module andPis bounded (i.e.,P_n=0 for all but a finite number of n). Letᏼ(RG) denote the subcategory of finitely generated projectiveRG-chain complexes.

Definition 3.35. AnRG-chain mapf :C→Dis a family{fn:Cn→Dn}of natural transformations such thatd_n◦f_n= f_n₋1◦d_nfor alln, where thed_nare the diﬀerentials ofC and thed_nare the diﬀerentials ofD.

Definition 3.36. TwoRG-chain maps f :C→Dandg:C→DareRG-chain homotopic, denoted by f ∼chg, if there exists a family{sn:Cn→Dn−1}of natural transformations such that

fn−gn=d_n+1◦sn+sn−1◦dn. (3.28) Definition 3.37. TwoRG-chain complexesCandDare chain homotopy equivalent if there existRG-chain maps f :C→Dandg:D→Csuch that f◦g∼chid_Dandg◦

f ∼chid_C. In this case, f is said to be a chain homotopy equivalence.

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3.1.5. Everythingα-twisted. For the remainder of the paper, letα:G→Gbe a functor.

We can useαto create an “α-twisted” version of many of our algebraic objects.

Definition 3.38. Define anRG-bimoduleαRG:G×G^op→R-mod by

αRG(x,y)=RGy,α(x) (3.29) forx,y∈Ob(G), and

αRG(g,h)=α(g)◦(−)◦h (3.30)

forg a map inGand ha map inG^op. This is theRG-bimoduleRG, but with the left module structure twisted byα.

Definition 3.39. LetM andN beRG-modules. Anα-linear homomorphismM→N is defined to be a natural transformationη:M→N◦α. A chain map f :C→DofRG- chain complexes is calledα-linear if for eachn, f_nisα-linear.

Lemma 3.40. Given leftRG-modulesPandQ, there is an isomorphism

Hom_RG(P,Q◦α)^∼₌Hom_RGP,_αRG⊗RGQ . (3.31) Definition 3.41. LetMbe anRG-module. Theα-dual ofMis

M^α=HomRG

M,αRG . (3.32)

Proposition 3.42. LetPandQbeRG-modules andNanRG-bimodule. Then there is a natural equivalence ofRG-modules

HomRG(P⊕Q,N)^∼₌HomRG(P,N)⊕HomRG(Q,N). (3.33) Corollary 3.43. LetPandQbe leftRG-modules. Then there is a natural equivalence

(P⊕Q)^α^∼₌P^α⊕Q^α. (3.34)

3.2. Generalized Hattori-Stallings trace. In this section, we define anα-twisted Hattori- Stallings trace forRG-modules. One can define a more general Hattori-Stallings trace for RG-modules, in the same manner as the classical definition given inSection 2.3. However, as we will not need this more general form, we will concern ourselves only with the special α-twisted case. We also extend the trace toRG-chain complexes.

3.2.1. Definition and commutativity. Given leftRG-modulesNandP, define anR-module homomorphism

φP=φP,N:P^α⊗RGN−→HomRG(P,N◦α) (3.35) by letting:φP(τ⊗n) :P→N◦αbe the natural transformation given by

φP(τ⊗n)y(p)=Nτy(p) (n), (3.36) whereτ∈P^α(x),m∈N(x),p∈P(y), andx,y∈Ob(G).

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Proposition 3.44. IfPis a finitely generated projectiveRG-module, thenφPis an isomor- phism.

The proof will use the following three lemmas.

Lemma 3.45. Givenx∈Ob(G), thenφ_RG_xis an isomorphism.

Proof. Writeφforφ_RG_x. Define ψ: HomRG

RG_x,N_◦α _−→RG^α_x_⊗_RGN (3.37) by

η −→α⊗η_xidx , (3.38)

whereη:RGx→N◦αis a natural transformation. Here,α∈P^α(x) is the natural transformation induced byα, that is,αy(f)=α(f) fory∈Ob(G) and f ∈RG(x,y).

It is easy to show thatφ◦ψ=id andψ◦φ=id.

Lemma 3.46. IfPandQare leftRG-modules, thenφ_P_⊕_Q=φ_P⊕φ_Q. Proof. Consider the following diagram:

(P⊕Q)^α⊗RGN ^φ^P^⊕^Q

∼=

HomRG(P⊕Q,N◦α)

∼=

P^α⊕Q^α ⊗RGN

∼=

P^α⊗RGN ⊕

Q^α⊗RGN

φP⊕φQ HomRG(P,N◦α)⊕HomRG(Q,N◦α) (3.39)

The vertical isomorphisms are as in Propositions3.8and3.9andCorollary 3.43. Using those isomorphism, one can see that the diagram commutes.

Lemma 3.47. LetPandQbe leftRG-modules and letN=P⊕Q. IfφNis an isomorphism, thenφPis an isomorphism also.

Proof. By the previous lemma,φN=φP⊕φQ. The result follows immediately.

Proof ofProposition 3.44. The proof is in two steps.

Step 1. Suppose thatPis a finitely generated freeRG-module. ThenPis naturally equivalent toRGB=

b∈ᏮRGβ(b)for some Ob(G)-setB. ByLemma 3.46,φP=

b∈Ꮾφ_RG_β(b), and byLemma 3.45, it is an isomorphism.

Step 2. Suppose thatPis a finitely generated projectiveRG-modules and soPis a direct summand of a finitely generated freeRG-module. CombiningStep 1andLemma 3.47we see thatφPis an isomorphism.

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ForPa leftRG-module, define anR-module homomorphism

P^α⊗RGP−→αRG/RG,αRG (3.40)

byτ⊗p →τ_x(p) whereτ∈P^α(x) andp∈P(x).

Definition 3.48. Let P be a finitely generated projective leftRG-module. The Hattori- Stallings trace, denoted by tr, is the composition

HomRG(P,P◦α)

tr

P^α⊗RGP

∼=

αRG/RG,αRG

HH0

RG;_αRG

(3.41)

where the isomorphism is the mapφ_Pand the unadorned arrow is the homomorphism described above.

Proposition 3.49 (commutativity). LetPandQbe finitely generated projective leftRG- modules. If f ∈HomRG(P,Q◦α) andg∈HomRG(Q,P), then

tr(f◦g)=tr(g◦α◦f). (3.42) Proof. The result follows from commutativity of three diagrams.

The first diagram is

HomRG(P,Q◦α)×HomRG(Q,P) P^α⊗RGQ ×

Q^∗⊗RGP

B

HomRG(P,P◦α) ^φ^P P^α⊗RGP

(3.43)

whereBis given by (η⊗p)×(τ⊗q) →(α◦η)⊗Q(τy(p))(q), the unlabelled vertical map is given by (f,g) →g◦α◦f and the unlabelled horizontal map isφP^α,Q×φQ,P.

The second diagram is gotten by transposing the products in the first diagram.

The third diagram is

Q^α⊗RGQ

Q^∗⊗RGP ×

P^α⊗RGQ

B

P^α⊗RGQ ×

Q^∗⊗RGP

B

HH0

RG;αRG

P^α⊗RGP

(3.44)

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where the unlabelled arrow is transposition,Bis analogous toB, and the other maps are

defined in the obvious ways.

3.2.2. For connected groupoids. Consider the following setup. LetGbe a connected groupoid, that is, one for which there exists a map between any two objects. Letα:G→Gbe a functor and letPbe a finitely generated projective leftRG-module. Choose an object∗ ofGand choose a mapτ:∗ →α(∗) inG.

Let RG(∗) be the subcategory ofRGwith a single object,∗, and with maps given by the maps inRGfrom∗to∗. Then the inclusionRG(∗)→RGis an equivalence of categories. The proof amounts to choosing a map μx:∗ →xfor eachx∈Ob(G). For eachx, we fix a choice ofμ_x.

The functorα induces a functor ατ:RG(∗)→RG(∗) which maps the object∗ to itself. Ifg:∗ → ∗, let ατ(g)=τ⁻¹◦α(g)◦τ. In the obvious way, theRG-moduleP induces a finitely generated projective leftRG(∗)-module, denotedP(∗). A natural transformation β∈Hom_RG(P,P◦α) induces a natural transformation β_τ=P(τ⁻¹)◦β_∗∈ HomRG(∗)(P(∗),P(∗)◦ατ).

Lemma 3.50. There is an isomorphism of groups A:HH0

RG(∗);ατRG(∗) −→HH0

RG;αRG (3.45)

given byA(m)=τ◦mform∈HH0(RG(∗);_α_τRG(∗)).

Proposition 3.51. The Hattori-Stallings trace ofβ_τandβare equivalent, that is,

Atr(β_τ) =tr(β). (3.46)

Proof. Givenη∈P^α(x) for somex∈Ob(G), define η:P(∗)→RG(∗,∗)∈P(∗)^α^τ by η(p)=τ⁻¹◦η_∗(p)◦μ_x, wherep∈P(∗). This gives us a mapP^α→P(∗)^α^τ.

Define a mapB:P^α⊗RGP→P(∗)^α^τ⊗RG(∗)P(∗) byη⊗p →η⊗P(μ⁻_x¹)(p), where η∈P^α(x) and p∈P(x) for some x∈Ob(G). Define a map C: HomRG(P,P◦α)→ HomRG(∗)(P(∗),P(∗)◦α_τ) byγ →γ_τ=P(τ⁻¹)◦γ_∗forγ∈HomRG(P,P◦α).

Commutativity of the following two diagrams implies thatA(tr(β_τ))=tr(β).

HomRG(∗)

P(∗),P(∗)◦ατ ^φ^P(^∗⁾ P(∗)^α^τ⊗RG(∗)P(∗)

[3pt] HomRG(P,P◦α)

C

P^α⊗RGP

φP

B

P(∗)^α^τ⊗RG(∗)P(∗) HH0

RG(∗);ατRG(∗)

A

[3pt]P^α⊗RGP

B

HH0(RG;_αRG)

(3.47)

Notice thatA(tr(βτ)) is independent of the choices of mapsμx.

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3.2.3. For chain complexes. We begin with the general case.

Definition 3.52. LetPbe a finitely generated projectiveRG-chain complex. Define the Hattori-Stallings trace

Tr : Hom_ᏼ(RG)(P,P◦α)−→HH0

RG;αRG (3.48)

by

f −→

i

(−1)ⁱtrfi , (3.49)

where f :P→P◦αis given by the family{fi∈HomRG(Pi,Pi◦α)}.

Commutativity follows from commutativity of the Hattori-Stallings trace for RG- modules.

Proposition 3.53 (commutativity). Let Pand Q be finitely generated projectiveRG- chain complexes, and let f ∈Homᏼ(RG)(P,Q◦α) andg∈Homᏼ(RG)(Q,P). Then

Tr(f◦g)=Tr(g◦α◦f). (3.50) The Hattori-Stallings trace is also invariant up to chain homotopy.

Proposition 3.54. LetPbe a finitely generated projectiveRG-chain complex. If f :P→ P◦αandg:P→P◦αare chain homotopic, then Tr(f)=Tr(g).

Proof. Let{sn:Pn→Pn+1◦α}be a chain homotopy from f tog. Then Tr(f)−Tr(g)=

i

(−1)ⁱtrfi−gi

=

i

(−1)ⁱtrdi+1◦α◦si+si−1◦di

=

i

(−1)ⁱtrsi◦di+1) + tr(si−1◦di .

(3.51)

The last equality comes from applying commutativity. Rearranging the terms in the last

sum gives Tr(f)−Tr(g)=0.

Now suppose thatCis anRG-chain complex which is chain homotopy equivalent to a finitely generated projectiveRG-chain complex. Suppose further thatφ:C→C◦α is a chain map. Choose a finitely generated projectiveRG-chain complexP, choose a chain homotopy equivalence f :C→P, and choose a liftψ:P→P◦αofφ. We get the diagram

P ^ψ P◦α

C

f

φ C◦α

f (3.52)

which commutes up to chain homotopy.

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Definition 3.55. The Hattori-Stallings trace ofφ:C→C◦αis defined to be the trace of ψ:P→P◦α:

Tr(φ)=Tr(ψ). (3.53)

We must show that Tr is independent of the choices we made. First, suppose thatφis another lift ofφ. Thenψ∼ch f◦φ◦f⁻¹∼chψand byProposition 3.54, Tr(ψ)=Tr(ψ).

Second, suppose thatQis another finitely generated projectiveRG-chain complex and g:C→Qis a chain homotopy equivalence. Then

Trg◦φ◦g⁻¹ =Trg◦f◦f⁻¹◦φ◦f⁻¹◦f◦g⁻¹

=Trf ◦g⁻¹◦g◦f⁻¹◦f◦φ◦f⁻¹

=Trf ◦φ◦f⁻¹ .

(3.54)

4. Base-point-free Lefschetz-Nielsen invariants

In this section, we present our base-point-free refinements of the classical geometric and algebraic Lefschetz-Nielsen invariants. We begin by defining the fundamental groupoid, and describing the way in which we think of the universal cover.

4.1. Fundamental groupoid. An important example of a groupoid is the fundamental groupoid. LetXbe a topological space.

Definition 4.1. The fundamental groupoid ΠX is the category whose objects are the points inX, whose maps are the homotopy classes rel endpoints of paths inX. Com- position is given by concatenation of paths. To be precise, if f andgare paths inXsuch that f(1)=g(0), then

[g]◦[f]=[f ·g]. (4.1)

For each morphism, an inverse is given by traversing a representative path backwards.

This groupoid deserves to be called the fundamental groupoid since for a given point x∈X, the subcategory ofΠXgenerated byxisπ1(X,x). The subcategory generated byx is the category with one object,x, and whose morphism set isΠX(x,x). In a sense, then, the fundamental groupoid is a way of encoding in one object the fundamental groups with all possible choices of base point.

Let f :X→Xbe a continuous map. Then f induces a functorΠ f :ΠX→ΠXgiven byΠ f(x)=f(x) andΠ f(g)=f ◦gwherex∈Xandgis a path inX.

4.2. Universal cover. LetXbe a path connected, locally path connected, semilocally sim- ply connected space. For eachx∈X, one can describe the universal cover [5, page 64] of Xas the space

X_x=(X,x)^(I,0)/∼, (4.2)