Relative fixed point theory

RELA TIVE FIXED POINT THEOR Y KA TE PONTO Abstract The Lefschetz ﬁxed point theorem and its con verse hav e many g eneralizatio ns . One of these generaliz ations is t o e ndo morphisms of a space relative to a ﬁxed subspace. In this pap er w e deﬁne relative Lefsc hetz n umbers and Reidemeister traces using tr a ces in bicategorie s with s ha dows. W e use the functor iality of this trace to identify diﬀerent forms of these inv aria n ts and to prove a relative Le fschet z ﬁxed point theorem and its conv erse. Introduction The g oal of top ologic al ﬁxed p oint theory is to ﬁnd inv a r iants that detect if a given endomorphism of a space has an y ﬁxed p oints. The Lefschetz ﬁxed p oint theorem iden tiﬁes one such inv aria nt. Theorem (Lefsc hetz ﬁxed p oint theore m) . L et B b e a close d smo oth manifold and f : B → B b e a c ontinu ou s map. If f has no ﬁx e d p oints then the L efschetz numb er of f is zer o. The Lefschetz num b er is the alternating sum of the levelwise traces of the map induced b y f o n the rational homology of B . This is a rela tively computable inv aria nt . It gives a necessa r y , but no t suﬃcient , condition for determining if a contin uo us map do es not hav e any ﬁxed po int s. If w e put additional restrictions on the map f , such as requiring it to pr eserve a subspace of B , the Lefschetz num b er still gives a neces sary condition for f to be ﬁxed po in t free. How ever, this inv ariant igno res the r elative structure and so is not the b est p os sible in v a riant. F or example, the identit y map of the circle is homotopic to a map with no ﬁxed po int s and so the Lefschetz num b er is zero. If this ma p is requir e d to pr eserve a prop er subinterv al it is no longer homo topic to a map w ith no ﬁxed p oints. There is a reﬁnement of the L e fschetz nu mber deﬁned using the induced maps on the r ational ho mology of the subs pa ce and the relative ratio na l ho mology . Theorem A (Relative Lefschet z ﬁxed p oint theorem) . L et A ⊂ B b e close d s m o oth manifolds and f : B → B b e a c ontinuous map such that f ( A ) ⊂ A . If f has no ﬁxe d p oints then the r elative L efschetz numb er of f is zer o. Date : No vem ber 3, 2018 The author was supp orted by a National Science F oundation postdo ctoral fellowship. 1 2 KA TE P ONTO The relative Lefschetz num b er of the identit y map of the c ircle relative to a prop er s ubin terv al is not zero . Both of these theor ems g ive a c o ndition that implies that a contin uous endomo r- phism f : B → B has a ﬁxed po int. In most cases they do not give a c o ndition that implies f has no ﬁxed p oints. T o address this question a reﬁned in v ar iant and some restrictions on the spaces hav e to b e intro duced. This reﬁned inv a riant is called the Nielsen nu mber or the Reidemeister tr ace. Theorem (Conv erse to the Lefsc hetz ﬁxed p o in t theorem) . L et B b e a close d smo oth manifold of dimension at le ast 3 and f : B → B b e a c ontinuous map. The map f is homotopic to a map with n o ﬁx e d p oints if and only if the R eidemeister t r ac e of f is zer o. The Reidemeister trace is a partitioning of the Lefschetz num b er to r eﬂect the wa ys ﬁxed p oints can b e changed by a homotopy of the original map. There is a generalization of the Reidemeister tra c e to a relative Reidemeister tr ace that is very similar to the ge neralization of the Lefschetz num b e r to the relative Lefschetz nu mber. Theorem B (Converse to the Relative Lefschetz ﬁxed p oint theor em) . Supp ose A ⊂ B ar e close d smo oth manifolds of dimension at le ast 3 and the c o dimension of A in B is at le ast 2. A map f : B → B such that f ( A ) ⊂ A is homotopic to a map with no ﬁxe d p oints thr ough maps pr eserving the subset A if and only if the r elative R eidemeister tr ac e of f is zer o. The go al of this pap e r is to provide deﬁnitions of the r elative Lefschetz num- ber and r elative Reidemeister tra ce a nd pro ofs of Theorems A and B that satisfy several require ments. First, the relative Reidemeister trace should detect if a map is relatively homo to pic to a map with no ﬁxed p oints. It is not nece ssary for the relative Reidemeister trace to provide a low er bound for the num b er of ﬁxed p oint of a given map. Second, the inv aria n ts should be tra ce-like. This means that they can b e desc r ib ed using the duality and trace in bicateg ories deﬁned in [26]. The relative Reidemeister trace should to be compatible with the appro ach of [19, 1 8]. Those pap ers give a pro of of the converse to the Lefschetz ﬁxed p oint theor em that is diﬀerent from the standa rd simplicial pro of. Finally , the r elative Reidemeister trace should be compatible with an eq uiv ariant gene r alization of the Reidemeister trace des crib ed in [25]. While the Lefschetz num b er and the Reidemeister trace hav e long established deﬁnitions, the rela tive for ms of these inv ar iants are less settled. V er sions o f the relative Lefschetz num b er hav e b een deﬁned in [1, 15] and of the r elative Reide- meister tra ce in [24, 28, 34, 35, 36]. The pap ers [28, 34, 35] ar e prima rily interested in determining lower b ounds for the num ber of ﬁxed p oints and so are generaliza - tions of the Nielsen n umber. T he inv a r iants deﬁned in [24, 36] are more trace- like, but the deﬁnition ar e still motiv a ted by connections to the Nielsen num b er. None of these inv a riants s atisfy a ll of o ur conditions ab ov e, and none of them exactly coincide with the deﬁnitions given here. RELA TIVE FIXE D POINT THEOR Y 3 In this pape r we give pro ofs of Theorems A a nd B follo wing the outline of [26]. W e use dualit y and trace in bicategories with shado ws to deﬁne t wo forms of the relative Lefschetz num b er and the rela tive Re idemeis ter trace. Then, using functoriality , we show diﬀerent inv a riants coincide. Finally , we generalize Klein and Williams’s pro of of the conv erse to the Lefschetz ﬁxed p oint theor e m in [19] to c o mplete the pro of of the conv erse to the rela tive Lefschetz ﬁxed point theorem. In the ﬁrst tw o sections w e will recall the necessary deﬁnitions o f duality and tra ce in symmetric mono idal categor ies and in bicategor ies with shadows. I n Section 3 we will describ e some examples of bicategor ie s with shadows and generalize some results from [2 6] that de s crib e speciﬁc exa mples o f duals . In Section 4 w e apply this category theory to the relative Lefschetz num be r . In Sections 5 a nd 6 we deﬁne the relative Reidemeister trace . W e describ e how to compare these inv a riants to each other a nd how to compare them to the r elative Nielsen nu mber deﬁned by Schirmer and Zhao . In Sections 7 and 8 w e give a pro of of the converse to the relative Lefschetz ﬁxed po int theorem following the pro ofs given b y Kle in and Williams in [1 9, 18]. In Sectio n 9 we include so me forma l results omitted from the third sectio n. W e assume the reader is familiar with the ba sic deﬁnitions of Nielsen theor y . References for this topic include [3, 16]. Ac knowledgements. I would like to thank Peter May , Gun Suny eekhan, and Bruce Williams for many helpful conversations and comments o n prev ious dra fts of this pap er. I would als o like to thank Xuezhi Zha o for sending me one of his preprints. Preliminaries . W e ﬁx some conv entions and reca ll a fact ab out coﬁbrations. Deﬁnition 0.1. Let A ⊂ B a nd X ⊂ Y . A map f : B → Y is a r elative map if f ( A ) ⊂ X . W e will write this f : ( B , A ) → ( Y , X ) . A homotopy H : B × I → Y is a r elative homotopy if H | A × I factors throug h the inclusion X ⊂ Y . Deﬁnition 0.2. [34, 3.1] A r elative map f : ( B , A ) → ( B , A ) is taut if ther e is a neighborho o d N ( A ) o f A in B such that f ( N ( A )) ⊂ A . W e will use this co ndition to is o late the ﬁx e d points o f A from those of B \ A . Lemma 0.3. [3 4, 3.2 ] If A ⊂ B is a c oﬁbr ation then any r elative map f : ( B , A ) → ( B , A ) is r elatively homotopic to a t aut map. W e will assume A ⊂ B is a co ﬁbration and a ll r elative maps f : ( B , A ) → ( B , A ) . are taut. If a relative map is no t taut it is implicitly r eplaced b y a relatively homotopic map that is taut. Since all inv ar iants deﬁned here ar e in v aria nts of the relative homotopy class, the c hoice of r eplacement do es not matter. 4 KA TE P ONTO 1. Duality and trace in symmetric monoidal ca tegories Dualit y and trace in symmetric monoida l ca teg ories is a generaliza tion of the trace in linear algebra that retains many imp ortant prop erties. The trace in a symmetric monoidal ca tegory satisﬁes a g eneralizatio n of inv a riance of bas is and has nice functor ial prop erties. The Le fschetz ﬁxed p oint theorem is one applica tio n of the functor iality prop erties of the trac e . This is a very brief summar y of [9]. F or more deta ils s ee [9 ], [2 1, Chapter I I I], or [27]. Let C b e a s y mmetric monoidal catego ry with mono idal pro duct ⊗ , unit S , and symmetry isomorphism γ : X ⊗ Y → Y ⊗ X . Deﬁnition 1.1. An ob ject A in C is dualizable with dua l B if there are maps η : S → A ⊗ B and ǫ : B ⊗ A → S such that the compos ites A ∼ = S ⊗ A η ⊗ i d / / A ⊗ B ⊗ A id ⊗ ǫ / / A ⊗ S ∼ = A and B ∼ = B ⊗ S id ⊗ η / / B ⊗ A ⊗ B ǫ ⊗ id / / S ⊗ B ∼ = B are the identit y maps of A a nd B re s pe c tively . The mo st familia r exa mple o f a sy mmetr ic mo noidal ca teg ory is the category o f mo dules o ver a comm utative ring R . The tensor pr o duct is the monoidal pro duct. If M is a ﬁnitely genera ted pro jective R -mo dule, M is dualizable and the dual of M is Hom R ( M , R ). The ev aluation map ǫ : Hom R ( M , R ) ⊗ R M → R is deﬁned b y ǫ ( φ, m ) = φ ( m ). Since M is ﬁnitely generated and pr o jective the dual bas is theo r em implies there is a ‘basis’ { m 1 , m 2 , . . . , m n } with dua l ‘bas is’ { m ′ 1 , m ′ 2 , . . . , m ′ n } . The co ev a lua tion is given by linea rly extending η (1) = X m i ⊗ m ′ i . The ca tegory of c hain complexes o f mo dules ov e r a commutativ e ring R is a lso symmetric monoidal. The dualizable ob jects ar e the chain complexes that are pro jective in each degr e e and ﬁnitely genera ted. Deﬁnition 1.2. If A is dualizable and f : A → A is a n endomorphism in C , the tr ac e of f , tr( f ), is the comp osite S η / / A ⊗ B f ⊗ id / / A ⊗ B γ / / B ⊗ A ǫ / / S . The trace of a n endomo rphism in the s ymmetric mo noidal category of vector spaces ov er a ﬁeld is the sum of the dia gonal elements in a ma trix representation. The trace of an endomorphism in the c a tegory o f chain complexes of mo dules o ver a comm utative ring R is called the L efschetz n umb er . RELA TIVE FIXE D POINT THEOR Y 5 Prop ositio n 1.3. L et F : C → D b e a symmetric monoidal functor, A b e an obje ct of C with dual B , and F ( A ) ⊗ F ( B ) → F ( A ⊗ B ) and S D → F ( S C ) b e isomorph isms. Then F ( A ) is dualizable with dual F ( B ) . If f : A → A is an endomorphi sm in C , F (tr( f )) = tr( F ( f )) . The stable ho motopy ca tegory is a symmetr ic monoidal categor y . Ther e is a lso a w ay to e x press dua lit y for spaces without us ing sp ectra . Deﬁnition 1. 4. A based space X is n - dualizable if there is a based space Y and contin uo us maps η : S n → X ∧ Y and ǫ : Y ∧ X → S n such that the diagra ms S n ∧ X η ∧ i d / / γ ' ' N N N N N N N N N N N X ∧ Y ∧ X id ∧ ǫ   Y ∧ S n id ∧ η / / ( σ ∧ 1) γ ' ' N N N N N N N N N N N Y ∧ X ∧ Y ǫ ∧ id   X ∧ S n S n ∧ Y commute up to stable homotopy . The map σ : S n → S n is deﬁned b y σ ( v ) = − v . Prop ositio n 1. 5. [21, II I.4.1, I I I.5.1][23, 18.6.5 ] (1) If M is a close d smo oth m anifold that emb e ds in R m , then M + is dualizable with dual T ν , t he Tho m sp ac e of the n ormal bund le of the emb e dding of M in R m . (2) If L is a close d submanifold of a close d smo oth manifold M that emb e ds in R m , then M ∪ C L is dualizable with dual T ν M ∪ C T ν L . (3) If B is a c omp act ENR that emb e ds in R n , B + is dualizable with du al the c one on the inclusion R n \ B → R n . (4) If B is a c omp act ENR t hat emb e ds in R n and A is a sub ENR of B , then B ∪ C A is dualizable with dual ( R n \ A ) ∪ C ( R n \ B ) . Here C denotes the cone . If A ⊂ B then B ∪ C A is the mapping co ne o n the inclusion A → B . The base point of B ∪ C A is the cone p o int. The tra ce of an endomor phism of s paces reg arded a s a map in the stable ho- motopy ca tegory is called the ﬁ xe d p oint index . The index is the stable homotopy class o f a ma p S n → S n and so is a n element of the 0 th stable homo topy gro up o f S 0 , π s 0 . F or other descr ip- tions of the ﬁxed p oint index see [3, 8]. The index of the identit y map of a space X is ca lled the Euler char acteristic o f X a nd it is denoted χ ( X ). This is consistent with the c la ssical deﬁnition of the Euler character is tic. Since the rational homolo gy functor is stro ng symmetric mono ida l, P rop osition 1.3 implies that the ﬁxed point index of a map f is equal to the Lefschetz num ber of H ∗ ( f ). Since the ﬁxed po in t index of a map with no ﬁxed p oints is zero , this implies the Lefsc hetz ﬁxed point theor em. 6 KA TE P ONTO 2. Duality and trace f or bica tegories with shadows Unfortunately , the Reidemeister tr ace ca nno t b e deﬁned as a trace in a s ymmetric monoidal catego ry . It can b e deﬁned using the more g e neral trace in a bicategor y with shadows. Duality and tr ace in a bicateg ory are v ery similar to dualit y and trace in a symmetric monoidal catego ry but are mor e ﬂe x ible. This is a brief s ummary of the relev ant sections of [23] and [26]. F or more details see [23, Chapter 1 6 ], [26], or [2 7]. Deﬁnition 2.1. [20, 1.0] A bic ate gory B co nsists of (1) A c o llection ob B . (2) Categories B ( A, B ) for each A, B ∈ ob B . (3) F unctors ⊙ : B ( B , C ) × B ( A, B ) → B ( A, C ) U A : ∗ → B ( A, A ) for A , B and C in ob B . Here ∗ denotes the catego r y with one ob ject and one morphism. The functor s ⊙ are requir ed to satisfy unit and asso ciativit y conditions up to natur al isomo rphism 2-cells. The elements of ob B a re called 0 -cells. The ob jects of B ( A, B ) are calle d 1-cells. The mo rphisms of B ( A, B ) are called 2 -cells. The mos t familiar example of a bicategory is the bicategory Mo d with 0- cells rings, 1 -cells bimo dules, and 2 -cells homo morphisms. The bicategory co mp os ition is tens or pro duct. Deﬁnition 2 . 2. [23, 1 6.4.1] A 1-cell X ∈ B ( B , A ) is right dualizable with dua l Y ∈ B ( A, B ) if there are 2-cells η : U A → X ⊙ Y ǫ : Y ⊙ X → U B such that Y ∼ / / id & & L L L L L L L L L L L L L L L L L L L L L L L Y ⊙ U A id ⊙ η / / Y ⊙ X ⊙ Y ǫ ⊙ id   X ∼ / / id & & M M M M M M M M M M M M M M M M M M M M M M M U A ⊙ X η ⊙ i d / / X ⊙ Y ⊙ X id ⊙ ǫ   U B ⊙ Y ≀   X ⊙ U B ≀   Y X commute. The map η is calle d the c o evaluation and ǫ is called the evaluation . If R is a (not necessarily comm utative) ring and M is a ﬁnitely generated pro jective r ight R - mo dule then M is a right dualizable 1 -cell in M o d with dual Hom R ( M , R ). The ev aluation map ǫ : Hom R ( M , R ) ⊗ Z M → R is deﬁned b y ǫ ( φ, m ) = φ ( m ). This is a map of R - R - bimo dules . Since M is ﬁnitely generated and pro jective ther e are elements { m 1 , m 2 , . . . , m n } and dual element s { m ′ 1 , m ′ 2 , . . . , m ′ n } of Hom R ( M , R ) so that the co ev aluatio n map η : Z → M ⊗ R Hom R ( M , R ) RELA TIVE FIXE D POINT THEOR Y 7 is deﬁned by linearly extending η (1) = P m i ⊗ m ′ i . This is a map of ab elia n groups. Unlik e the symmetric mo noidal case, we need more s tructure before we can deﬁne trace. The additional structure is a sha dow. Deﬁnition 2.3. [26, 4.4.1] A shado w for B is a functor h h − i i : a A ∈ ob B B ( A, A ) → T to a categ ory T and a natur al isomorphism θ : h h X ⊙ Y i i ∼ = h h Y ⊙ X i i for ev ery pair of 1-ce lls X ∈ B ( A, B ) and Y ∈ B ( B , A ) such that h h ( X ⊙ Y ) ⊙ Z i i θ / /   h h Z ⊙ ( X ⊙ Y ) i i / / h h ( Z ⊙ X ) ⊙ Y i i h h X ⊙ ( Y ⊙ Z ) i i θ / / h h ( Y ⊙ Z ) ⊙ X i i / / h h Y ⊙ ( Z ⊙ X ) i i θ O O h h Z ⊙ U A i i θ / / & & L L L L L L L L L L L h h U A ⊙ Z i i   θ / / h h Z ⊙ U A i i x x r r r r r r r r r r r h h Z i i commute. Let P b e a n R - R -bimo dule. Let N ( P ) be the subgro up of P gener a ted by elements of the form rp − pr for p ∈ P and r ∈ R . Then the shadow of P is P / N ( P ). Deﬁnition 2. 4. [26, 4.5 .1] Let X b e a dualizable 1-cell and f : Q ⊙ X → X ⊙ P be a 2-cell in B . The tr ac e o f f is the co mp os ite h h Q i i ∼ = h h Q ⊙ U A i i id ⊙ η / / h h Q ⊙ X ⊙ Y i i f ⊙ id   h h X ⊙ P ⊙ Y i i θ / / h h P ⊙ Y ⊙ X i i id ⊙ ǫ / / h h P ⊙ U B i i ∼ = h h P i i . If M is a ﬁnitely gener ated pro jective right R -mo dule and f : M → M is a map of r ight R -mo dules the trace of f is the trace deﬁned by Stallings in [30]. A functor of bicategor ies F is a s hadow fun ctor if there is a natural transforma - tion ψ : h h F ( − ) i i → F ( h h − i i ) 8 KA TE P ONTO such that h h F X ⊙ F Y i i θ / /   h h F Y ⊙ F ( X ) i i   h h F ( X ⊙ Y ) i i ψ   h h F ( Y ⊙ X ) i i ψ   F ( h h X ⊙ Y i i ) θ / / F ( h h Y ⊙ X i i ) commutes for all 1 -cells X and Y where X ⊙ Y and Y ⊙ X a re both deﬁned. Prop ositio n 2. 5. [26, 4.5.7] L et X b e a right dualizable 1-c el l in B with dual Y , f : Q ⊙ X → X ⊙ P b e a 2-c el l in B a nd F : B → B ′ b e a shadow functor. I f F ( X ) ⊙ F ( Y ) → F ( X ⊙ Y ) , F ( X ) ⊙ F ( P ) → F ( X ⊙ P ) , and U F ( B ) → F ( U B ) ar e isomorphi sms and ˆ f is the c omp osite F Q ⊙ F X φ / / F ( Q ⊙ X ) F ( f ) / / F ( X ⊙ P ) φ − 1 / / F X ⊙ F P then h h F Q i i tr( ˆ f ) / / ψ   h h F P i i ψ   F h h Q i i F (tr( f )) / / F h h P i i c ommut es. W e will use this prop osition to compare diﬀerent for ms of the Lefschetz num b er and Reidemeister tra ce. 3. Some examples of bica tegories with shadows The c la ssical descr iptions of ﬁxed p oint inv ar iants require a choice of base p oint. When working with a single s pace this isn’t a pro blem. With ﬁb erwise spaces, equi- v aria nt spaces, or pairs of spaces, choos ing base p oints requir es addition conditions on the spa ce. W e can av oid these choices by using gr o up o ids ra ther than gro ups. In this section we descr ib e a genera lization of the bica tegory Mod that we will use to deﬁned ﬁxed p oint in v aria nts without choos ing base po ints. In this bicategor y w e replace rings by categ ories, mo dules by functors, and homomor phisms by na tural transformatio ns. Let V b e a symmetric mono idal categ ory with monoidal pro duct ⊗ and unit S . Deﬁnition 3.1. A categor y A is enriche d in V if for each a, b ∈ ob( A ), A ( a, b ) is a n o b ject o f V and the compo s ition fo r A , A ( b , c ) ⊗ A ( a, b ) → A ( a, c ) , is a morphis m in V . RELA TIVE FIXE D POINT THEOR Y 9 F or pairs of enriched categorie s A and B deﬁne an enriched ca tegory A ⊗ B with ob jects pairs ( a, b ) where a ∈ ob A and b ∈ ob B . If a, a ′ ∈ ob A and b, b ′ ∈ ob B , then ( A ⊗ B )(( a, b ) , ( a ′ , b ′ )) = ( A ( a, a ′ )) ⊗ ( B ( b, b ′ )) . Deﬁnition 3.2. An enriche d distributor is a functor X : A ⊗ B op → V such that the action of morphisms of A and B on X a r e maps in V . This type of functor is als o called an A - B -bimo dule. If F : A → C is an enriched functor and Y : C ⊗ B op → V is a distributor deﬁne a ne w distributo r Y F : A ⊗ B op → V b y Y F ( a, b ) = Y ( F ( a ) , b ). Deﬁnition 3 .3. An enriche d natur al tr ans formation η : X → Y is a natural transformatio n where the maps η a,b : X ( a, b ) → Y ( a, b ) are ma ps in V for all a ∈ ob A and b ∈ ob B . Enriched catego ries are the 0-cells of a bicategory w e denote by E V . The 1-cells are the distr ibutors enriched in V . The 2-cells are enriched natural transforma tions. If X : A ⊗ B op → V and Y : B ⊗ C op → V are tw o distributors, X ⊙ Y is a distributor A ⊗ C op → V . F or a ∈ ob( A ) and c ∈ ob( C ), X ⊙ Y ( a, c ) is the co equalizer of the actions of B on X and Y , a b,b ′ ∈ ob B X ( a, b ) ⊗ B ( b ′ , b ) ⊗ Y ( b ′ , c ) / / / / a b ∈ ob B X ( a, b ) ⊗ Y ( b, c ) . If Z : A ⊙ A op → V is an enriched functor, the shadow o f Z , h h Z i i , is the co equalizer of the tw o actions of A on Z , a a,a ′ ∈ ob( A ) A ( a, a ′ ) ⊗ Z ( a, a ′ ) / / / / a a ∈ ob( A ) Z ( a, a ) . In [26, Chapter 9 ] we o bserved that if A is a c onnected group oid, a distributor X : A → V is dualizable if and o nly if X ( a ) is dualizable ov er A ( a, a ) for any a ∈ ob A . The categorie s we will use here and in [25] to deﬁne rela tive and equiv ar iant ﬁxed p oint inv aria nt s a re not usually group oids , but we can extend the results from [2 6] to describ e these exa mples. Deﬁnition 3.4. [22, I I.9.2 ] A catego ry A is an EI-c ate gory if a ll endomorphis ms are is omorphisms. In an EI-ca tegory there is a partial order on the set of ob jects: x < y if A ( x, y ) 6 = ∅ . Let Ch R be the symmetric monoidal categ ory of chain co mplexes of modules ov er a commutativ e ring R and chain maps. Let A be a category enriched in the category of mo dules ov er R . This can b e reg arded as a category enriched in chain complexes c o ncentrated in degree zero. Deﬁnition 3. 5. A functor X : A → Ch R is su pp orte d on isomorphi sms if X ( f ) is the zer o map if f is not an isomorphism. 10 KA TE P ONTO If X is supp or ted on isomorphisms it only ‘sees’ a disjoint c o llection of group oids rather than the entire ca tegory A . Let B ( A ) b e a choice of representativ e for each isomorphism class of ob jects in A . Lemma 3.6. If X : A op → Ch R and Y : A → Ch R ar e supp orte d on isomorphisms then X ⊙ Y ∼ = M c ∈ B ( A ) X ( c ) ⊗ A ( c ,c ) Y ( c ) . The pr o of o f this lemma is in Section 9 . The idea of the pro of is to use Deﬁnition 3.5 to s how that M c ∈ B ( A ) X ( c ) ⊗ A ( c ,c ) Y ( c ) satisﬁes the universal prop erty that deﬁnes X ⊙ Y . Lemma 3.7. L et X and Y satisfy the c onditions of L emma 3.6. If X ( c ) is dualizable as an A ( c, c ) -mo dule with dual Y ( c ) for e ach c ∈ B ( A ) then X is dualizable with dual Y . The idea of this pro of is to use Lemma 3.6 and the co ev aluation and ev aluation maps for each X ( c ) to deﬁne co ev a luation a nd ev aluation maps for X . This pr o of can also b e found in Section 9. Another choice for V is the sy mmetr ic mo noidal ca tegory o f p ointed top ologica l spaces, T op ∗ . The bicategory E T op ∗ has 0- cells categor ies enriched in based spac es and 1-cells distributors enr iched in based spa c e s. The 2 -cells in E T op ∗ are natural transformatio ns enriched in T op ∗ . If X op : A → T op ∗ and Y : A → T op ∗ are e nr iched functors X ⊙ Y is the bar resolution B ( X , A , Y ). This is the geo metric re a lization o f the s implicial space with n -simplices a a 0 ,a 1 ,...,a n ∈ ob A X ( a 0 ) ∧ A ( a 1 , a 0 ) + . . . ∧ A ( a n , a n − 1 ) + ∧ Y ( a n ) . The deﬁnition of the s hadow is similar. If Z : A ⊗ A op → T o p ∗ is a e nr iched functor, the s ha dow of Z , h h Z i i , is the cyclic bar resolution C ( A , Z ). This is the geometric realization o f the simplicial spa ce with n -s implices a a 0 ,a 1 ,...,a n ∈ ob A Z ( a n , a 0 ) ∧ A ( a 1 , a 0 ) + . . . ∧ A ( a n , a n − 1 ) + . Let A b e a category enriched in based spaces . Let U A : A ⊗ A op → T op ∗ be deﬁned by U A ( a, a ′ ) = A ( a ′ , a ). Compo sition in A deﬁnes the action o f A and A op on U A . Deﬁnition 3. 8. An enriched functor X : A → T op ∗ is n - dualizable if there is a functor Y : A op → T o p ∗ , a map η : S n → B ( X , A , Y ), and a n A - A -equiv ar ia nt map ǫ : Y ∧ X → S n ∧ U A such tha t the us ua l triangle dia grams commute up to A -equiv ariant homotopy . W e will us e the ideas of Lemma 3.7 to pro duced dual pa irs in this bicatego r y , but w e will not prove a gener al characteriza tion. RELA TIVE FIXE D POINT THEOR Y 11 Deﬁnition 3.9. If X : A → T op ∗ is dualizable, P : A ⊗ A op → T op ∗ is a n enriched functor and f : X → X ⊙ P is an enriched natura l transformatio n the tr ac e of f is the stable ho motopy cla ss of the comp osite S n η / / h h X ⊙ Y i i f ⊙ id / / h h X ⊙ P ⊙ Y i i θ / / h h P ⊙ Y ⊙ X i i id ⊙ ǫ / / S n ∧ h h P i i . 4. The rela tive Lefschetz number The globa l and geometr ic relative Le fschet z num ber s can b o th be describ ed us- ing a clas sical appr oach, but we will describ e them using duality and trace in a bicategory . The formal structure gives a diﬀerent p ersp ective on thes e inv a riants and is a star ting p oint for the more complicated inv ariants we will consider in the later sections. Deﬁnition 4 . 1. The r elative c omp onent c ate gory Π 0 ( B , A ) of a pair ( B , A ) has ob jects the p oints of B . The mor phisms of Π 0 ( B , A ) are Π 0 ( B , A )( x, y ) =            ∗ if x ∈ B \ A and [ x ] = [ y ] ∈ π 0 ( B ) ∅ if x ∈ B \ A and [ x ] 6 = [ y ] ∈ π 0 ( B ) ∗ if x, y ∈ A and [ x ] = [ y ] ∈ π 0 ( A ) ∅ if x, y ∈ A and [ x ] 6 = [ y ] ∈ π 0 ( A ) ∅ if x ∈ A, y 6∈ A The co mpo sition is deﬁned by the rule s ∗ ◦ ∗ = ∗ ∅ ◦ ∅ = ∅ ∅ ◦ ∗ = ∅ ∗ ◦ ∅ = ∅ . F or most pa irs of spaces A ⊂ B this ca tegory is an EI-ca tegory but not a group oid. F or example, if x ∈ B \ A , y ∈ A , a nd [ x ] = [ y ] ∈ π 0 ( B ) then Π 0 ( B , A )( x, y ) = ∗ and Π 0 ( B , A )( y , x ) = ∅ . The r e la tive comp onent catego ry is s imilar to the equiv ariant comp onent categ o ry in [32, I.10.3]. If A a nd B are connected and B \ A is no nempt y this category has tw o isomor- phism classes o f o b jects. If x ∈ A , let A ( x ) b e the comp one nt of A that contains x . If y ∈ B , let B ( y ) be the compo nent o f B that cont ains y . Deﬁnition 4.2. The r elative c omp onent sp ac e , B | A , of the pair ( B , A ) is the func- tor Π 0 ( B , A ) op → T op ∗ deﬁned b y B | A ( x ) =  A ( x ) + if x ∈ A B ( x ) / ( A ∩ B ( x )) if x 6∈ A The mo rphisms A ( x ) → A ( x ) and B ( x ) / ( A ∩ B ( x )) → B ( x ) / ( A ∩ B ( x )) ar e the ident ity maps. The map A ( x ) → B ( x ) / ( A ∩ B ( x )) is the inclusio n of A ( x ) as the base point. Recall that C ( A ∩ B ( x )) is the co ne on A ∩ B ( x ) . The base p oint is the cone po int . Since A ⊂ B is ass umed to b e a coﬁbra tion B ( x ) ∪ C ( A ∩ B ( x )) is homotopy equiv alent to B ( x ) / ( A ∩ B ( x )). Lemma 4.3. If A and B ar e b oth c omp act ENR’s or close d smo oth manifolds then B | A is dualizable. 12 KA TE P ONTO R emark 4 .4 . Star ting with the pro of of this theor e m w e will fo cus o n the case o f closed smo oth ma nifolds. The res ults in this s ection a nd Sections 5 and 6 hav e versions for compact ENR’s as well. The statements and pro ofs for compact ENR’s are very similar to thos e for closed smo oth manifolds. Some of the re s ults in Section 7 hav e only b een shown for manifolds. Pr o of. Deﬁne a functor D ( B | A ) : Π 0 ( B , A ) → T o p ∗ by D ( B | A )( x ) =  D ( A ( x ) + ) if x ∈ A D ( B ( x ) ∪ C ( A ∩ B ( x ))) if x 6∈ A where D ( A ( x ) + ) and D ( B ( x ) ∪ C ( A ∩ B ( x ))) denote the duals o f A ( x ) + and B ( x ) ∪ C ( A ∩ B ( x ) ) with resp ect to a n em b edding of B in R n as describ ed in Prop ositio n 1.5. The morphisms are the iden tity maps or the inclus ion. T o simplify no tation, consider the case wher e A a nd B are b oth connected. The general ca se is s imilar. The ev aluatio n for this dual pair is a natura l tra ns formation ǫ : D ( B | A ) ⊙ B | A → S n ∧ (Π 0 ( B , A )) + . Let x ∈ A and y ∈ B \ A represent the is omorphism class es of ob jects of Π 0 ( B , A ). Then ǫ co nsists of four maps D ( B ∪ C ( A )) ∧ ( B / A ) → S n D ( A + ) ∧ ( B / A ) → S n D ( B ∪ C ( A )) ∧ A + → ∗ D ( A + ) ∧ A + → S n By natura lity , the second map must b e the constant map to a p oint. Since A + and B ∪ C ( A ) are both dualizable the ev aluations for these dual pairs are the ﬁrst and fourth maps. Note that B ( B | A , Π 0 ( B , A ) , D ( B | A )) is eq uiv alent to ( A + ∧ D ( A + )) ∨ [( B / A ) ∧ D ( B ∪ C A )] . The dua lizability of A + and B ∪ C ( A ) pr ovide co ev a lua tion maps η A : S n → A + ∧ D ( A + ) η B / A : S n → B / A ∧ D ( B ∪ C ( A )) . The co ev a luation for this dual pair is the compo site S n △ / / S n ∨ S n η A ∨ η B/A   ( A + ∧ D ( A + )) ∨ [( B / A ) ∧ D ( B ∪ C A + )] . V erifying that these maps de s crib e a dual pair can b e chec ked on x and y sepa - rately . T he conditions reduce to conditions c heck ed for Pr op osition 1.5.  Let Π f 0 ( B , A ) b e the functor Π 0 ( B , A ) × Π 0 ( B , A ) op → T op ∗ deﬁned by Π f 0 ( B , A )( x, y ) = Π 0 ( B , A )( f ( y ) , x ) + . The left a ction is comp osition. The r ight action is given by applying f and then the compos itio n. RELA TIVE FIXE D POINT THEOR Y 13 A relative map f : ( B , A ) → ( B , A ) induces a natural transformation f : B | A → B | A ⊙ Π f 0 ( B , A ) . Since B | A is dualiza ble, the tra ce o f f is deﬁned. Deﬁnition 4.5. The r elative ge ometric L efschetz num b er of f , i B | A ( f ), is the trace of f . The relative g eometric Lefschetz nu mber is the stable ho motopy class of a map S 0 → h h Π f 0 ( B , A ) i i and so it is an element of the 0 th stable homotopy group of h h Π f 0 ( B , A ) i i . This group is denoted π s 0 ( h h Π f 0 ( B , A ) i i ). It is the free ab elia n gro up on the set h h Π f 0 ( B , A ) i i . Since the rela tive geo metric Lefschetz nu mber is deﬁned to b e the trace of f it is an inv aria nt of the relative homotopy class of f . If Π 0 ( A ) is the set of comp onent of A , h h Π f 0 ( A ) i i is { [ x ] ∈ π 0 ( A ) | [ f | A ( x )] = [ x ] } . If X is a closed smo oth manifold, f : X → X is a c ontin uous map a nd F is an isolated subse t of the set o f ﬁxed p o ints of f let i ( F, f ) be the sum o f the ﬁxed po int indices of the ﬁxed p oints in F . See [8] for the deﬁnition of the ﬁxed p oint index o f an iso lated set of ﬁxed p oints. Lemma 4.6. Ther e is an isomorphism h h Π f 0 ( B , A ) i i ∼ = h h Π f 0 ( A ) i i ∐ h h Π f 0 ( B ) i i and the image of i B | A ( f ) under this isomorphism is X x ∈ h h Π f 0 ( A ) i i i (Fix( f ) ∩ A ( x ) , f )[ x ] + X y ∈ h h Π f 0 ( B ) i i i (Fix( f ) ∩ ( B ( y ) \ A ) , f ))[ y ] . Since f is taut i (Fix( f ) ∩ A, f ) = i (Fix( f ) ∩ A, f | A ) and i (Fix( f ) ∩ ( B \ A ) , f ) = i (Fix( f ) , f ) − i (Fix( f ) ∩ A, f | A ) . The ﬁr st equa lity fo llows from co mm utativity of the index and the de ﬁnitio n of a taut ma p. See [35, 3.5] for a pro o f of the second equalit y . Pr o of. Assume A and B are connected and A is a prop er subset of B . T hes e assumptions restrict the num b er of co mpo nen ts of h h Π f 0 ( B , A ) i i . The pro of is similar for the g eneral case. The set h h Π f 0 ( B , A ) i i is de ﬁned to b e the coe qualizer ` x,y Π 0 ( B , A )( x, y ) + ∧ Π f 0 ( B , A )( y , x ) / / / / ` x Π f 0 ( B , A )( x, x ) / / h h Π f 0 ( B , A ) i i . Since Π 0 ( B , A )( x, y ) is empt y if x ∈ A and y 6∈ A , this co equalizer splits as tw o co equalizers . One is ov er pairs ( x, y ) where x, y ∈ A and the other is ov er pairs ( x, y ) wher e x, y 6∈ A . Each of these c o equalizers consis ts o f a sing le e lement . This isomorphism is compatible with L emma 4.3 so the imag e o f i B | A ( f ) under the pro jection to the ﬁrst s umma nd is the tr a ce of f restricted to A . As observed after Prop osition 1.5 this is the ﬁxed po in t index of f | A . The image of i B | A ( f ) under the pro jection to the second summand is the trace of f / A : B / A → B / A . The ﬁxed p oints of f / A ar e the ﬁxed p oints of f | B \ A and 14 KA TE P ONTO the p oint that represents A . The p oint that r epresents A is the base p oint and so its index do es not contribute to the trace of f / A , see [21, I I I.8.5].  The second compo nen t of i B | A ( f ) is the index deﬁned in [15, 1.1]. Example 4.7. Let J be a nonempty , prop er, connected subinterv a l of S 1 . Let f : ( S 1 , J ) → ( S 1 , J ) b e the identit y map. Then i S 1 | J ( f ) = (1 , − 1). Corollary 4.8. If f : ( B , A ) → ( B , A ) has no ﬁxe d p oints then i B | A ( f ) = 0 . Pr o of. Since f has no ﬁxed p o ints i (Fix( f ) , f ) = 0. T o compute the relative geo- metric Lefschetz num b e r of f we repla c e f by a relatively homoto pic map g that is taut. The map g ca n b e chosen so tha t f | A = g | A . T hen i (Fix( g ) , g ) = 0 a nd i (Fix( g ) ∩ A, g | A ) = i (Fix( f ) ∩ A, f | A ) = 0. Since f has no ﬁxed p oints and f | A = g | A , i (Fix( g ) ∩ A, g ) = 0. Since g is taut i (Fix( g ) ∩ ( B \ A ) , g ) = i (Fix( g ) , g ) − i (Fix( g ) ∩ A, g | A ) = 0 .  Let Z Π 0 ( B , A ) b e the category with the same ob jects as Π 0 ( B , A ). F or ob jects x and y of Π 0 ( B , A ) Z Π 0 ( B , A )( x, y ) is the fr ee ab elian group on Π 0 ( B , A )( x, y ). Comp osing B | A with the rational homology functor deﬁnes a functor H ∗ ( B | A ) : Z Π 0 ( B , A ) → Ch Q . Prop ositio n 4.9. If A ⊂ B ar e close d smo oth manifolds, t hen H ∗ ( B | A ) is dualiz- able. Pr o of. There are tw o wa ys to prove this pr op osition. First, the ra tional homology functor is strong symmetric monoidal, so this follows fro m Prop osition 2.5. W e can also show H ∗ ( B | A ) is dualizable dire c tly by describing the co ev a lua tion and ev aluation. The functor H ∗ ( B | A ) is supp or ted o n isomor phisms and so it is enough to co nstruct a co ev aluation and ev aluation for the chain complexes of vector spaces H ∗ ( A ) and H ∗ ( B , A ). These are b o th ﬁnite dimensional, a nd so they are bo th dualizable with duals as in Section 1.  A relative map f : ( B , A ) → ( B , A ) induces a map H ∗ ( f ) : H ∗ ( B | A ) → H ∗ ( B | A ) ⊙ Z Π f 0 ( B , A ) by a pplying the r ational homology functor to f . Deﬁnition 4.1 0. The r elative glob al L efschetz numb er of f , L B | A ( f ), is the trace of H ∗ ( f ). Lemma 4.11. The image of L B | A ( f ) u nder the isomorphism in L emma 4.6 is X x ∈ h h Π f 0 ( A ) i i L A ( x ) ( f )[ x ] + X y ∈ h h Π f 0 ( B ) i i L B ( y ) / ( A ∩ B ( y )) ( f )[ y ] . Here L A ( x ) ( f ) a nd L B ( y ) / ( A ∩ B ( y )) ( f ) denote the tra ces of the maps induced by f on H ∗ ( A ( x )) and H ∗ ( B ( y ) / ( A ∩ B ( y ))). Pr o of. Using Prop osition 4.9 this pr o of is similar to the pro of of Lemma 4.6.  RELA TIVE FIXE D POINT THEOR Y 15 The in v a riant L B / A ( f ) is the relative Lefschetz num b er of [1]. Prop ositio n 4. 12. In Z h h Π f 0 ( B , A ) i i , L B | A ( f ) = i B | A ( f ) . Pr o of. This pro po sition follows fr om Prop ositio n 2.5 and the o bserv a tion that the rational homology functor is stro ng symmetric monoidal.  The r e la tive Lefschetz ﬁxed p oint theorem follows from this prop ositio n and Corollar y 4.8. Theorem A (Relative Lefschet z ﬁxed p oint theorem) . L et A ⊂ B b e close d s m o oth manifolds and f : ( B , A ) → ( B , A ) b e a r elative map. If f has no ﬁxe d p oints then L B | A ( f ) = 0 . F urther, if L B | A ( f ) 6 = 0 all ma ps relatively homotopic to f hav e a ﬁxed p oint. 5. The geometric Reidemeister trace T o prov e a conv erse to Theorem A it is necessa ry to introduce r eﬁnement s of the inv aria nt s deﬁned in the previous sec tio n. The ﬁrst of these inv a r iants is the geo- metric Reidemeister trace. This is a reﬁnement of the geometric Lefschetz num b er and it will ser ve as a transitio n b etw e e n the global Reidemeister trace in Section 6 and the inv ariant deﬁned in Section 7. As for the inv ariants in the pr evious sectio n, it is p os sible to deﬁne the ge o metric Reidemeister trace using a generaliza tion of the sta ndard appro a ch o f ﬁxed p oint indices and ﬁxed p oint clas s es. Also as in the previous s ection, w e do not use that a pproach her e. Instea d we use duality and trace in bicatego ries with shadows. This pers pe c tive g ives simple compariso ns of diﬀerent in v a r iants and also uniﬁes the descriptions of diﬀerent fo r ms of the Reidemeister trace with the Lefschetz num ber . Deﬁnition 5. 1. The r elative fundamental c ate gory , Π 1 ( B , A ), of the pair ( B , A ) has ob jects the points of B . The morphis ms Π 1 ( B , A )( x, y ) are the homotopy classes of paths from x to y in A if x ∈ A and homotopy cla s ses of paths from x to y in B if x ∈ B \ A . The rela tive fundamen tal categor y is a sub categ o ry of the fundamental gr o up o id of B . In most cases it is not a gro upo id. F or example, if A and B a re b oth path connected, x ∈ A , and y ∈ B \ A then Π 1 ( B , A )( x, y ) is empty and Π 1 ( B , A )( y , x ) is nonempty . This category is an EI-categor y . This catego ry is s imilar to the equiv ar iant fundamental ca tegory , see [32, I.10.7]. F or x ∈ A , let ˜ A x be the univ ersa l cov er of A based at x . W e think o f p oints in ˜ A x as homoto p y cla sses of paths in A that star t at x . F or y ∈ B \ A let ˜ B y be the universal cover of B based at y . Let p : ˜ B y → B b e the quotient map and ¯ A y = p − 1 ( A ) ⊂ ˜ B y . Deﬁnition 5.2. The r elative universal c over o f the pair ( B , A ) is the functor g B | A : Π 1 ( B , A ) op → T op ∗ deﬁned b y g B | A ( x ) =  ( ˜ A x ) + if x ∈ A ˜ B x / ¯ A x if x 6∈ A on ob jects and b y comp osition of pa ths on mo rphisms. 16 KA TE P ONTO Lemma 5.3. If A ⊂ B is a c oﬁbr ation ˜ B x / ¯ A x is π 1 ( B ) -homotopy e quivalent to ˜ B x ∪ C ¯ A x . Pr o of. There is a π 1 ( B )-eq uiv ariant map φ : ˜ B x ∪ C ¯ A x → ˜ B x / ¯ A x deﬁned b y collapsing the co ne to the base p oint. Since A ⊂ B is a coﬁbration there is a map u : B → I such that u − 1 (0) = A and a ho motopy h : B × I → B such that h ( b, 0) = b for a ll b ∈ B , h ( a, t ) = a for all a ∈ A and t ∈ I a nd h ( b, 1) ∈ A if u ( b ) < 1. The map ψ : ˜ B x / ¯ A x → ˜ B x ∪ C ¯ A x is de ﬁned by ψ ( γ ) =  h ( γ (1) , t ) | [0 , 2(1 − u ( γ (1)))] ◦ γ if 1 2 ≤ u ( γ (1)) ≤ 1 ( h ( γ (1) , t ) ◦ γ , 1 − 2 u ( γ (1))) if 0 ≤ u ( γ (1)) ≤ 1 2 The ma p ψ is π 1 ( B ) equiv ariant. Up to homotopy it is an inv erse for φ .  Theorem 5.4. If A ⊂ B ar e close d smo oth manifolds the r elative universal c over g B | A is dualizable as a mo du le over Π 1 ( B , A ) . Pr o of. The pro o f of this lemma is very similar to the pro of of Lemma 4.3. W e will deﬁne this dual pair by deﬁning a dual pair for ea ch isomorphism clas s of ob jects in Π 1 ( B , A ). T o s implify notation, co nsider the case where A a nd B are connected. Let S ν A be the ﬁber wise one p oint co mpactiﬁcation of the nor mal bundle o f A . This is a space over A and has a s ection given by the p oints at inﬁnity . Let D ( ˜ A + ) b e the space ( ˜ A × A S ν A ) / ∼ where all p oints of the for m ( γ , ∞ γ (1) ) are ident iﬁed to a single point. This is the dual of ˜ A + as a distr ibutor o ver π 1 ( A ), see [26, 5 .3.3]. Let C B ( S ν B , S ν A ) be ( B × { 0 } ) ∪ ( S ν A × I ) ∪ ( S ν B × { 1 } ) . This is the ﬁb erwis e co ne of the map S ν A → S ν B ov er B . Let D ( ˜ B ∪ C ¯ A ) be the space ( ˜ B × B C B ( S ν B , S ν A )) / ∼ where a ll p oints of the form ( γ , γ (1) × { 1 } ) ar e iden tiﬁed to a single p oint. This is the ⊙ comp osition of the ﬁb erwis e spaces ( ˜ B , p ) + and C B ( S ν B , S ν A ) deﬁned in [23, 17.1 .3]. An argument like that in [26, 5.3.3] for ˜ A + shows this is the dual of ˜ B ∪ C ¯ A as a distributor ov er π 1 ( B ). The dual o f g B | A , denoted D ( g B | A ), is D ( g B | A )( x ) =  D ( ˜ A + ) if x ∈ A D ( ˜ B ∪ C ¯ A ) if x ∈ B \ A The a ction of the mor phisms in Π 1 ( B , A ) is b y compo sition. RELA TIVE FIXE D POINT THEOR Y 17 Using the assumption that A and B ar e connected there ar e tw o isomor phism classes of ob jects in Π 1 ( B , A ). As in Lemma 4.3, there are four maps that deﬁne the natural transformatio n ǫ . E xactly as in that case there are only tw o that are nontrivial. These maps are the ev alua tion maps for the dual pa irs ( ˜ A + , D ( ˜ A + )) and ( ˜ B / ¯ A, D ( ˜ B ∪ C ¯ A )). Also as in Lemma 4.3, B ( g B | A , Π 1 ( B , A ) , D ( g B | A )) is equiv alent to  ˜ A + ∧ π 1 ( A ) D ( ˜ A + )  ∨  ( ˜ B / ¯ A ) ∧ π 1 ( B ) D ( ˜ B ∪ C ¯ A )  . The co ev aluatio n map is the comp osite of the fold map S n → S n ∨ S n and the co e v aluations for the pairs ( ˜ A + , D ( ˜ A + )) and ( ˜ B ∪ C ¯ A, D ( ˜ B ∪ C ¯ A )),. The requir ed diagr ams c ommut e s ince the co ev a lua tion and ev aluation maps are deﬁned using co ev aluatio n and ev aluatio n maps from the dual pairs ( ˜ A + , D ( ˜ A + )) and ( ˜ B / ¯ A, D ( ˜ B ∪ C ¯ A )).  R emark 5.5 . W e can als o give more explicit descr iptions of the co e v aluation and ev aluatio n maps for the pa irs ( ˜ A + , D ( ˜ A + )) and ( ˜ B / ¯ A, D ( ˜ B ∪ C ¯ A )). The co ev aluatio n for the pair ( ˜ A + , D ( ˜ A + )) is the co mpo site S n / / T ν A / / ˜ A + ∧ π 1 A D ( ˜ A + ) of the Pon tryagin-Thom map for an embedding of A in S n with the map v 7→ ( γ , γ , v ) where γ is a n y element o f ˜ A that ends at the base of v . Since A is lo cally contractible there is a neighborho od U of the diag o nal in A × A and a map H : V → A I that s atisﬁes H ( x, x )( t ) = x , H ( x, y , 0) = x , a nd H ( x, y , 1 ) = y . The ev a luation for the pair ( ˜ A + , D ( ˜ A + )), D ( ˜ A + ) ∧ ˜ A + → S n ∧ π 1 A + is de ﬁned by ( v , γ , δ ) = ( ǫ ( v , δ (1 )) , γ − 1 H ( δ (1) , γ (1)) δ ) where ǫ is the ev a luation for the dual pair ( A + , D ( A + )). The co ev aluation and ev aluation for the dual pair ( ˜ B / ¯ A, D ( ˜ B ∪ C ¯ A )) ar e similar. A relative map f : ( B , A ) → ( B , A ) induces a map f ∗ : g B | A → g B | A ⊙ Π f 1 ( B , A ) where Π f 1 ( B , A )( x, y ) = Π 1 ( B , A )( f ( y ) , x ) + . The left a ction is the usual left action. The r ight action is given by applying f and then compo sition. Deﬁnition 5 .6. The r elative ge ometric R eidemeister tra c e of f : ( B , A ) → ( B , A ), ge R B | A ( f ), is the tra ce o f the map f ∗ : g B | A → g B | A ⊙ Π f 1 ( B , A ) . The relative geo metric Reidemeister tr ace is the stable ho mo topy cla ss of a ma p S 0 → h h Π f 1 ( B , A ) i i 18 KA TE P ONTO and so it is an element of the 0 th stable homotopy group of h h Π f 1 ( B , A ) i i . By def- inition, the r elative geo metr ic Reidemeister trace is an inv aria nt of the r elative homotopy class of the map. Let X b e a dua lizable space. F or a space U and a map △ : X → X ∧ U the tr ansfer of an endo morphism f : X → X with resp ect to △ is the co mpo site S n η → X ∧ DX γ → D X ∧ X id ∧ f → D X ∧ X id ∧△ → D X ∧ X ∧ U ǫ ∧ id → S n ∧ U. Let Λ f | A A : = { γ ∈ A I | f ( γ (0)) = γ (1) } and Λ f B : = { γ ∈ B I | f ( γ (0)) = γ (1) } . Since A and B are lo cally contractible and f is taut there are neighbor ho o ds U A of the ﬁxed p oints of A and U B of the ﬁxed p oints of B \ A and maps ι A : U A → Λ f | A A ι B : U B → Λ f B that take ﬁxed p oints of f to the constant path at that p oint. Note that tw o ﬁxed po int s of f are in the same ﬁxed p oint class if and only if their imag es ar e in the same connected comp onent of Λ f | A A or Λ f B . See [3] or [16] for the deﬁnition of ﬁxed p o int classes. Let τ U A ( f | A ) denote the trans fer of f with resp ect to the diagonal map A + → A + ∧ U A /∂ ( U A ) and similarly for B . Lemma 5.7. If A is a pr op er subset of B ther e is an isomorphism π s 0 ( h h Π f 1 ( B , A ) i i ) ∼ = π s 0 (Λ f | A A ) ⊕ π s 0 (Λ f B ) . The image of the r elative ge ometric R eidemeister t r ac e of f u nder t his isomorphism is ( ι A ) ∗ ( τ U A ( f | A )) + ( ι B ) ∗ ( τ U B ( f )) . Pr o of. W e ﬁrst deﬁne the isomorphism. Note that π s 0 ( X ) ∼ = Z π 0 ( X ) for a ny s pace X , so it is enough to show π 0 (Λ f | A A ) ⊕ π s 0 (Λ f B ) satisﬁe s the universal prop er ty that deﬁnes the shadow of Π f 1 ( B , A ). The shadow o f Π f 1 ( B , A ) is deﬁned to be the co equalizer of the maps ∐ x,y Π 1 ( B , A )( x, y ) × Π 1 ( B , A )( f ( y ) , x ) / / / / ∐ x Π 1 ( B , A )( f ( x ) , x ) . The inclus ion maps ( ∐ x ∈ A Π 1 ( B , A )( f ( x ) , x )) ∐ ( ∐ x 6∈ A Π 1 ( B , A )( f ( x ) , x )) → π 0 (Λ f | A A ) ⊕ π s 0 (Λ f B ) deﬁne a ma p θ : ∐ x Π 1 ( B , A )( f ( x ) , x ) → π 0 (Λ f | A A ) ⊕ π s 0 (Λ f B ) . Let α ∈ Π 1 ( B , A )( x, y ) and β ∈ Π 1 ( B , A )( f ( y ) , x ). If x, y ∈ A then β α and f ( α ) β represent the same elements in π 0 (Λ f | A A ). If x, y ∈ B \ A , β α and f ( α ) β represent the same element s in π 0 (Λ f B ). If x a nd y do not satisfy these conditions , there is no condition to chec k o n the paths. So θ co eq ualizes. RELA TIVE FIXE D POINT THEOR Y 19 If φ : ∐ x Π 1 ( B , A )( f ( x ) , x ) → M is a map that co equalizes the maps ab ov e deﬁne a map ¯ φ : π 0 (Λ f | A A ) ⊕ π s 0 (Λ f B ) → M by ¯ φ ( γ ) = φ ( β ) where β is a ny element of Π 1 ( B , A )( f ( x ) , x ) that maps to γ in π 0 (Λ f | A A ) ⊕ π s 0 (Λ f B ). This is indep endent of choices s ince if α is a nother lift of γ then there are paths µ a nd ν such that f ( µ ) ν is homotopic to β and ν µ is homotopic to α . Then ¯ φ is unique and π 0 (Λ f | A A ) ⊕ π s 0 (Λ f B ) is the co equalizer . T o describ e the image o f the geometric Reidemeister trace under this isomor- phism it is enoug h to show the trace of ˜ f | A : ˜ A → ˜ A is ( ι A ) ∗ ( τ U A ( f | A )) and the tr a ce of ˜ f : ˜ B / ¯ A → ˜ B / ¯ A is ( ι B ) ∗ ( τ U B ( f )). W e will describ e the ﬁr st, the sec o nd is simila r. In Remark 5.5 we gave explicit des criptions of the co ev aluation and ev aluation for the dual pair ( ˜ A + , D ( ˜ A + )). Let q : D ( ˜ A + ) ∧ ˜ A + → D A ∧ A + be the quotient map. If η 1 and ǫ 1 are the co ev a luation and ev aluation for the dual pair ( ˜ A + , D ( ˜ A + )) and η 2 and ǫ 2 are the co ev alua tion and ev a luation for the dual pair ( A + , DA + ) then the explicit descr iptions of η 1 and ǫ 1 show S n η 1 / / η 2 & & M M M M M M M M M M M M ˜ A + ∧ π 1 A D ( ˜ A + ) q   A + ∧ D A and h h D ( ˜ A + ) ∧ ˜ A + ∧ Π f 1 ( A ) i i ǫ 1 / / q   S n ∧ h h Π f 1 ( A ) i i + D A ∧ A + id ∧△ / / D A ∧ A + ∧ U A /∂ U A ǫ 2 ∧ id / / S n ∧ U A /∂ U A id ∧ ι A O O commute. T ogether these diagrams show ( ǫ 1 ∧ id)( ˜ f ∧ id) γ η 1 = (id ∧ ι A )( ǫ 2 ∧ id)(id ∧△ ) q ( ˜ f ∧ id) γ η 1 = (id ∧ ι A )( ǫ 2 ∧ id)(id ∧△ )( f ∧ id) γ q η 1 = (id ∧ ι A )( ǫ 2 ∧ id)(id ∧△ )( f ∧ id) γ η 2 The ﬁr st co mpo site is the trace of g f | A . The la st c omp o site is ( ι A ) ∗ ( τ U A ( f | A )).  F or a ﬁxe d p oint class β of f : B → B let i r el β be the index of the ﬁxed p oints asso ciated to β that are contained in B \ A . F or a ﬁxed p oint class α of f | A : A → A , let i α be the index of the ﬁxed p oints asso cia ted to α . Since the map f is taut, i α is the ﬁxed p oint index of the ﬁxed points in A with resp ect to either f | A or f . The following coro llary is a consequence o f Lemma 5.7 and is the g eneralizatio n of Lemma 4.6. This co rollary identiﬁes the relative geometric Reidemeister trace with the g eneraliza tio n o f the c la ssical description o f the Reidemeister tra ce. 20 KA TE P ONTO Corollary 5.8. Under the isomorphism in L emma 5.7, ge R B | A ( f ) =  X i α α  +  X i r el β β  ∈ π 0 (Λ f | A A ) ⊕ π s 0 (Λ f B ) . The following tw o examples were descr ibed in [2 4]. In that pap er the g eneralized Lefschetz num b er and one form o f the r elative Niels e n n umber are computed. Example 5.9. [24, 5.1] Let B = D 2 × S 1 and A = S 1 × S 1 . Let f : B → B b e f ( re iθ , e it ) = ( f 1 ( r ) e − iθ , e 3 it ) where f 1 : [0 , 1] → [0 , 1] is a contin uous function s uch that f 1 (0) = 0, f 1 (1) = 1 and f 1 has no o ther ﬁxed p oints. Then f is a relative map with six ﬁxed p oints. There ar e fo ur ﬁxed po int s in A . These ﬁxed p oints all r epresent diﬀerent ﬁxe d po int cla s ses and all hav e index − 1. The tw o ﬁxed po in ts o utside of A represent diﬀerent ﬁx e d point classes in B and a lso ha ve index − 1. Since A is a torus, π 1 ( A ) = h a, b | abab = 1 i . The relatio n imp osed on the shadow implies h h π 1 ( A ) φ i i consists of 4 elements, 1 , a, b , ab. F or B , π 1 ( b ) = h b i and h h π 1 ( B ) φ i i consists o f 2 elements, 1 , b. Then ge R B | A ( f ) = − 1(1 A + a A + b A + ab A + 1 B + b B ) . Example 5.10. [24, 5.2] Let B = S 1 × S 1 and A = 1 × S 1 . Let f : B → B b e f ( e iθ , e it ) = ( e 3 iθ , e 4 it ) . There are three ﬁx ed po ints of f in A and three additional ﬁxed p oints o f f in B \ A . The three ﬁx ed points of f in A r e present each o f the three p ossible ﬁxe d p oint classes. These ﬁxed p oints all have index 1 . The three ﬁxed p oints in B that ar e not in A also represe nt three distinct ﬁxed p oint classes, but these are only three of the six p ossible ﬁxed po int classes . These ﬁxed points also hav e index 1 . Let π 1 ( B ) = h a, b | abab = 1 i . Then π 1 ( A ) = h a i . The set h h π 1 ( B ) φ i i consists of 1 , a, a 2 , b, ab , a 2 b. The s et h h π 1 ( A ) φ i i consists of 1 , a, a 2 . Then ge R B | A ( f ) = 1 A + a A + a 2 A + b B + ( ab ) B + ( a 2 b ) B . The relative Nie lsen n umber. One of the ex pec ta tions for the Reidemeister trace is that it can detect when a map has no ﬁx ed p oints but it does no t hav e to provide a lower b ound for the nu mber of ﬁxed p oints. This is very diﬀeren t from the Nielsen num b er. The go al of the Nielsen n umber is to pr ovide a low er b o und. In the classical ca s e, the Nielsen num b er is the num b er of no nz e r o co e ﬃcient s in the Reidemeister trace. This implies the Nielsen n umber is zer o if and o nly if the Reidemeis ter trace is zero . F or more genera l s ituations the connection b etw een nonzero coeﬃcients of the Reidemeister trace and the Niels e n n umber do e s not hold. It r emains true that the Nielsen num be r is zer o if and only if the Reidemeister trace is z ero. RELA TIVE FIXE D POINT THEOR Y 21 The inclusion o f A int o B induces a map π 1 ( A ) → π 1 ( B ) and als o induces a map Φ from the ﬁxed p o int cla sses of A to the ﬁxed p oint c la sses of B . A ﬁxed p oint class of f o r f | A is essential if its co eﬃcient in the clas sical Reidemeister trace is nonzero. Let N ( f , f | A ) be the n umber of essential ﬁxed p oint clas ses of B that are in the image o f an essential c la ss of A . Let N ( f ) be the clas sical Nielsen num b er of f and N ( f | A ) be the classical Nielsen num ber of f | A . Deﬁnition 5.11. [35, 2.5] The r elative Nielsen numb er , N ( f ; B , A ), is N ( f | A ) + ( N ( f ) − N ( f , f | A )) . Lemma 5.12. The r elative Nielsen num b er of f is zer o if and only if the r elative ge ometric R eidemeister tr ac e of f is zer o. Pr o of. If the rela tive geometric Reidemeister tr ace of f is zer o Coro llary 5.8 implies  P i r el β β  + ( P i α α ) is zer o. Since Z h h Π f 1 ( B , A ) i i is a free group g enerated b y the α ’s and β ’s eac h i r el β and i α are z ero. Since the i α ’s are zero , N ( f | A ) and N ( f , f | A ) are z ero. Since each of the i α ’s are zero i β = i r el β = 0 for every β . This implies N ( f ) is also zer o. By deﬁnition N ( f | A ), N ( f ), a nd N ( f , f | A ) ar e all greater than o r equa l to zero and N ( f , f | A ) ≤ N ( f ). If the relative Nielsen num b er of f is zero N ( f | A ) = 0 and N ( f ) = N ( f , f | A ). Since N ( f | A ) = 0 , N ( f , f | A ) = 0 and so N ( f ) = 0. Since N ( f | A ) = 0 all of the i α ’s are zero and i r el β = i β . Since N ( f ) = 0, i β = 0 for all β .  The rela tive Niels e n num ber s for the maps in the examples ab ov e were computed in [24]. The re la tive Niels en num be r for Example 5.9 is 4. This is no t the num b er of non zero co eﬃcients in the relative Reidemeister tr ace. The re la tive Nielsen num b er for Ex a mple 5 .10 is 6 . This do es ha pp en to b e the n umber o f nonze r o co eﬃcients in the relative Reidemeister tra ce. These num b ers co incide b eca use N ( f , f | A ) is zero for this example. Other references for rela tive Nielsen theory include [15, 2 8, 29, 33, 34]. Thes e inv aria nt s are a lso r elated to the Nielsen num b ers for stra tiﬁed spa ces deﬁned in [17]. 6. The global Reidemeister trace In this section we deﬁne the relative globa l Reidemeister tra ce. This in v a r iant is a generalization of the relative g lobal Lefschetz num b er and c a n b e identiﬁed with the relative geometr ic Reidemeister trace. T he relative global Reidemeister tr a ce a relative genera lization of the inv ar iant deﬁned in [13]. It is related to the inv ar iants deﬁned in [24 , 3 6], but it is not the same a s either of these inv a riants. Let Z Π 1 ( B , A ) b e the catego ry with the same ob jects a s Π 1 ( B , A ). The mor- phism set Z Π 1 ( B , A )( x, y ) is the free ab elian gro up on the set Π 1 ( B , A )( x, y ). There is a functor C ∗ ( g B | A ) : Z Π 1 ( B , A ) op → Ch Q 22 KA TE P ONTO deﬁned by C ∗ ( g B | A )( x ) = C ∗ ( g B | A ( x ); Q ) where the seco nd C ∗ indicates the cellular chain co mplex. The action of the morphisms of Π 1 ( B , A ) is induced from the action on g B | A . This functor is deﬁned in the same wa y that the functor H ∗ ( B | A ) is deﬁned from the functor B | A except w e r eplace the ra tional homolog y functor with the r ational c hain complex functor . Prop ositio n 6.1 . If A ⊂ B ar e close d sm o oth manifolds the Z Π 1 ( B , A ) -mo dule C ∗ ( g B | A ) is dualizable. Pr o of. Like Pro po sition 4.9 there are tw o p o ssible pro ofs of this theo rem. The ra tional cellular chain complex functor induces a functor o n bicategor ies, and for A ⊂ B clos e d smo oth manifolds, Theore m 5.4 shows that g B | A is dualizable. Prop ositio n 2.5 then implies that C ∗ ( g B | A ) is dualizable. There is a second approach using Lemma 3.7. If x ∈ A , C ∗ ( g B | A )( x ) = C ∗ ( ˜ A x ) as a mo dule ov er π 1 ( A, x ). This is a ﬁnitely g enerated free mo dule and so is dualizable with dual Hom Z π 1 ( A,x ) ( C ∗ ( ˜ A x ) , Z π 1 ( A, x )) . If x ∈ B \ A , C ∗ ( g B | A )( x ) = C ∗ ( ˜ B x / ¯ A x ) as a mo dule over π 1 ( B , x ). This is also a ﬁnitely generated fr ee mo dule a nd so is dualizable with dual Hom Z π 1 ( B ,x ) ( C ∗ ( ˜ B x / ¯ A x ) , Z π 1 ( B , x )) . Since C ∗ ( g B | A ) is supp orted on iso morphisms, Lemma 3.7 implies C ∗ ( g B | A ) is dual- izable.  A map f : ( B , A ) → ( B , A ) induces a map f ∗ : C ∗ ( g B | A ) → C ∗ ( g B | A ) ⊙ Z Π f 1 ( B , A ) . Since C ∗ ( g B | A ) is dualiz able, the tr a ce of f ∗ is de ﬁned. Deﬁnition 6.2. The r elative glob al R eidemeister tr ac e of f : ( B , A ) → ( B , A ), gl R B | A ( f ), is the trace of f ∗ : C ∗ ( g B | A ) → C ∗ ( g B | A ) ⊙ Z Π f 1 ( B , A ) . The relative g lobal Reidemeister trace of f is a map Z → Z h h Π f 1 ( B , A ) i i . Lemma 6.3. If A is a pr op er subset of B then h h Π f 1 ( B , A ) i i ∼ = h h Π f 1 ( B ) i i ∐ h h Π f 1 ( A ) i i . The image of gl R B | A ( f ) u nder this isomorphism is X x ∈ h h Π f 0 ( A ) i i gl R ( f | A ( x ) )[ x ] + X y ∈ h h Π f 0 ( B ) i i gl R ( f | B ( y ) / ( B ( y ) ∩ A ) )[ y ] . Here gl R ( f | A ( x ) ) deno tes the usual global Reidemeister tr ace of f | A ( x ) as deﬁned by Huss eini in [13 ]. The in v a riant gl R ( f | B ( y ) ∪ C f | B ( y ) ∩ A ) is the tr ace of f ∗ : C ∗ ( B ( y ) / ( B ( y ) ∩ A )) → C ∗ ( B ( y ) / ( B ( y ) ∩ A )) ⊗ π f 1 ( B , y ) as a mo dule over π 1 ( B , y ). RELA TIVE FIXE D POINT THEOR Y 23 Pr o of. T o simplify notation, co ns ider the ca se where A and B are connected. The pro of is simila r if A and B are not connected. The shadow is deﬁned to b e the coeq ualizer o f the maps ∐ x,y Π 1 ( B , A )( x, y ) × Π 1 ( B , A )( f ( y ) , x ) / / / / ∐ x Π 1 ( B , A )( f ( x ) , x ) . Instead of indexing these copro ducts ov er all ob jects in Π 1 ( B , A ) we can index ov er repr esentativ es of each isomor phism class of ob jects in Π 1 ( B , A ). This gives four terms in the ﬁrst copro duct. The tw o cro ss terms ar e both empty and so this co equalizer splits into the co equa lizer tha t deﬁnes h h Π f 1 ( B ) i i a nd the co equalizer that deﬁnes h h Π f 1 ( A ) i i . F or the se c o nd statement, note that this is omorphism is compatible with the description of the dual pair . Then the trace is the pair of c la ssical traces.  This de s cription o f the relative globa l Reidemeister trace sho ws that the second co ordinate is the the relative Reidemeister trace o f [36]. This also shows that this in v a r iant is rela ted to, but not the same as, the genera lized Lefsc hetz num b er deﬁned in [24 ]. Prop ositio n 6.4. If A ⊂ B ar e close d smo oth manifold s and f : ( B , A ) → ( B , A ) is a r elative map then ge R B | A ( f ) = gl R B | A ( f ) . Pr o of. Since both ge R B | A ( f ) a nd gl R B | A ( f ) are deﬁned as traces and the rationa l cellular chain co mplex functor is stro ng symmetric monoidal this pr op osition follows from Prop osition 2.5.  7. A converse to the rel a tive Lefschetz fixed point theorem There are several pro ofs of the conv erse to the relative Lefschetz ﬁxed p oint theorem. Some, like [15, 28, 2 9, 3 4], are generaliz a tions of the simplicial arg umen ts used in the standa rd pr o of of the co n verse to the classical Lefschetz ﬁxed p oint theorem, see [3 ]. In this se c tion and the next sectio n w e describ e a pro of of the co nv er se to the relative Lefschetz ﬁxed p oint theorem that follows the outline of [19, 18]. This approach is not simplicial and it eas ily gener alizes. F o r exa mple, see [1 8] for the equiv ar iant gener alization a nd [2 6] for the ﬁb erwise genera lization. The approach of [19] is based on inv ariants that detect sectio ns of ﬁbr ations. In the next sectio n we prov e r elative genera lizations of the results in [1 9]. In this section w e apply thos e results to re lative ﬁxed point in v aria nt s. The main r esult of this sectio n is: Theorem B (The conv erse to the Relativ e Lefschetz ﬁxed p oint theorem) . Su pp ose A ⊂ B ar e close d smo oth manifolds of dimension at le ast 3 and the c o dimension of A in B is at le ast 2. The re lative glob al R eidemeister tr ac e of a map f : ( B , A ) → ( B , A ) is zer o if and only if f is r elatively homotopic to a map with no ﬁx e d p oints. The ﬁrst step in the pro of of Theo r em B is to descr ibe relative maps without ﬁxed p o ints in terms o f relative sectio ns. 24 KA TE P ONTO Lemma 7.1 . Le t A ⊂ B b e close d smo oth manifolds. R elative homotopies of a map f : ( B , A ) → ( B , A ) t o a re lative map with no ﬁ xe d p oints c orr esp ond to liftings that make the diagr am b elow c ommut e up to r elative homotopy ( B × B \ △ , A × A \ △ )   ( B , A ) Γ f / / 6 6 ( B × B , A × A ) . The function Γ f is the gr aph of f a nd Γ f ( m ) = ( m, f ( m )). Pr o of. If f is relatively homotopic to a ﬁxed p oint free map g via a relative homo - topy H , then Γ H is a relative ho mo topy fro m Γ f to Γ g . F or the conv erse, supp os e ther e is a rela tive map k : ( B , A ) → ( B × B \ △ , A × A \ △ ) and a rela tive homotopy K from k to Γ f . If A is a smo oth manifold the ﬁrst co ordinate pr o jection pro j 1 : A × A \ △ → A is a ﬁb er bundle and there is a lift J A in the diagram A k / / i 0   A × A \ △ pro j 1   A × I pro j 1 K / / J A 9 9 A. Since A ⊂ B is a coﬁbr ation and pro j 1 : B × B \ △ → B is a ﬁbration the diagra m B ∪ A × I k ∪ J A / / i 0   B × B \ △ pro j 1   B × I pro j 1 ◦ K / / J 4 4 B has a lift J extending the lift J A ab ov e, see [31, Theorem 4 ]. Note that pro j 1 ◦ J ( − , 1) = id. Let g = pr o j 2 J ( − , 1). This map has no ﬁxed p oints. The homotopies K and J deﬁne a relative homotopy fro m Γ f to Γ g .  Given a map f : V → Y , let r ( f ) : N ( f ) → Y denote a Hurewicz ﬁbration such that V / / f ! ! D D D D D D D D D N ( f ) r ( f )   Y commutes and V → N ( f ) is an equiv alence. Lemma 7.2. L et X ⊂ Y , p : M Y → Y b e a sp ac e over Y and M X ⊂ p − 1 ( X ) . RELA TIVE FIXE D POINT THEOR Y 25 Liftings up to r elative homotopy in the diagr am ( M Y , M X ) f   ( B , A ) 9 9 g / / ( Y , X ) c orr esp ond to r elative se ctions of the p air of ﬁbr ations ( g ∗ N ( f Y ) , g ∗ N ( f X )) → ( B , A ) . If p : E → B is a Hurewicz ﬁbra tion the unr e duc e d ﬁb erwise susp ens ion of p is the double mapping cylinder S B E : = B × { 0 } ∪ p E × [0 , 1] ∪ p B × { 1 } . The ma p p : E → B deﬁnes a ﬁbra tio n S B E → B . There are tw o sections of this ﬁbr ation, σ 1 , σ 2 : B → S B E , deﬁned by the inclusions of B × { 0 } a nd B × { 1 } . If S 0 B : = B ∐ B , these sections deﬁne an elemen t of [ S 0 B , S B E ] B . Let i B : B × B \ △ → B × B b e the inclusion. Then the pair of ﬁbr ations (Γ f ∗ ( N ( i B )) , Γ f ∗ ( N ( i A ))) → ( B , A ) determine an elemen t in [ S 0 B , S B Γ f ∗ ( N ( i B ))] B ⊕ [ S 0 A , S A Γ f ∗ ( N ( i A ))] A . This e lement will b e deno ted K W R B | A ( f ). Prop ositio n 7 .3. L et A ⊂ B b e close d smo oth manifolds of dimension at le ast 3 such that the c o dimension of A in B is at le ast 2. A c ont inuous map f : ( B , A ) → ( B , A ) is r elatively homotopic to a map with no ﬁx e d p oints if and only if K W R B | A ( f ) = 0 . The pro o f of this prop osition, except fo r one key step proved in the next section, follows the prelimina ry lemma b elow. Lemma 7. 4. [19, 6.1 , 6.2] L et M b e a manifold of dimension n and i : M × M \ △ → M × M b e the inclusion. Then Γ f ∗ ( N ( i )) → M is ( n − 1) -c onne cte d. Pr o of of Pr op osition 7.3 . Lemma 7.1 a nd Lemma 7 .2 con vert the question of ﬁnd- ing a lift of a relative map f : ( B , A ) → ( B , A ) to the question of ﬁnding a s ection of the ﬁbra tion (Γ f ∗ ( N ( i B )) , (Γ f | A ∗ ( N ( i A ))) → ( B , A ) . If the dimension o f A is n A and the dimension of B is n B then Lemma 7.4 implies that Γ f ∗ ( N ( i B )) → B is ( n B − 1)-connected a nd Γ f | A ∗ ( N ( i A )) → A is ( n A − 1)- connected. If n A and n B are at lea st 3 a nd n B − n A is at lea st 2, P rop osition 8.6 implies tha t (Γ f ∗ ( N ( i B )) , Γ f | A ∗ ( N ( i A ))) → ( B , A ) has a r e lative sectio n if a nd o nly if K W R B | A ( f ) = 0.  26 KA TE P ONTO The h yp otheses in this prop osition a re not the standar d hypo theses used in the conv erse to the r elative Lefschetz ﬁxed po int theorem. The standard co ndition is that π 1 ( B \ A ) → π 1 ( B ) is surjective. The co dimension condition implies this condition. W e use a co dimen- sion condition since it is compatible with the approach o f the pro of. I do n’t know if the s urjectivity condition can be used in this approach. T o complete the pro o f of Theo r em B we need to c o mpare K W R B | A ( f ) and the relative geometric Reidemeister trace. Prop ositio n 7. 5. L et A ⊂ B b e close d smo oth manifolds and f : ( B , A ) → ( B , A ) b e a r elative map. Then K W R B | A ( f ) = 0 if and only if l R B | A ( f ) = 0 . W e recall a lemma from [1 8]. Lemma 7.6. [18, 7.1 ][26, 8.3.1 ] L et M b e a close d smo oth manifold with normal bund le ν M . Then ther e is a we ak e quivalenc e S ν M ⊙ Γ f ∗ S M × M N ( i M ) → S n ∧ Λ f M . Pr o of of Pr op osition 7.5 . Suppo se X and Y are ex-s paces over B , the pro jection maps ar e ﬁbrations and the se ctions are ﬁberwise coﬁbration. Let { X, Y } B denote the ﬁberwis e s table homotopy c lasses of maps from X to Y . If A and B are b oth closed smo o th manifolds of dimensio n a t least three, then the dimensio n assumption, Lemma 7.4, a nd the ﬁber wise F reudenthal susp ension theorem in [14, 4.2] imply that the maps [ S 0 A , S A Γ f | A ∗ ( N ( i A ))] A → { S 0 A , S A Γ f ∗ ( N ( i A )) } A [ S 0 B , S B Γ f ∗ ( N ( i B ))] B → { S 0 B , S B Γ f ∗ ( N ( i B )) } B are iso morphisms. Costenoble- W aner dualit y [23, 18.5.5, 1 8.6.3] a nd Lemma 7.6 imply there a re isomorphisms { S 0 A , S A Γ f | A ∗ ( N ( i A )) } A ∼ = { S n , S ν A ⊙ S A Γ f ∗ ( N ( i A )) } ∼ = { S n , S n ∧ Λ f | A A + } . and { S 0 B , S B Γ f ∗ ( N ( i B )) } B ∼ = { S n , S ν B ⊙ S B Γ f ∗ ( N ( i B )) } ∼ = { S n , S n ∧ Λ f B + } . Let U A be a neig hborho o d o f the ﬁxed p oints o f f | A such that there is a map ι A : U A → Λ f | A A that takes ﬁxed p oints to the constant path at tha t point. In [26, 6.3.2] it is shown that the imag e of K W R B | A ( f ) in π s 0 (Λ f | A A + ) is ι A ( τ ( f | U A )). Let U B be a neighbor ho o d o f the ﬁxed p oints of f in B \ A such that there is a map ι B : U B → Λ f B which ta kes ﬁxed points to cons tant pa ths. RELA TIVE FIXE D POINT THEOR Y 27 The image of K W R B | A ( f ) in π s 0 (Λ f B + ) is the co mpo site of the transfer of f with resp ect to the diago nal map B + → B + ∧ ( U B ∐ U A ) /∂ ( U B ∐ U A ) with the ma p ι : = ι A ∐ ι B : U A ∐ U B → Λ f B . Since the tra nsfer is additive, [7], the image of K W R B | A ( f ) in π s 0 (Λ f B + ) is ι ( τ U B ∐ U A ( f )) = ι ( τ U B ( f ) + τ U A ( f | A )) = ι ( τ U B ( f )) + ι ( τ U A ( f | A )) . Then K W R B | A ( f ) is zero if and only if ι A ( τ ( f | U A )) and ι ( τ U B ( f )) + ι ( τ U A ( f | A )) are bo th zer o . Using Lemma 5.7 these elemen ts are zer o if and only if l R B | A ( f ) is zero.  Pr o of of The or em B. Pr op osition 7 .3 implies that f is relatively homo to pic to a ﬁxed point free map if and only if K W R B | A ( f ) = 0. Pr op osition 7.5 implies K W R B | A ( f ) = 0 if a nd only if ge R B | A ( f ) = 0. Prop osition 6.4 implies gl R B | A ( f ) = ge R B | A ( f ).  R emark 7.7 . Pro p o sition 7.3 and the pr o of of P rop osition 7.5 show if dim( A ) ≥ 3 and dim( B ) ≥ dim( A ) + 2 K W R B | A ( f ) is zero if and only if the t wo nonrelativ e inv aria nt s for A and B are zero. Using these tw o inv ariants to deﬁne a relative inv ar iant would b e a nalogo us to deﬁning the rela tive inv aria nts in the previous sections as the pair of class ic al in- v aria nt s for the spaces A a nd B . This alterna te deﬁnition would satisfy the require- men ts of the intro duction fo r a ﬁx e d po int in v a riant. Howev e r , there are several reasons why the cor resp onding deﬁnition in the equiv ariant case is not accepta ble. The deﬁnitions in the pr evious sec tions were ch osen bec ause they are consistent with the choices in [25]. 8. Rela tive sections In this s e ction we g eneralize the re s ult from [19] o n sections of ﬁbrations to relative ﬁbrations. If the dimension of B is 2 n and the ﬁbration p : E → B is n + 1 - connected, it is shown in [19] that the t wo sections σ 1 , σ 2 : B → S B E are ho motopic over B if and only there is a section of p . W e can gener alize this result to rela tive sections. If A ⊂ B let E A be a subspace o f E B such that the image of p r estricted to E A is c ontained in A . Let S A,B E A be B × { 0 } ∪ E A × I ∪ A × { 1 } . Let [( S 0 B , A ∐ B ) , ( S B E B , S A,B E A )] B be the relative ﬁb erwise homotopy classes of maps from ( S 0 B , A ∐ B ) to ( S B E B , S A,B E A ). Deﬁnition 8. 1. Let A ⊂ B , p : E B → B be a ﬁbration, and E A ⊂ p − 1 ( A ) such that E A → A is a ﬁbra tion. The r elative homotopy Euler class χ ∈ [( S 0 B , A ∐ B ) , ( S B E B , S A,B E A )] B is σ 1 ∐ σ 2 : S 0 B → S B E B . 28 KA TE P ONTO Prop ositio n 8.2. If ( E B , E A ) → ( B , A ) a dmits a r elative se ction ˜ ς t hen χ is trivial. Conversely, assume p : E A → A is ( m + 1) -c onne ct e d, A is a 2 m -dimensional CW-c omplex, p : E B → B is ( n +1) -c onne cte d and ( B , A ) is a r elative 2 n -dimensional CW-c omplex. If χ is trivial then p has a r elative se ction. Before w e prov e this prop os ition we r ecall a pr eliminary le mma . Lemma 8 .3. [1 9, 3.1] L et p : E → B b e a ( j + 1) -c onne cte d ﬁbr ation and P b e the homotopy pul lb ack P / /   B   B / / S B E . A ﬁb erwise homotopy fr om σ 1 to σ 2 deﬁnes a 2 j -e quivalenc e q : E → P . Pr o of of Pr op osition 8.2 . If there is a relative section ˜ ς then the homotopy H : ( S 0 B , A ∐ B ) × I → ( S B E B , S A,B E A ) deﬁned b y H ( b, t ) = ( ˜ ς ( b ) , t ) s hows χ is trivial. If χ is trivia l ther e is a relative ﬁb erwise homotopy K : ( S 0 B , A ∐ B ) × I → ( S B E B , S A,B E A ) from σ 2 to σ 1 . The restriction of K to S 0 A deﬁnes a homotopy b e t ween σ 1 | A : A → S A E A and σ 2 | A . Lemma 8.3, Whitehead’s theor em, and the homotopy K | S 0 A imply q A ∗ : [ A, E A ] → [ A, P A ] is a bijection. The space P A is as in Lemma 8.3. The r estriction K | S 0 A induces a map h A : A → P A such that p h A = id. Since q A ∗ is a bijection there is a map k A : A → E A and a homotopy J A from q A k A to h A . Then pk A = p ( q A k A ) ≃ ph A = id A via the ho motopy p ( J A ). The diagra m A k A / / i 0   E A p   A × I p ( J A ) / / L A ; ; A has a lift L A , and p ( L A ( a, 1)) = a . Then L A ( − , 1) is a section of p − 1 ( A ) → A that is c ontained in E A . The ho motopy K deﬁnes a map h B : B → P B extending the map h A . The space P B is a s in Le mma 8.3. The homotopy extensio n and lifting prop erty implies the dotted maps in the following diagr am can b e ﬁlled in A i 1 / /   A × I   J A { { x x x x x x x x A k A ~ ~ } } } } } } } }   i 0 o o P B E B q o o B i 1 / / h B > > } } } } } } } } B × I J B c c B i 0 o o k B ` ` RELA TIVE FIXE D POINT THEOR Y 29 deﬁning maps k B and J B extending k A and J A . Since the pair ( B , A ) has the relative homotopy lifting prop erty there is a lift L B in the diagram B ∪ ( A × I ) k B ∪ L A / /   E B p   B × I p ( J B ) / / L B 7 7 B . Ev alua ting at 1, p ( L B ( b, 1)) = pJ B ( b, 1) = ph B ( b ) = b . Since L B ( a, 1) ∈ E A for a ∈ A , L B ( − , 1) is the required section.  Lemma 7.1, Lemma 7.2, and Prop osition 8.2 imply χ is a complete o bstruction to determining if a relative ﬁbr ation has a s ection. In the examples we ar e interested in, it is ea sier to work with inv ar iants deﬁned fro m χ than with χ itself. Under some additional hypotheses, these asso c iated inv aria nt s are zero if and only if χ is zero. If A ⊂ B , deﬁne C B ( B , A ) : = B × { 0 } ∪ A × [0 , 1] ∪ B × { 1 } . This is a n ex-space o ver B with section given by the inclusion of B into C B ( B , A ) as B × { 0 } . In the dia gram b elow the vertical maps ar e induced b y coﬁb er sequences, [4, II.2 .4] and so the co lumns are exact. The hor izontal maps are fo rgetful maps. The diagram comm utes. χ B ,A ∈ [( C B ( B , A ) , S B A ) , ( S B E B , S A E A )] B φ   ψ / / [ C B ( B , A ) , S B E B ] B ρ   ¯ χ B ,A ∋ χ ∈ [( S 0 B , S 0 A ) , ( S B E B , S A E A )] B / /   [ S 0 B , S B E B ] B   χ B ∋ χ A ∈ [ A ∐ B , S A,B E A ] B / / [ A ∐ B , S B E B ] B ¯ χ A ∋ The element s χ A , χ B , and ¯ χ A are the images of χ . The element χ B ,A is deﬁned if χ A = 0. Then χ B ,A is the preimag e o f χ . The element ¯ χ B ,A is deﬁned if ¯ χ A = 0. Then ¯ χ B ,A is the pre image of χ B . Lemma 8.4. If ¯ χ B ,A = 0 then χ B ,A = 0 . Pr o of. Suppo se ¯ χ B ,A = 0. Then there is a ﬁb er wise homotopy L : C B ( B , A ) × I → S B E B such that L ( b, 1 , 0) = σ 2 ( b ) L ( b, 1 , 1) = σ 1 ( b ) L ( b, 0 , t ) = σ 1 ( b ) L ( a, s, 0 ) = χ B ,A ( a, s ) ∈ S A E A L ( a, s, 1 ) = σ 1 ( a ) for all a ∈ A , b ∈ B , and s, t ∈ I . 30 KA TE P ONTO Let J : = ( { 0 } × I ) ∪ ( I × { 1 } ) ∪ ( { 1 } × I ). Deﬁne a map ¯ L : B × J → S B E B by ¯ L ( b, 0 , t ) = σ 1 ( b ) ¯ L ( b, s, 1) = σ 1 ( b ) ¯ L ( b, 1 , t ) = L ( b, 1 , t ) . The dia gram ( B × J ) ∪ i ( A × I × I ) ¯ L ∪ L | A × I × I / /   S B E B   B × I × I K 5 5 pro j / / B commutes and there is a lift K since S B E B → B is a ﬁbr ation. Then K 0 : = K ( − , − , 0) : B × I → S B E B satisﬁes K 0 ( b, 0) = K ( b, 0 , 0) = L ( b, 0 , 0 ) = σ 1 ( b ) K 0 ( b, 1) = K ( b, 1 , 0) = L ( b, 1 , 0 ) = σ 2 ( b ) K 0 ( a, s ) = K ( a, s, 0) = L ( a, s, 0) ∈ S A E A Deﬁne a map ˜ K : C B ( B , A ) × I → S B E B by ˜ K ( b, 1 , t ) = K 0 ( b, 1 − t ) ˜ K ( b, 0 , t ) = σ 1 ( b ) ˜ K ( a, s, t ) = K 0 ( a, s (1 − t )) ˜ K shows χ B ,A is tr ivial in [( C B ( B , A ) , S B A ) , ( S B E B , S A E A )] B .  Lemma 8.5. If the map E B → B is a (dim ( A ) + 1) - e quivalenc e then ρ is inje ctive. Pr o of. In this pr o of le t i denote the inclusion of A in B . Let Σ B ( A ∐ B ) : = (( A × I ) ∐ B ) / ∼ where ( a, 0) ∼ i ( a ) ∼ ( a, 1). Then ρ is part of a long exact sequence [Σ B ( A ∐ B ) , S B E B ] B / / [ C B ( B , A ) , S B E B ] B ρ / / [ S 0 B , S B E B ] B / / [ A ∐ B , S B E B ] B . T o show that ρ is injective it is enoug h to show [Σ B ( A ∐ B ) , S B E B ] B is tr ivial. RELA TIVE FIXE D POINT THEOR Y 31 Let α b e an element of [Σ B ( A ∐ B ) , S B E B ] B . T he n α deﬁnes a map S 1 × A → S B E B also de no ted α . This map satisﬁes pα ( t, a ) = i ( a ). Consider the diagr a m S 1 × A i 0 / /   S 1 × A × I   i ◦ pro j y y t t t t t t t t t t S 1 × A   i 1 o o α y y t t t t t t t t t B S B E B o o D 2 × A i ◦ pro j ; ; x x x x x x x x x i 0 / / D 2 × A × I H e e D 2 × A i 1 o o β e e Since S B E B → B is a (dim( A ) + 2)-equiv alence, the homotopy extensio n and lifting prop erty implies there are ma ps β and H that make the diagram comm ute. The diagram ( D 2 × A ) ∐ i S 1 × A × I β ∐ ( α ◦ pro j) / /   S B E B p   D 2 × A × I H / / K 4 4 B commutes. Since S B E B → B is a ﬁbration there is a lift K that makes the diagr am commute. Then K 0 : = K ( − , − , 0) : D 2 × A → S B E B satisﬁes pK 0 ( v , a ) = H ( v , a, 0 ) = i ( a ) and K 0 ( w, a ) = α ( w , a ) if w ∈ S 1 . Then K 0 ∐ id : (( D 2 × A ) ∐ B ) / ∼ → S B E B deﬁnes a map that shows α is trivial.  The following prop osition is a consequence of P rop osition 8.2, Le mma 8 .4, and Lemma 8.5. Prop ositio n 8.6. If p : E A → A is ( m + 1) -c onne ct e d, A is a 2 m -dimensional CW- c omplex, p : E B → B is (2 m + 1) -c onne cte d and ( B , A ) is a r elative 4 m -dimensional CW-c omplex ( E B , E A ) → ( B , A ) admits a r elative se ction if and only if χ A and χ B ar e b oth zer o. 9. Other descriptions of ⊙ in special cases These ar e the pr o ofs omitted from Section 3. L e t A b e an EI-c ategory e nr iched in the categ ory o f abe lia n g roups. Lemma 9.1 (Lemma 3.6) . If X : A → Ch R and Y : A op → Ch R ar e supp orte d on isomorphisms X ⊙ Y ∼ = M c ∈ B ( A ) X ( c ) ⊗ A ( c ,c ) Y ( c ) . 32 KA TE P ONTO Pr o of. W e will show that ⊕ X ( c ) ⊗ A ( c,c ) Y ( c ) satisﬁes the universal prop erty that deﬁnes X ⊙ Y . By deﬁnition of B ( A ), for any ob ject a in A ther e is e xactly one o b ject c ∈ B ( A ) such that there is a n is o morphism f : a → c in A . Deﬁne a ma p θ a : X ( a ) ⊗ Z Y ( a ) → X ( c ) ⊗ A ( c,c ) Y ( c ) as the c omp o site of X ( f ) ⊗ Y ( f − 1 ) : X ( a ) ⊗ Z Y ( a ) → X ( c ) ⊗ Z Y ( c ) with the q uotient ma p X ( c ) ⊗ Z Y ( c ) → X ( c ) ⊗ A ( c,c ) Y ( c ) . If g is another is omorphism in A from a to c , then ( X ( f )( A ) , Y ( f − 1 )( B )) is ident iﬁed with ( X ( g )( A ) , Y ( g − 1 )( B )) and the map θ a is well deﬁned. Let θ : M a ∈ ob( A ) X ( a ) ⊗ Z Y ( a ) → M c ∈ B ( A ) X ( c ) ⊗ A ( c,c ) Y ( c ) be the sum of the maps θ a . If ( A, f , B ) ∈ X ( a ) ⊗ Z A ( a, b ) ⊗ Z Y ( b ) the images of this element in ⊕ a ∈ ob( A ) X ( a ) ⊗ Z Y ( a ) are ( A, Y ( f )( B )) a nd ( X ( f )( A ) , B ). The images of these elemen ts are ident iﬁed under θ . Let φ : M a ∈ ob A X ( a ) ⊗ Z Y ( a ) → M be a ma p that co equalizes the tw o maps fr om ⊕ a,b ∈ ob A X ( a ) ⊗ Z A ( a, b ) ⊗ Z Y ( b ) to ⊕ a ∈ ob A X ( a ) ⊗ Z Y ( a ) . Deﬁne a map ψ : M c ∈ B ( A ) X ( c ) ⊗ A ( c ,c ) Y ( c ) → M by cho osing lifts o f elemen ts in X ( c ) ⊗ A ( c ,c ) Y ( c ) to elements of X ( c ) ⊗ Z Y ( c ) . Since φ coe q ualizes, the choices do not matter and ψ is unique.  Lemma 9.2 (Lemma 3.7) . L et X and Y satisfy the c onditions of L emma 3.6. If X ( c ) is dualizable as a A ( c, c ) -mo dule with dual Y ( c ) for e ach c ∈ B ( A ) then X is dualizable with dual Y . Pr o of. If X ( c ) is dualiz able as an A ( c, c )-mo dule with dual Y ( c ) then there is a map o f c hain complexes of a belia n g roups η c : Z → X ( c ) ⊙ Y ( c ) and a map of chain complexes of A ( c, c )-bimodules ǫ c : Y ( c ) ⊙ X ( c ) → A ( c, c ) for each c ∈ B ( A ). RELA TIVE FIXE D POINT THEOR Y 33 Let η : Z → X ⊙ Y be the comp os ite Z △ → M B ( A ) Z ⊕ η c → M B ( A ) X ( c ) ⊗ A ( c ,c ) Y ( c ) ∼ = X ⊙ Y where △ : Z → ⊕ B ( A ) Z is the ma p that takes 1 to (1 , 1 , . . . , 1). Let a and b b e isomorphic ob jects of A . Let c b e an ob ject of B ( A ) that is isomorphic to a a nd let h b e a n isomorphism in A from a to c and g be an isomorphism from b to c . Then ǫ a,b is the compo site Y ( c ) ⊗ Z X ( c ) ǫ c / / A ( c, c ) A ( g ,h − 1 )   Y ( b ) ⊗ Z X ( a ) Y ( g − 1 ) ⊗ X ( h ) O O A ( b, a ) . If a and b ar e not iso morphic in A ǫ a,b is zero. Since c is unique and the maps ǫ c are ma ps o f A ( c, c )- bimo dules , ǫ is a natural tra nsformation. This also implies that ǫ is indep endent o f the choice o f g and h . Let η c (1) = X i e c,i ⊗ f c,i for each c ∈ B ( A ). If x ∈ X ( a ) the v a lue o f the comp osite X ( a ) ∼ = Z ⊗ X ( a ) η ⊗ 1 → X ⊙ Y ⊗ X ( a ) 1 ⊙ ǫ → X ⊙ A ( − , a ) ∼ = X ( a ) applied to x is X c ∈ B ( A ) X i X ( ǫ ( f c,i , x ))( e c,i ) . The only nonzero terms in this s um a re tho s e where there is an isomorphism h fr o m x to c . By deﬁnition, ǫ ( f c,i , x ) = h − 1 ǫ c ( f c,i , X ( h )( x )) and X i X ( ǫ ( f c,i , x ))( e c,i ) = X ( h − 1 ) X i X ( ǫ c ( f c,i , X ( h )( x )))( e c,i ) = X ( h − 1 ) X ( h )( x ) = x. The o ther diagram is simila r.  References [1] C. Bo wszyc. Fixed p oint theorems f or the pairs of spaces. Bul l. Ac ad. Polo n. Sci . S´ er. Sci. Math. Ast r onom. Phys. , 16:845–850, 1968. [2] Theo dor Br¨ oc ker and Klaus J¨ anich. Intr o duction t o diﬀer ential top olo gy . Cam bridge U ni - v ersity Press, Cam bridge, 1982. T ranslated f rom the German by C. B. Thomas and M. J. Thomas. [3] R ob ert F. Brown. The Lefschetz ﬁxed p oint the or em . 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