aa r X i v : . [ m a t h . A T ] A ug ONE-PARAMETER LEFSCHETZ CLASS OFHOMOTOPIES ON TORUS
WESLEM L. SILVA
Abstract.
The main result this paper states that if F : T × I → T is a homotopy on torus then the one-parameter Lefschetz class L ( F ) of F is given by L ( F ) = ± N ( F ) α , where N ( F ) is the one-parameter Nielsen number of F and α is one of the two generatorsin H ( π ( T ) , Z ). Introduction
Let F : T × I → T be a homotopy on torus and G = π ( T, x ).R.Geoghegan and A. Nicas in [5] developed an one-parameter theoryand defined the one-parameter trace R ( F ) of F . The element R ( F )is a 1-chain in HH ( Z G, ( Z G ) φ )), where the structure of the bimodule( Z G ) φ ) is given in section 2. This 1-chain gives information aboutthe fixed points of F , that is, using R ( F ) is possible to define the one-parameter Nielsen number N ( F ) of F and the one-parameter Lefschetzclass L ( F ) of F . N(F) is the number of non-zero C-components in R ( F ) and L ( F ) is the image of R ( F ) in H ( G ) by homomorfism ¯ j C : H ( Z ( g C )) → H ( G ), induced by inclusion j C : Z ( g C ) → G , where Z ( g C ) is the semicentralizer of an element g C which represents thesemiconjugacy class C . The precise definition is given in [5].The main purpose this paper is show that for which homotopy ontorus then L ( F ) = ± N ( F ) α , where α is on of the two generators in H ( G ).In [1] R.B.S.Brooks et al. showed that if f : X → X is any map on ak-dimensional torus X then N ( f ) = | L ( f ) | , where N ( f ) is the Nielsen Date : September 11, 2018.2010
Mathematics Subject Classification.
Primary 55M20; Secondary 57Q40,57M05.
Key words and phrases.
One-parameter fixed point theory, one-parameter Lef-schetz class, one-parameter Nielsen number . number and L ( f ) the Lefschetz number of f . In some sense our resultis a version of this result for one-parameter case when k = 2.This paper is organized into five sections, besides this one. In Section2 contain a review of one-parameter fixed point theory. In section 3 wepresented some results of semiconjugacy classes on torus. In Section 4have the proof of the main result which is the Theorem 4.1.2. One-parameter Fixed Point Theory
Let X be a finite connected CW complex and F : X × I → X acellular homotopy. We consider I = [0 ,
1] with the usual CW structureand orientation of cells, and X × I with the product CW structure,where its cells are given the product orientation. Pick a basepoint( v, ∈ X × I , and a basepath τ in X from v to F ( v, π ( X × I, ( v, ≡ G with π ( X, v ) via the isomorphism induced byprojection p : X × I → X . We write φ : G → G for the homomorphism; π ( X × I, ( v, F → π ( X, F ( v, c τ → π ( X, v )We choose a lift ˜ E in the universal cover, ˜ X , of X for each cell E and we orient ˜ E compatibly with E . Let ˜ τ be the lift of the basepath τ which starts at in the basepoint ˜ v ∈ ˜ X and ˜ F : ˜ X × I → ˜ X theunique lift of F satisfying ˜ F (˜ v,
0) = ˜ τ (1). We can regard C ∗ ( ˜ X ) as aright Z G chain complex as follows: if ω is a loop at v which lifts to apath ˜ ω starting at ˜ v then ˜ E [ ω ] − = h [ w ] ( ˜ E ), where h [ ω ] is the coveringtransformation sending ˜ v to ˜ ω (1).The homotopy ˜ F induces a chain homotopy ˜ D k : C k ( ˜ X ) → C k +1 ( ˜ X )given by ˜ D k ( ˜ E ) = ( − k +1 F k ( ˜ E × I ) ∈ C k +1 ( ˜ X ), for each cell ˜ E ∈ ˜ X .This chain homotopy satisfies; ˜ D ( ˜ Eg ) = ˜ D ( ˜ E ) φ ( g ) and the boundaryoperator ˜ ∂ k : C k ( ˜ X ) → C k − ( ˜ X ) satisfies; ˜ ∂ ( ˜ Eg ) = ˜ ∂ ( ˜ E ) g .Define endomorphism of, ⊕ k C k ( ˜ X ), by ˜ D ∗ = ⊕ k ( − k +1 ˜ D k , ˜ ∂ ∗ = ⊕ k ˜ ∂ k , ˜ F ∗ = ⊕ k ( − k ˜ F k and ˜ F ∗ = ⊕ k ( − k ˜ F k . We consider trace( ˜ ∂ ∗ ⊗ ˜ D ∗ ) ∈ HH ( Z G, ( Z G ) φ ). This is a Hochschild 1-chain whose boundaryis: trace( ˜ D ∗ φ ( ˜ ∂ ∗ ) − ˜ ∂ ∗ ˜ D ∗ ) . We denote by G φ ( ∂ ( F )) the subset of G φ consisting of semiconjugacy classes associated to fixed points of F or F . NE-PARAMETER LEFSCHETZ CLASS OF HOMOTOPIES ON TORUS 3
Definition 2.1.
The one-parameter trace of homotopy F is: R ( F ) ≡ T ( ˜ ∂ ∗ ⊗ ˜ D ∗ ; G φ ( ∂ ( F ))) ∈ M C ∈ G φ − G φ ( ∂ ( F )) HH ( Z G, ( Z G ) φ ) C ∼ = M C ∈ G φ − G φ ( ∂ ( F )) H ( Z ( g C )) . Definition 2.2.
The C − component of R ( F ) is denoted by i ( F, C ) ∈ HH ( Z G, ( Z G ) φ ) C . We call it the fixed point index of F correspondingto semiconjugacy class C ∈ G φ . A fixed point index i ( F, C ) of F iszero if the all cycle in i ( F, C ) is homologous to zero. Definition 2.3.
Given a cellular homotopy F : X × I → X the one-parameter Nielsen number, N ( F ) , of F is the number of nonzero fixedpoint indices. Definition 2.4.
The one-parameter Lefschetz class, L ( F ) , of F is de-fined by; L ( F ) = X C ∈ G φ − G φ ( ∂F ) j C ( i ( F, C )) where j C : H ( Z ( g C )) → H ( G ) is induced by the inclusion Z ( g C ) ⊂ G . From [5] we have the following theorems.
Theorem 2.1 (Invariance) . Let
F, G : X × I → X be cellular; if F ishomotopic to G relative to X × { , } then R ( F ) = R ( G ) . Theorem 2.2 (one-parameter Lefschetz fixed point theorem) . If L ( F ) =0 then every map homotopic to F relative to X ×{ , } has a fixed pointnot in the same fixed point class as any fixed point in X × { , } . Inparticular, if F and F are fixed point free, every map homotopic to F relative to X × { , } has a fixed point. Theorem 2.3 (one-parameter Nielsen fixed point theorem) . Everymap homotopic to F relative to X × { , } has at least N ( F ) fixedpoint classes other than the fixed point classes which meet X × { , } .In particular, if F and F are fixed point free maps, then every maphomotopic to F relative to X × { , } has at least N ( F ) path compo-nents. For a complete description of the one-parameter fixed point theorysee [5].
WESLEM L. SILVA Semiconjugacy classes on torus
In this subsection we describe some results about the semiconjugacyclasses in the torus related to a homotopy F : T × I → T . We willconsider the homomorphism φ = c τ ◦ F given above.We take w = [(0 , ∈ T and G = π ( T, w ) = { u, v | uvu − v − = 1 } ,where u ≡ a and v ≡ b . Thus, given the homomorphism φ : G → G we have φ ( u ) = u b v b and φ ( v ) = u b v b . Therefore, φ ( u m v n ) = u mb + nb v mb + nb , for all m, n ∈ Z . We denote this homomorphism bythe matrix: [ φ ] = b b b b ! Proposition 3.1.
Two elements g = u m v n and g = u m v n in G belong to the same semiconjugacy class, if and only if there are integers m, n satisfying the following equations: ( m ( b −
1) + nb = m − m mb + n ( b −
1) = n − n Proof.
If there is g = u m v n ∈ G satisfying g = gg φ ( g ) − then weobtain the equation of the proposition. The other direction is analo-gous. (cid:3) (cid:3) We take the isomorphism Θ : G → Z × Z such that Θ( u m v n ) =( m, n ). By Proposition 3.1 two elements g = u m v n and g = u m v n in G belong to the same semiconjugacy class, if and only if there is z ∈ Z × Z satisfying: ([ φ ] − I ) z = Θ( g g − ), where I is the identitymatrix. If determinant of the matrix ([ φ ] − I ) is zero then will have aninfinite amount of elements in a semiconjugacy class. Corollary 3.2.
For each g ∈ G the semicentralizer Z ( g ) is isomorphicto the kernel of [ φ ] − I . Lemma 3.1.
The 1-chain, u k v l ⊗ u m v n , is a cycle if and only if theelement ( k, l ) ∈ Z × Z belongs to the kernel of [ φ ] − I .Proof. If u k v l ⊗ u m v n is a cycle, then 0 = d ( u k v l ⊗ u m v n ) = u m v n φ ( u k v l ) − u k v l u m v n = u m v n u kb + lb v kb + lb − u k v l u m v n = u m + kb + lb v kb + lb + n − u k + m v l + n . This implies k ( b −
1) + lb = 0 and kb + l ( b −
1) = 0. Theother direction is analogous. (cid:3) (cid:3)
NE-PARAMETER LEFSCHETZ CLASS OF HOMOTOPIES ON TORUS 5
Corollary 3.3.
If the matrix of the homomorphism φ is given by [ φ ] = b b ! with b = 0 or b = 1 , then the 1-chain, u k v l ⊗ u m v n , is a cycle if andonly if l = 0 . By definition given a 2-chain u s v t ⊗ u k v l ⊗ u m v n ∈ C ( Z G, ( Z G ) φ ) C then d ( u s v t ⊗ u k v l ⊗ u m v n ) = u k v l ⊗ u m + sb + tb v n + sb + tb − u k + s v l + t ⊗ u m v n + u s v t ⊗ u k + m v l + n . Proposition 3.4.
The 1-chain, u k ⊗ u m v n ∈ C ( Z G, ( Z G ) φ ) C , is ho-mologous to the 1-chain, ku ⊗ u m + k − v n , for all k, m, n ∈ Z .Proof. Note that for k = 0 and 1 the proposition is true. We supposethat for some s > ∈ Z , the 1-chain u s ⊗ u m v n is homologous tothe 1-chain su ⊗ u m + s − v n , we will write u s ⊗ u m v n ∼ su ⊗ u m + s − v n .Considering to the 2-chain u s ⊗ u ⊗ u m v n ∈ C ( Z G, ( Z G ) φ ) we have d ( u s ⊗ u ⊗ u m v n ) = u ⊗ u m + s v n − u s +1 ⊗ u m v n + u s ⊗ u m v n ∼ u ⊗ u m + s v n − u s +1 ⊗ u m v n + su ⊗ u m + s − v n = ( s + 1) u ⊗ u m +( s +1) − v n − u s +1 ⊗ u m v n . Therefore ( s + 1) u ⊗ u m +( s +1) − v n ∼ u s +1 ⊗ u m v n . Using induction, weobtain the result. The case in which k < (cid:3) (cid:3)
The proof of following results can be found in [8].
Proposition 3.5.
In the case b = 1 and b = 0 each 1-cycle t X i =1 a i u k i v l i ⊗ u m i v n i ∈ C ( Z G, ( Z G ) φ ) is homologous to a 1-cycle the following form: ¯ t X i =1 ¯ a i u ⊗ u ¯ m i v ¯ n i . Proposition 3.6.
Each 1-cycle u ⊗ u m v n ∈ HH ( Z G, ( Z G ) φ ) C is nottrivial, that is, is not homologous to zero. WESLEM L. SILVA
Corollary 3.7.
Let t X i =1 u ⊗ u m i v n i ∈ HH ( Z G, ( Z G ) φ ) , m i , n i ∈ Z be a cycle. If the cycles u ⊗ u m i v n i and u ⊗ u m j v n j are in differentsemiconjugacy classes for i = j , i, j ∈ { , ..., t } , then t X i =1 u ⊗ u m i v n i isa nontrivial cycle. Each cycle u ⊗ u m i v n i projects to the class [ u ] thatis one of the two generators of H ( G ) . Homotopies on torus
Let F : T × I → T be a homotopy on torus T . Proposition 4.1.
Let F : T × I → T be a homotopy. Suppose that L ( F t ) = 0 for each t ∈ I . Then F is homotopic to a homotopy H with H transverse the projection P : T × I → T such that F ix ( H | T ×{ , } ) = ptyset .Proof. We can choose a homotopy F homotopic to F with F trans-verse the projection P . Therefore, F ix ( F ) is transverse, that is, F ix ( F ) ∩ ( T × { t } ) is finite. Since L ( F | T ) = L ( F | T ) = 0 thenfor each > ǫ > F to a homotopy F such that F ( x, t ) = F ( x, t ) for each ( x, t ) ∈ T × [ ǫ, − ǫ ] and F has no fixedpoints in T × { , } . In fact, take A : T × I × I → T defined by A (( x, y ) , t, s ) = F ( x, y, if ≤ t ≤ sǫF ( x, y, − sǫ ( t − sǫ )) if sǫ ≤ t ≤ − sǫF ( x, y, if − sǫ ≤ t ≤ L ( F | T ) = 0, there are two homotopies H , H : T × I → T suchthat H ( x, y,
1) = F ( x, y, H ( x, y,
0) = F ( x, y,
1) and H ( x, y, H ( x, y,
1) are fixed points free maps. Considere the homotopy B : T × I × I → T defined by; B (( x, y ) , t, s ) = H ( x, y, tǫ s ) if ≤ t ≤ ǫF ( x, y, − ǫ ( t − ǫ )) if ǫ ≤ t ≤ − ǫH ( x, y, ( t − ǫ ) ǫ s ) if − ǫ ≤ t ≤ J (( x, y ) , t, s ) = ( A (( x, y ) , t, s ) if ≤ s ≤ B (( x, y ) , t ) , s − if ≤ s ≤ NE-PARAMETER LEFSCHETZ CLASS OF HOMOTOPIES ON TORUS 7 we have a homotopy between F and a map H where H satisfying thehypothesis of the theorem. Note that we can choose > ǫ > F ix ( F ) ⊂ T × [ ǫ, − ǫ ] because F ix ( F ) is contained in int ( T × I ).Thus, F ix ( H ) is transverse. (cid:3) (cid:3) Figure 1.
Circles in
F ix ( F ) . Let
F ix ( F, ∂ ) be the subset of
F ix ( F ) consisting of those circles offixed points which are not in the same fixed point class as any fixedpoint of F or F . From [5] F ix ( F ) consists of oriented arcs and circles.From Proposition 4.1 if F : T × I → T is a homotopy and P : T × I → T the projection then we can choose F such that F ix ( F ) istransverse the projection P . Thus, F ix ( F, ∂ ) is a closed oriented 1-manifold in the interior of T × I × T . Let E F be space of all paths ω ( t )in T × I × T from the graph Γ F = { ( x, t, F ( x, t )) | ( x, t ) ∈ T × I } of F tothe graph Γ P = { ( x, t, x ) | ( x, t ) ∈ T × I } of P with the compact-opentopology, that is, maps ω : [0 , → T × I × T such that ω (0) ∈ Γ( F )and ω (1) ∈ Γ( P ).Let C , ..., C k be isolated circles in F ix ( F ) ∩ int ( T × I ), oriented bythe natural orientations, and V = S C j . Then V determines a familyof circles V ′ in E F via constant paths, i.e. each oriented isolated circleof fixed points C : S → T × I of F determines an oriented circle C ′ : S → E defined by con ( C ( z )) where con ( C ( z ) is the constantpath at C ( z ) = ( x, t ), that is, con ( C ( z ))( t ) = ( x, t , x ) for each t ∈ [0 , X i ( F, C j ) . [ C ′ j ] ∈ H ( E F ). Since C j istransverse then i ( F, C j ) = 1 for all j , see [2]. From [5] we have; Proposition 4.2.
Since π ( T ) = 0 then there is a isomorphism Ψ : H ( E F ) → HH ( Z G, ( Z G ) φ ) , where G = π ( T, x ) . WESLEM L. SILVA
Remarks 4.1.
From [2] , section IV , given F : T × I → T a homotopythen we can to deform F to a homotopy G such that in each fixedpoint class of G has an unique circle, and this circle is transverse theprojection. Now we are going to proof the main result.
Theorem 4.1 (Main Theorem) . If F : T × I → T is a homotopy thenthe one-parameter Lefschetz class L ( F ) of F satisfies L ( F ) = ± N ( F ) α where α is one of the two generators of H ( π ( T ) , Z ) .Proof. The proof this theorem will be done in two cases. Case I when det ([ φ ] − I ) = L ( F | T ) = 0 and case II when det ([ φ ] − I ) = L ( F | T ) = 0. Case I
Let us suppose that the homomorphism φ is induced by a homotopy F satisfies det ([ φ ] − I ) = 0. Using the notation above we can supposewhich φ is given by [ φ ] = b b ! , and [ φ ] = I ≡ (Identity), that is, b = 1 and b = 0, with b = 0 or b − = 0. This is done choosing a base { v, w } for T = R / Z , where v is a eigenvector of [ φ ] associated to 1.Note that if [ φ ] = I then R ( F ) = 0 because any F can be deformed toa fixed point free map. For example, take the homotopy F : T × I → T defined by; F (( x, y ) , t ) = ( x + c t + ǫ, y + c t )with ǫ any irrational number between 0 and 1. We will have [ F ] =[ φ ] = I , but F is a fixed point free map. Thus R ( F ) = 0, which implies L ( F ) = N ( F ) = 0. Therefore, henceforth we suppose [ φ ] = I. Since T is a polyhedron then T is a regular CW-complex. Thus,for any cellular decomposition of the torus the entries of matrices ofthe operators ˜ ∂ and ˜ ∂ will be composed by elements 0 , ± , ± u, ± v, because the incindence number of a 2-cell in a 1-cell is ± ±
1, see chapter II of [10].
NE-PARAMETER LEFSCHETZ CLASS OF HOMOTOPIES ON TORUS 9
Therefore chosen an orientation to each cell in a decomposition cel-lular to the torus then the one-parameter trace R ( F ) will be the formof the following matrix: R ( F ) = tr [ − ˜ ∂ ] ⊗ [ ˜ D ] 00 [ ˜ ∂ ] ⊗ [ ˜ D ] ! where [ ˜ ∂ ] ij , [ ˜ ∂ ] kl ∈ { , ± , ± u, ± v, } . Thus, we can write R ( F ) = ± ⊗ ( m X i =1 g i ) + u ⊗ ( n X j =1 h j ) + v ⊗ ( p X k =1 t k )or only − u or − v , where g i = u m i v n i , h j = u x j v y j and t k = u z k v w k . Wewill suppose which R ( F ) is write like above. The case with − u or − v the proof is analogous.From Lemma 4.1 of [8] the element ± ⊗ ( m X i g i ) is homologous tozero. By Proposition 4.1 we can suppose that F has no fixed points in T ×{ , } . In this situation R ( F ) will be a 1-cycle in HH ( Z G, ( Z G ) φ ).Thus, By Proposition 3.5, the sum v ⊗ ( p X k t k ) can not be appear inone-parameter trace R ( F ) of F . Therefore, in this case the trace R ( F )has the form: R ( F ) = ± ⊗ ( m X i =1 g i ) + u ⊗ ( n X j =1 h j )From Proposition 4.2 each C-component nonzero in R ( F ) will repre-sent by one unique cycle. Therefore the one-parameter Nielsen numberin this case will be N ( F ) = n .From section 2 the one-parameter Lefschetz class is the image of R ( F ) in H ( G ) by induced of inclusion i : Z ( g C ) → G . Thus, eachelement u ⊗ h j is sending in H ( G ) in the class [ u ], that is, the imageof R ( F ) in H ( G ) will be L ( F ) = n X j =1 [ u ] = n [ u ] = N ( F )[ u ]Take α = [ u ], which is one of the two generators of H ( G ). If weconsider left action instead right action in the covering space we will obtain L ( F ) = − N ( F )[ u ]. Therefore, L ( F ) = ± N ( F ) α Case II
In this case we have det ([ φ ] − I ) = L ( F | T ) = 0. Therefore, byCorollary 3.2, for each element g ∈ G the semicentralizer, Z ( g ), of g in G is trivial. Thus, H ( Z ( g C )) = 0 for each semiconjugacy class C ,that is, HH ( Z G, ( Z G ) φ ) = 0 which implies R ( F ) = 0 . In this case wehave L ( F ) = N ( F ) = 0 . (cid:3) (cid:3) We have other interpretation in Case II. Note that by definition of R ( F ) in section 2 we are not considering in trace R ( F ) the semiconju-gacy classes represented by fixed point classes which meet T × { , } .If we consider all fixed points classes then the trace R ( F ) has the form: R ( F ) = ± ⊗ ( m X i =1 g i ) + v ⊗ ( p X k =1 t k because in this situation can not be appear circles in F ix ( F ), but onlyarcs join T × { } to T × { } . By Proposition 3.5 R ( F ) can not bea 1-cycle. Since for each t the map F t can be deformed to a mapwith L ( F | T ) fixed points, then from Theorem 3.3 of [4] we will have p = L ( F | T ) = det ([ φ ] − I ) , i.e. in this case F ix ( F ) will be compose by det ([ φ ] − I ) = L ( F | T ) arcs join T × { } to T × { } . Figure 2.
Arcs in
F ix ( F ) . NE-PARAMETER LEFSCHETZ CLASS OF HOMOTOPIES ON TORUS 11
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