A system of unstable higher Toda brackets
aa r X i v : . [ m a t h . A T ] A p r A SYSTEM OF UNSTABLE HIGHER TODA BRACKETS
HIDEAKI ¯OSHIMA AND KATSUMI ¯OSHIMA
Abstract.
We show that a system of unstable higher Toda brackets can be defined in-ductively. Introduction
The Toda bracket is one of the basic tools in homotopy theory. After [ T1 , T2 ] a num-ber of definitions of higher Toda brackets have appeared in the literature. Nowadays theo-ries of higher Toda brackets become considerably abstract and categorical (see for example[ BBG , BBS , BJM ] and their References), and few of them deal in subscripted brackets. Wewould like to study topological and not so abstract theory of subscripted brackets. Even fortopological higher Toda brackets, it seems difficult to nominate one of known theories as thestandard one. In [
OO2 , OO3 ], we defined two systems of unstable higher Toda brackets ascandidates for the standard system. Our definitions were basically just postulated and so onemay feel that they are not really defined. The purpose of the present paper is to clear a littlesuch a doubt by proving that one of two systems can be defined inductively.Given the data(1.1) n, an integer ≥ , ( X n +1 , . . . , X ) , a sequence of well-pointed spaces ,~ m = ( m n , . . . , m ) , a sequence of non-negative integers ,~ f = ( f n , . . . , f ) , a sequence of pointed maps f k : Σ m k X k → X k +1 , we defined the unstable n -fold Toda bracket { ~ f } (¨ s t ) ~ m ⊂ [Σ n − Σ m n · · · Σ m X , X n +1 ] in [ OO2 , OO3 ] (see Section 2 below). We expand [
OO2 , Theorem 6.7.1] and [
OO3 , Theorem 8.1] into
Theorem 1.1.
For n ≥ , we have { f n , . . . , f } (¨ s t )( m n ,...,m ) = [ A ,A { f n , . . . , f , [ f , A , Σ m f ] , (Σ m f , e Σ m A , Σ m Σ m f ) } (¨ s t )( m n ,...,m , , ◦ (1 Σ m Σ m Σ m X ∧ τ (S m ∧ · · · ∧ S m n , S ) ∧ (S ) ∧ ( n − )= [ ~ A { f n , . . . , f , [ f , A , Σ m f ] , (Σ m f , e Σ m A , Σ m Σ m f ) } (¨ s t )( m n ,...,m , , ◦ (1 Σ m Σ m Σ m X ∧ τ (S m ∧ · · · ∧ S m n , S ) ∧ (S ) ∧ ( n − ) where the union S A ,A is taken over all pairs ( A , A ) of null homotopies A : f ◦ Σ m f ≃ ∗ and A : f ◦ Σ m f ≃ ∗ , and the union S ~ A is taken over all sequences ~ A = ( A n − , . . . , A ) Mathematics Subject Classification.
Primary 55P99; Secondary 55Q05.
Key words and phrases.
Unstable higher Toda brackets. of null homotopies A k : f k +1 ◦ Σ m k +1 f k ≃ ∗ such that [ f k +2 , A k +1 , Σ m k +2 f k +1 ] ◦ (Σ m k +2 f k +1 , e Σ m k +2 A k , Σ m [ k +2 ,k +1] f k ) ≃ ∗ (1 ≤ k ≤ n − . In the above description,[ f , A , Σ m f ] : Σ m X ∪ Σ m f C Σ m Σ m X → X is an extension of f with respect to Σ m f , and(Σ m f , e Σ m A , Σ m Σ m f ) : ΣΣ m Σ m Σ m X → Σ m X ∪ Σ m f C Σ m Σ m X is a coextension of Σ m Σ m f with respect to Σ m f (see [ T2 , Og , OO1 , OO2 ]). For othernotations see Section 2. If A , A , or ~ A does not exist, then { ~ f } (¨ s t ) ~ m denotes the empty set.We call a sequence ~ A = ( A n − , . . . , A ) in the above theorem an admissible sequence of nullhomotopies for ~ f = ( f n , . . . , f ).From the definition of { ~ f } (¨ s t ) ~ m (see Section 2 below), the equality { f , f , f } (¨ s t )( m ,m ,m ) = { f , f , Σ m f } m is obtained easily (cf.[ OO3 , Theorem 7.1]), where { f , f , Σ m f } m is theclassical Toda bracket denoted by { f , Σ m f , Σ m Σ m f } m in [ T2 ]. Hence { f , f , f } (¨ s t )( m ,m ,m ) = [ A ,A { [ f , A , Σ m f ] , (Σ m f , e Σ m A , Σ m Σ m f ) } (0 , , where { f, g } (0 , denotes the one point set consisting of the homotopy class of f ◦ g for anypointed maps Z f ← Y g ← X .The above theorem says that { ~ f } (¨ s t ) ~ m is the union of classical Toda brackets and that wecan define ¨ s t -brackets inductively even in the category of pointed spaces TOP ∗ .In Section 2, we review the definition of { ~ f } (¨ s t ) ~ m . In Section 3, we prove Theorem 1.1. InSection 4, we give an inductive definition of n -fold brackets in TOP ∗ . In Appendix A, we givea remark to [ OO3 , (4.3)] which shall be used in the proof of Theorem 1.1. In Appendix B,we write diagrams which shall be needed in Section 3.2.
Review of the definition of { ~ f } (¨ s t ) ~ m Let TOP be the category of topological spaces (spaces for short) and continuous maps(maps for short), TOP ∗ the category of spaces with the base point (based spaces for short)and maps preserving the base point (based maps for short), and TOP w the full subcategoryof TOP ∗ of well-pointed spaces, that is, based spaces such that the inclusion of the base pointto the based space is a cofibration in TOP.Given a space A , let TOP A be the category of spaces under A and maps under A . That is,its object consists of maps j : A → X and TOP A ( j, j ′ ), the set of morphisms from j : A → X to j ′ : A → X ′ , consists of maps f : X → X ′ such that f ◦ j = j ′ . The following well-knownresult of Dold is useful: if j : A → X and j ′ : A → X ′ are cofibrations and f ∈ TOP A ( j, j ′ )is a homotopy equivalence in TOP, then f is a homotopy equivalence in TOP A . In the lastcase, we denote by f − ∈ TOP A ( j ′ , j ) a homotopy inverse of f in TOP A .We list up some notations in TOP ∗ . The base point of the based space X is denoted by x or ∗ . Let f : X → Y be a based map. I = [0 , , the closed unit interval having 1 as its base point; X ∧ · · · ∧ X n = ( X × · · · × X n ) /T ( X , . . . , X n ) ,T ( X , . . . , X n ) is the subspace of X × · · · × X n consisting of points such thatat least one component is the base point; SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 3 CX = X ∧ I, Σ X = CX/ ( X ∧ { , } );S m = { ( t , . . . , t m +1 ) ∈ R m +1 | X t k = 1 } , the m -sphere having (1 , , . . . ,
0) as its base point , we identify S m with S ∧ · · · ∧ S | {z } m = (S ) ∧ m for m ≥ OO2 ];1 X : X → X, the identity map of the space X ;Σ m X = X ∧ S m , we identify Σ X with Σ X ;Σ m f = f ∧ S m : X ∧ S m → Y ∧ S m ;S m [ s + r,s ] = S m s ∧ · · · ∧ S m s + r , Σ m [ s + r,s ] X = X ∧ S m [ s + r,s ] , Σ m [ s + r,s ] f = f ∧ S m [ s + r,s ] , for non-negative integers r and m s + r , . . . , m s ;1 f : X × I → Y, the constant homotopy ( x, t ) f ( x ); e Σ m H : Σ m X × I → Σ m Y, ( x ∧ s, t ) H ( x, t ) ∧ s, for a homotopy H : X × I → Y ; Y ∪ f CX, the mapping cone of f ; ψ mf : Σ m Y ∪ Σ m f C Σ m X −→ ≈ Σ m ( Y ∪ f CX ) , y ∧ s y ∧ s, x ∧ s ∧ t x ∧ t ∧ s ; i f : Y → Y ∪ f CX, the inclusion; q f : Y ∪ f CX → ( Y ∪ f CX ) /Y = Σ X, the quotient; q ′ f : ( Y ∪ f CX ) ∪ i f CY −→ ≃ (( Y ∪ f CX ) ∪ i f CY ) /CY = Σ X, the quotient; τ ( X, Y ) : X ∧ Y → Y ∧ X, the switching map , x ∧ y y ∧ x. Given a homotopy commutative square and a homotopy X a (cid:15) (cid:15) f / / Y b (cid:15) (cid:15) X ′ f ′ / / Y ′ , H : b ◦ f ≃ f ′ ◦ a, we define Φ( f, f ′ , a, b ; H ) : Y ∪ f CX → Y ′ ∪ f ′ CX ′ by y b ( y ) , x ∧ t ( H ( x, t ) 0 ≤ t ≤ / a ( x ) ∧ (2 t −
1) 1 / ≤ t ≤ . Then Φ( f, f ′ , a, b ; H ) ◦ i f = i f ′ ◦ b . Consider the following diagram: X a (cid:15) (cid:15) f / / Y b (cid:15) (cid:15) i f / / Y ∪ f CX Φ( H ) (cid:15) (cid:15) i if / / ( Y ∪ f CX ) ∪ i f CY Φ ′ ( H ) (cid:15) (cid:15) q ′ f / / Σ X Σ a (cid:15) (cid:15) X ′ f ′ / / Y ′ i f ′ / / Y ′ ∪ f ′ CX ′ i if ′ / / ( Y ′ ∪ f ′ CX ′ ) ∪ i f ′ CY ′ q ′ f ′ / / Σ X ′ where Φ( H ) = Φ( f, f ′ , a, b ; H ) and Φ ′ ( H ) = Φ( i f , i f ′ , b, Φ( H ); 1 i f ′ ◦ b ). Lemma 2.1 ( § P ]) . In the above diagram, we have (1)
The second and the third squares are commutative; the first and the fourth squaresare homotopy commutative. (2) Φ ′ ( H ) ≃ Φ( H ) ∪ Cb . (3) q ′ f ′ ◦ (Φ( H ) ∪ Cb ) ≃ Σ a ◦ q ′ f . HIDEAKI ¯OSHIMA AND KATSUMI ¯OSHIMA (4) If a and b are homotopy equivalences, then Φ( H ) is a homotopy equivalence. (5) If the first square is strictly commutative and H = 1 b ◦ f , then Φ( H ) ≃ b ∪ Ca . From now on, we will work in TOP w and so we will omit the word “based”.Suppose that (1.1) is given. An ¨ s t - presentation of ~ f is a collection { S r , f r , A r | ≤ r ≤ n } satisfying (i)-(v) below.(i) S = (Σ m X ; X , X ∪ f C Σ m X ; f ; i f ) is the diagramΣ m X f = g , (cid:15) (cid:15) X = C , i f = j , / / X ∪ f C Σ m X = C , . Note that i f is a free cofibration (see (ii) below for the terminology “free”) and so Σ m i f isan embedding for any m ≥ OO2 , Corollary 2.3(1),(2)]).(ii) S r (3 ≤ r ≤ n ) is a diagram Σ m r − X r − f r − = g r, (cid:15) (cid:15) · · · Σ s − Σ m [ r − ,r − s ] X r − sg r,s (cid:15) (cid:15) · · · Σ r − Σ m [ r − , X g r,r − (cid:15) (cid:15) X r = C r, j r, / / · · · j r,s − / / C r,s j r,s / / · · · j r,r − / / C r,r − j r,r − / / C r,r such that it is reduced , that is, C r, = X r ∪ f r − C Σ m r − X r − and j r, = i f r − , and the map j r,s in TOP w is a free cofibration, that is, j r,s is a cofibration in TOP so that j r,s is anembedding (Strøm) and so Σ m r j r,s is also an embedding by [ OO2 , Corollary 2.3(1)].(iii) f r is a map f r : ( Σ m r C r,r → X r +1 ≤ r ≤ n − m n C n,n − → X n +1 r = n which is an extension of f r . Let f rs : Σ m r C r,s → X r +1 be the restriction of f r to Σ m r C r,s for 1 ≤ s ≤ r if r ≤ n − ≤ s ≤ n − r = n (so that f r = f r ).(iv) A r consists of homotopy equivalences a r,s : C r,s +1 → ≃ C r,s ∪ g r,s C Σ s − Σ m [ r − ,r − s ] X r − s (1 ≤ s ≤ r − a r,s ◦ j r,s = i g r,s i.e. a r,s ∈ TOP C r,s ( j r,s , i g r,s ) and A r is reduced , that is, a r, = 1 C r, . A r is called a structure on S r .Before stating the condition (v), we explain some terminologies. The quasi-structure Ω( A r )derived from A r on S r consists of homotopy equivalences ω r,s = q ′ g r,s ◦ ( a r,s ∪ C C r,s ) : C r,s +1 ∪ j r,s CC r,s → ≃ ΣΣ s − Σ m [ r − ,r − s ] X r − s (1 ≤ s ≤ r − . For 1 ≤ s ≤ r −
1, we set e Σ m r ω r,s = (1 Σ m [ r − ,r − s ] X r − s ∧ τ (S s − ∧ S , S m r )) ◦ Σ m r ω r,s ◦ ψ m r j r,s (2.1) : Σ m r C r,s +1 ∪ Σ mr j r,s C Σ m r C r,s −→ ≃ ΣΣ s − Σ m [ r,r − s ] X r − s , e Σ m r g r,s = Σ m r g r,s ◦ (1 Σ m [ r − ,r − s ] X r − s ∧ τ (S m r , S s − ))(2.2) : Σ s − Σ m [ r,r − s ] X r − s −→ Σ m r C r,s , e Σ m r a r,s = (1 Σ mr C r,s ∪ C (1 Σ m [ r − ,r − s ] X r − s ∧ τ (S s − , S m r ))) ◦ ( ψ m r g r,s ) − ◦ Σ m r a r,s (2.3) : Σ m r C r,s +1 −→ ≃ Σ m r C r,s ∪ e Σ mr g r,s C Σ s − Σ m [ r,r − s ] X r − s . SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 5
Set e Σ m r A r = { e Σ m r a r,s | ≤ s ≤ r − } . We easily have(2.4) e Σ m r ω r,s = q ′ e Σ mr g r,s ◦ ( e Σ m r a r,s ∪ C Σ mr C r,s ) (1 ≤ s ≤ r − . Let e Σ m r S r be the following diagram: Σ m [ r,r − X r − mr fr − e Σ mr gr, (cid:15) (cid:15) · · · Σ s − Σ m [ r,r − s ] X r − s e Σ mr gr,s (cid:15) (cid:15) · · · Σ r − Σ m [ r, X e Σ mr gr,r − (cid:15) (cid:15) Σ mr X r = Σ mr C r, mr jr, / / · · · / / Σ mr C r,s Σ mr jr,s / / · · · / / Σ mr C r,r − mr jr,r − / / Σ mr C r,r We have e Σ m r a r,s ∈ TOP Σ mr C r,s (Σ m r j r,s , i e Σ mr g r,s ) is a homotopy equivalence in the categoryTOP Σ mr C r,s . Let(2.5) ( e Σ m r a r,s ) − ∈ TOP Σ mr C r,s ( i e Σ mr g r,s , Σ m r j r,s )be a homotopy inverse of e Σ m r a r,s . It follows from the proof of [ OO2 , Lemma 4.3(1)]that ( e Σ m r a r,s ) − ∪ C Σ mr C r,s : (Σ m r C r,s ∪ e Σ mr C Σ s − Σ m [ r,r − s ] X r − s ) ∪ i e Σ mr gr,s C Σ m r C r,s → Σ m r C r,s +1 ∪ Σ mr j r,s C Σ m r C r,s is a homotopy inverse of e Σ m r a r,s ∪ C Σ mr C r,s so that(2.6) ( ( e Σ m r ω r,s ) − ≃ (( e Σ m r a r,s ) − ∪ C Σ mr C r,s ) ◦ ( q ′ e Σ mr g r,s ) − , e Σ m r ω r,s ◦ (( e Σ m r a r,s ) − ∪ C Σ mr C r,s ) ≃ q ′ e Σ mr g r,s by (2.4), where ( e Σ m r ω r,s ) − is a homotopy inverse of e Σ m r ω r,s . At this stage we can define adiagram ( e Σ m r S r )( f r , e Σ m r A r ):Σ m r X rf r (cid:15) (cid:15) · · · Σ s Σ m [ r,r − s ] X r − sg ′ r +1 ,s +1 (cid:15) (cid:15) · · · Σ r − Σ m [ r, X g ′ r +1 ,r (cid:15) (cid:15) X r +1 j ′ r +1 , / / · · · j ′ r +1 ,s / / C ′ r +1 ,s +1 j ′ r +1 ,s +1 / / · · · j ′ r +1 ,r − / / C ′ r +1 ,r j ′ r +1 ,r / / C ′ r +1 ,r +1 where C ′ r +1 , = X r +1 , C ′ r +1 ,s +1 = X r +1 ∪ f rs C Σ m r C r,s (1 ≤ s ≤ r ) ,j ′ r +1 , = i f r , j ′ r +1 ,s +1 = 1 X r +1 ∪ C Σ m r j r,s (1 ≤ s ≤ r − ,g ′ r +1 , = f r , g ′ r +1 ,s +1 = ( f rs +1 ∪ C Σ mr C r,s ) ◦ ( e Σ m r ω r,s ) − (1 ≤ s ≤ r − . By [
OO2 , Proposition 2.2, Corollary 2.3], for 1 ≤ s ≤ r , C ′ r +1 ,s +1 is well-pointed and j ′ r +1 ,s is a free cofibration. It follows from [ OO2 , Lemma 5.3] (cf. [
OO3 , Section 3]) that( e Σ m r S r )( f r , e Σ m r A r ) has a reduced structure, that is, there exist homotopy equivalences a ′ r +1 ,s ∈ TOP C ′ r +1 ,s ( j ′ r +1 ,s , i g ′ r +1 ,s ) for 1 ≤ s ≤ r with a ′ r +1 , = 1 C ′ r +1 , . The last condition (v)we promised to state is the following.(v) S r +1 = ( e Σ m r S r )( f r , e Σ m r A r ) (2 ≤ r < n ), that is, C r +1 ,s = C ′ r +1 ,s , j r +1 ,s = j ′ r +1 ,s , and g r +1 ,s = g ′ r +1 ,s .We denote by { ~ f } (¨ s t ) ~ m the set of homotopy classes of f n ◦ e Σ m n g n,n − : Σ n − Σ m [ n, X → X n +1 for all ¨ s t -presentations { S r , f r , A r | ≤ r ≤ n } of ~ f . As seen in [ OO3 , Theorem 6.1], { ~ f } (¨ s t ) ~ m depends only on the homotopy classes of f k (1 ≤ k ≤ n ). HIDEAKI ¯OSHIMA AND KATSUMI ¯OSHIMA
Suppose that n ≥ s t -presentation { S r , f r , A r | ≤ r ≤ n } of ~ f is given. Since f r +12 ◦ ψ m r +1 f r : Σ m r +1 X r +1 ∪ Σ mr +1 f r C Σ m [ r +1 ,r ] X r → X r +2 is an extension of f r +1 , we cantake A r : f r +1 ◦ Σ m r +1 f r ≃ ∗ (1 ≤ r < n ) such that f r +12 ◦ ψ m r +1 f r = [ f r +1 , A r , Σ m r +1 f r ].Thus we have a sequence ( A n − , . . . , A ) of null homotopies A r : f r +1 ◦ Σ m r +1 f r ≃ ∗ such that f r +12 ◦ ψ m r +1 f r = [ f r +1 , A r , Σ m r +1 f r ] (1 ≤ r < n ) so that g r +2 , ≃ ( f r +1 , A r , Σ m r +1 f r ) (1 ≤ r ≤ n −
2) by [
OO2 , (4.2)]. By (2.5), for 2 ≤ r < n , f r +13 ◦ ( e Σ m r +1 a r +1 , ) − : Σ m r +1 C r +1 , ∪ e Σ mr +1 g r +1 , C ΣΣ m [ r +1 ,r − X r − → X r +2 is an extension of f r +12 so that f r +12 ◦ e Σ m r +1 g r +1 , ≃ ∗ . We have f r +12 ◦ e Σ m r +1 g r +1 , = f r +12 ◦ ψ m r +1 f r ◦ ( ψ m r +1 f r ) − ◦ Σ m r +1 g r +1 , ◦ (1 Σ m [ r,r − X r − ∧ τ (S m r +1 , S )) ≃ [ f r +1 , A r , Σ m r +1 f r ] ◦ ( ψ m r +1 f r ) − ◦ Σ m r +1 ( f r , A r − , Σ m r f r − ) ◦ (1 Σ m [ r,r − X r − ∧ τ (S m r +1 , S ))= [ f r +1 , A r , Σ m r +1 f r ] ◦ (Σ m r +1 f r , e Σ m r +1 A r − , Σ m [ r +1 ,r ] f r − ) (by [ OO1 , Lemma 2.4]) . Therefore ~ A = ( A n − , . . . , A ) is an admissible sequence of null homotopies for ~ f . Hence wehave Proposition 2.2. If n ≥ and { ~ f } (¨ s t ) ~ m is not empty, then for any element α of { ~ f } (¨ s t ) ~ m there exist an ¨ s t -presentation { S r , f r , A r | ≤ r ≤ n } of ~ f and an admissible sequence ~ A =( A n − , . . . , A ) of null homotopies for ~ f = ( f n , . . . , f ) such that α = f n ◦ e Σ m n g n,n − and f r ◦ ψ m r f r − = [ f r , A r − , Σ m r f r − ] (2 ≤ r ≤ n ) ,g r +1 , ≃ ( f r , A r − , Σ m r f r − ) (2 ≤ r ≤ n − . Proof of Theorem 1.1
Since { f n , . . . , f , [ f , A , Σ m f ] , (Σ m f , e Σ m A , Σ m [3 , f ) } (¨ s t )( m n ,...,m , , = { f n , . . . , f , [ f , A , Σ m f ] ◦ ( ψ m f ) − , ψ m f ◦ (Σ m f , e Σ m A , Σ m [3 , f ) } (¨ s t )( m n ,...,m , , by Corollary A.2 in Appendix A, it suffices to prove the following two containments for ourpurpose. { f n , . . . , f } (¨ s t ) ~ m ⊂ [ ~ A { f n , . . . , f , [ f , A , Σ m f ] ◦ ( ψ m f ) − ,ψ m f ◦ (Σ m f , e Σ m A , Σ m [3 , f ) } (¨ s t )( m n ,...,m , , ◦ (1 Σ m [3 , X ∧ τ (S m [ n, , S ) ∧ (S ) ∧ ( n − ) , (3.1) SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 7 { f n , . . . , f } (¨ s t ) ~ m ⊃ [ A ,A { f n , . . . , f , [ f , A , Σ m f ] ◦ ( ψ m f ) − ,ψ m f ◦ (Σ m f , e Σ m A , Σ m [3 , f ) } (¨ s t )( m n ,...,m , , ◦ (1 Σ m [3 , X ∧ τ (S m [ n, , S ) ∧ (S ) ∧ ( n − ) , (3.2)where S ~ A is the union taken over all admissible sequences ~ A = ( A n − , . . . , A ) of null homo-topies A k : f k +1 ◦ Σ m k +1 f k ≃ ∗ for ~ f , and S A ,A is the union taken over all pairs ( A , A )of null homotopies A : f ◦ Σ m f ≃ ∗ and A : f ◦ Σ m f ≃ ∗ .This section consists of two subsections § § § § X ∗ = ΣΣ m [3 , X , X ∗ = Σ m ( X ∪ f C Σ m X ) , X ∗ k = X k +1 (3 ≤ k ≤ n ) ,m ∗ = m ∗ = 0 , m ∗ k = m k +1 (3 ≤ k ≤ n − . Proof of (3.1).
Let n ≥
4. It suffices to prove (3.1) when { ~ f } (¨ s t ) ~ m is not empty. Take α ∈{ f n , . . . , f } (¨ s t ) ~ m . By Proposition 2.2, there exist an ¨ s t -presentation { S r , f r , A r | ≤ r ≤ n } of ~ f and an admissible sequence ~ A = ( A n − , . . . , A ) of null homotopies A k : f k +1 ◦ Σ m k +1 f k ≃ ∗ such that α = f n ◦ e Σ m n g n,n − and f r ◦ ψ m r f r − = [ f r , A r − , Σ m r f r − ] (2 ≤ r ≤ n ) ,g r +1 , ≃ ( f r , A r − , Σ m r f r − ) (2 ≤ r ≤ n − . Set f ∗ = e Σ m g , : X ∗ → X ∗ , f ∗ = f : X ∗ → X ∗ ,f ∗ k = f k +1 : Σ m ∗ k X ∗ k → X ∗ k +1 (3 ≤ k ≤ n − . Then f ∗ = [ f , A , Σ m f ] ◦ ( ψ m f ) − and f ∗ = e Σ m g , = Σ m g , ◦ (1 Σ m [2 , X ∧ τ (S m , S )) ≃ Σ m ( f , A , Σ m f ) ◦ (1 Σ m [2 , X ∧ τ (S m , S ))= ψ m f ◦ (Σ m f , e Σ m A , Σ m [3 , f ) (by [ OO1 , Lemma 2.4]) . Hence by [
OO3 , Theorem 6.1] we have { f ∗ n − , . . . , f ∗ , f ∗ , f ∗ } (¨ s t )( m ∗ n − ,...,m ∗ , , = { f n , . . . , f , [ f , A , Σ m f ] ◦ ( ψ m f ) − , ψ m f ◦ (Σ m f , e Σ m A , Σ m [3 , f ) } (¨ s t )( m n ,...,m , , . We will construct an ¨ s t -presentation { S ∗ r , f ∗ r , A ∗ r | ≤ r ≤ n − } of ~ f ∗ such that f ∗ n − = f n and e Σ m n g n,n − ≃ e Σ m ∗ n − g ∗ n − ,n − ◦ (1 Σ m [3 , X ∧ τ (S m [ n, , S ) ∧ (S ) ∧ ( n − ) . If this is done, then f n ◦ e Σ m n g n,n − ≃ f ∗ n − ◦ e Σ m ∗ n − g ∗ n − ,n − ◦ (1 Σ m [3 , X ∧ τ (S m [ n, , S ) ∧ (S ) ∧ ( n − )so that we obtain (3.1). HIDEAKI ¯OSHIMA AND KATSUMI ¯OSHIMA
First we set S ∗ = ( X ∗ ; X ∗ , X ∗ ∪ f ∗ CX ∗ ; f ∗ ; i f ∗ ) and A ∗ = { C ∗ , } . Then C ∗ , = Σ m C , , C ∗ , = Σ m C , ∪ e Σ m g , C ΣΣ m [3 , X , e Σ m a , : Σ m C , → C ∗ , is a homotopy equivalence in TOP Σ m C , (Σ m j , , i e Σ m g , ), and(3.1.1) ω ∗ , = q ′ e Σ m g , . Set f ∗ = f ◦ ( e Σ m a , ) − : C ∗ , → X ∗ .Secondly set S ∗ = S ∗ ( f ∗ , A ∗ ). Then g ∗ , = f ∗ and C ∗ , = C , , C ∗ , = C , , C ∗ , = X ∗ ∪ f ∗ CC ∗ , . Set f ∗ = ( f : Σ m ∗ C ∗ , → X ∗ n = 4 f ◦ Σ m ∗ (1 X ∪ C ( e Σ m a , ) − ) : Σ m ∗ C ∗ , → X ∗ n ≥ . By (2.6) and (3.1.1), we have( e Σ m ω , ) − ≃ (( e Σ m a , ) − ∪ C Σ m C , ) ◦ ( ω ∗ , ) − and so g , = ( f ∪ C Σ m C , ) ◦ ( e Σ m ω , ) − ≃ ( f ∪ C Σ m C , ) ◦ (( e Σ m a , ) − ∪ C Σ m C , ) ◦ ( ω ∗ , ) − = ( f ∗ ∪ C X ∗ ) ◦ ( ω ∗ , ) − = g ∗ , . Hence(3.1.2) e Σ m g , ≃ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ S ) . Take H : g , ≃ g ∗ , arbitrarily and setΦ( H ) = Φ( g , , g ∗ , , Σ Σ m [3 , X , C , ; H ): C , ∪ g , C Σ Σ m [3 , X → C ∗ , ∪ g ∗ , C ΣΣ m ∗ [2 , X ∗ ,a ∗ , = Φ( H ) ◦ a , ◦ (1 X ∪ C ( e Σ m a , ) − ) : C ∗ , → C ∗ , ∪ g ∗ , C ΣΣ m ∗ [2 , X ∗ . (3.1.3)Set A ∗ = { C ∗ , , a ∗ , } which is a reduced structure on S ∗ .When n = 4, { S ∗ r , f ∗ r , A ∗ r | r = 2 , } is an ¨ s t -presentation of ( f ∗ , f ∗ , f ∗ ) such that f ◦ e Σ m g , ≃ f ∗ ◦ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ S )by (3.1.2). Hence (3.1) holds for n = 4.Thirdly let n ≥ S ∗ = ( e Σ m ∗ S ∗ )( f ∗ , e Σ m ∗ A ∗ ). Then C ∗ ,s = C ,s (1 ≤ s ≤ , C ∗ , = C , , C ∗ , = X ∗ ∪ f ∗ C Σ m ∗ C ∗ , . Set f ∗ = ( f : Σ m ∗ C ∗ , = Σ m C , → X = X ∗ n = 5 f ◦ Σ m ∗ (1 X ∪ C Σ m ∗ (1 X ∪ C ( e Σ m a , ) − )) : Σ m ∗ C ∗ , → X ∗ n ≥ , Φ( e Σ m H ) = Φ(Σ m g , , Σ m ∗ g ∗ , , Σ m Σ Σ m [3 , X , Σ m C , ; e Σ m H ): Σ m C , ∪ Σ m g , C Σ m Σ Σ m [3 , X → Σ m ∗ C ∗ , ∪ Σ m ∗ g ∗ , C Σ m ∗ ΣΣ m ∗ [2 , X ∗ , Φ ′ ( e Σ m H ) = Φ( i Σ m g , , i Σ m ∗ g ∗ , , Σ m C , , Φ( e Σ m H ); 1 i Σ m ∗ g ∗ , ) SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 9 : (Σ m C , ∪ Σ m g , C Σ m ΣΣΣ m [3 , X ) ∪ C Σ m C , → (Σ m ∗ C ∗ , ∪ Σ m ∗ g ∗ , C Σ m ∗ ΣΣ m ∗ [2 , X ∗ ) ∪ C Σ m ∗ C ∗ , ,τ = 1 Σ m [3 , X ∧ τ (S m , (S ) ∧ ) : ΣΣΣ m [4 , X → Σ m ΣΣΣ m [3 , X ,τ ′ = 1 Σ m Σ Σ m [3 , X : Σ m Σ Σ m [3 , X → Σ m ∗ ΣΣ m ∗ [2 , X ∗ = Σ m Σ Σ m [3 , X ,τ ′′ = 1 ΣΣ m [3 , X ∧ τ (S , S m ) : Σ m ΣΣΣ m [3 , X → ΣΣ m ΣΣ m [3 , X . Then(3.1.4) τ ′′ ◦ τ ′ ◦ τ = 1 Σ m [3 , X ∧ τ (S m , S ) ∧ S . Here we consider Diagram D in Appendix B.1. Lemma 3.1.1. e Σ m ∗ a ∗ , = (1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 Σ m C , ∪ Cτ ) ◦ e Σ m a , ◦ Σ m (1 X ∪ C ( e Σ m a , ) − ) .Proof. We have e Σ m ∗ a ∗ , = (1 ∪ Cτ ′′ ) ◦ ( ψ m ∗ g ∗ , ) − ◦ Σ m ∗ a ∗ , = (1 ∪ Cτ ′′ ) ◦ ( ψ m ∗ g ∗ , ) − ◦ Σ m Φ( H ) ◦ Σ m a , ◦ Σ m (1 X ∪ C ( e Σ m a , ) − ) (by (3.1.3))= (1 ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ ( ψ m g , ) − ◦ Σ m a , ◦ Σ m (1 X ∪ C ( e Σ m a , ) − )= (1 ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 ∪ Cτ ) ◦ (1 ∪ Cτ ) − ◦ ( ψ m g , ) − ◦ Σ m a , ◦ Σ m (1 X ∪ C ( e Σ m a , ) − )= (1 ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 ∪ Cτ ) ◦ e Σ m a , ◦ Σ m (1 X ∪ C ( e Σ m a , ) − ) (by (2.3)) . This ends the proof. (cid:3)
We haveΣ( τ ′′ ◦ τ ′ ◦ τ ) ◦ e Σ m ω , = Σ τ ′′ ◦ Σ τ ′ ◦ q ′ Σ m g , ◦ ((1 Σ m C , ∪ Cτ ) ∪ C Σ m C , ) ◦ ( e Σ m a , ∪ C Σ m C , ) ≃ Σ τ ′′ ◦ q ′ Σ m ∗ g ∗ , ◦ (Φ( e Σ m H ) ∪ C Σ m C , ) ◦ ((1 Σ m C , ∪ Cτ ) ∪ C Σ m C , ) ◦ ( e Σ m a , ∪ C Σ m C , ) (by Lemma 2.1(2),(3))= q ′ e Σ m ∗ g ∗ , ◦ ((1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ∪ C Σ m ∗ C ∗ , ) ◦ (Φ( e Σ m H ) ∪ C Σ m C , ) ◦ ((1 Σ m C , ∪ Cτ ) ∪ C Σ m C , ) ◦ ( e Σ m a , ∪ C Σ m C , ) ≃ e Σ m ∗ ω ∗ , ◦ (( e Σ m ∗ a ∗ , ) − ∪ C Σ m ∗ C ∗ , ) ◦ ((1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ∪ C Σ m ∗ C ∗ , ) ◦ (Φ( e Σ m H ) ∪ C Σ m C , ) ◦ ((1 Σ m C , ∪ Cτ ) ∪ C Σ m C , ) ◦ ( e Σ m a , ∪ C Σ m C , ) (by (2.6))and so( e Σ m ω , ) − ◦ Σ( τ ′′ ◦ τ ′ ◦ τ ) − ≃ (cid:0) (Σ m a , ) − ◦ (1 ∪ Cτ ) − ◦ Φ( e Σ m H ) − ◦ (1 ∪ Cτ ′′ ) − ◦ e Σ m ∗ a ∗ , ∪ C Σ m C , (cid:1) ◦ ( e Σ m ∗ ω ∗ , ) −
10 HIDEAKI ¯OSHIMA AND KATSUMI ¯OSHIMA ≃ (Σ m (1 X ∪ C ( e Σ m a , ) − ) ∪ C Σ m C , ) ◦ ( e Σ m ∗ ω ∗ , ) − (by Lemma 3.1.1) . Hence( e Σ m ω , ) − ≃ (cid:0) Σ m (1 X ∪ C ( e Σ m a , ) − ) ∪ C Σ m C , (cid:1) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ Σ( τ ′′ ◦ τ ′ ◦ τ )and so g , = ( f ∪ C Σ m C , ) ◦ ( e Σ m ω , ) − ≃ ( f ∪ C Σ m C , ) ◦ (cid:0) Σ m (1 X ∪ C ( e Σ m a , ) − ) ∪ C Σ m C , (cid:1) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ Σ( τ ′′ ◦ τ ′ ◦ τ )= ( f ∗ ∪ C Σ m C , ) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ Σ( τ ′′ ◦ τ ′ ◦ τ )= g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ ) (by (3.1.4))that is, g , ≃ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ ) . By the last relation and (2.2), we easily have(3.1.5) e Σ m g , ≃ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ) . Take H : g , ≃ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ ) arbitrarily and setΦ( H ) = Φ( g , , g ∗ , , Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ , C , ; H ): C , ∪ g , C Σ ΣΣ m [4 , X → C ∗ , ∪ g ∗ , C Σ Σ m ∗ [3 , X ∗ ,a ∗ , = Φ( H ) ◦ a , ◦ (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − )): C ∗ , → C ∗ , ∪ g ∗ , C Σ Σ m ∗ [3 , X ∗ . (3.1.6)Let A ∗ be a reduced structure on S ∗ containing a ∗ , as a member.When n = 5, { S ∗ r , f ∗ , A ∗ | ≤ r ≤ } is an ¨ s t -presentation of ( f ∗ , . . . , f ∗ ) such that f ◦ e Σ m g , ≃ f ∗ ◦ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ) (by (3.1.5)) . Hence (3.1) holds for n = 5.Fourthly let n ≥ S ∗ = ( e Σ m ∗ S ∗ )( f ∗ , e Σ m ∗ A ∗ ). Then C ∗ ,s = C ,s (1 ≤ s ≤ , C ∗ , = C , , C ∗ , = X ∗ ∪ f ∗ C Σ m ∗ C ∗ , . Set f ∗ = ( f n = 6 f ◦ Σ m ∗ (1 X ∪ C Σ m ∗ (1 X ∪ C Σ m ∗ (1 X ∪ C ( e Σ m a , ) − ))) n ≥ ,τ = 1 Σ m [4 , X ∧ τ (S m , (S ) ∧ ) : Σ Σ m [5 , X → Σ m Σ Σ m [4 , X ,τ ′ = 1 Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ ∧ S m : Σ m Σ Σ m [4 , X → Σ m ∗ Σ Σ m ∗ [3 , X ∗ ,τ ′′ = 1 Σ m ∗ X ∗ ∧ τ ((S ) ∧ , S m ∗ ) : Σ m ∗ Σ Σ m ∗ [3 , X ∗ → Σ Σ m ∗ [4 , X ∗ . Then(3.1.7) τ ′′ ◦ τ ′ ◦ τ = 1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ . Also we setΦ( e Σ m H ) = Φ(Σ m g , , Σ m ∗ g ∗ , , τ ′ , Σ m C , ; e Σ m H ): Σ m C , ∪ Σ m g , C Σ m Σ Σ m [4 , X → Σ m ∗ C ∗ , ∪ Σ m ∗ g ∗ , C Σ m ∗ Σ Σ m ∗ [3 , X ∗ , SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 11 Φ ′ ( e Σ m H ) = Φ( i Σ m g , , i Σ m ∗ g ∗ , , Σ m C , , Φ( e Σ m H ); 1 i Σ m ∗ g ∗ , ): Σ m C , ∪ Σ m g , C Σ m Σ Σ m [4 , X → Σ m ∗ C ∗ , ∪ e Σ m ∗ g ∗ , C Σ m ∗ Σ Σ m ∗ [3 , X ∗ . Here we consider Diagram D in Appendix B.1. Lemma 3.1.2.
We have e Σ m ∗ a ∗ , = (1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 Σ m C , ∪ Cτ ) ◦ e Σ m a , ◦ Σ m (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − )) . Proof.
We have e Σ m ∗ a ∗ , = (1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ◦ ( ψ m ∗ g ∗ , ) − ◦ Σ m ∗ a ∗ , = (1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ◦ ( ψ m ∗ g ∗ , ) − ◦ Σ m ∗ Φ( H ) ◦ Σ m ∗ a , ◦ Σ m ∗ (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − )) (by (3.1.6))= (1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ ( ψ m g , ) − ◦ Σ m ∗ a , ◦ Σ m ∗ (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − ))= (1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 Σ m C , ∪ Cτ ) ◦ (1 Σ m C , ∪ Cτ ) − ◦ ( ψ m g , ) − ◦ Σ m ∗ a , ◦ Σ m ∗ (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − ))= (1 Σ m ∗ C ∗ , ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 Σ m C , ∪ Cτ ) ◦ e Σ m a , ◦ Σ m ∗ (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − )) (by (2.3)) . This ends the proof. (cid:3)
We haveΣ( τ ′′ ◦ τ ′ ◦ τ ) ◦ e Σ m ω , = Σ τ ′′ ◦ Σ τ ′ ◦ Σ τ ◦ q ′ e Σ m g , ◦ ( e Σ m a , ∪ C Σ m C , )= Σ τ ′′ ◦ Σ τ ′ ◦ q ′ Σ m g , ◦ ((1 Σ m C , ) ∪ Cτ ) ∪ C Σ m C , ) ◦ ( e Σ m a , ∪ C Σ m C , ) ≃ Σ τ ′′ ◦ q ′ Σ m ∗ g ∗ , ◦ Φ ′ ( e Σ m H ) ◦ ((1 Σ m C , ) ∪ Cτ ) ∪ C Σ m C , ) ◦ ( e Σ m a , ∪ C Σ m C , ) (by Lemma 2.1(1))= q ′ e Σ m ∗ g ∗ , ◦ ((1 Σ m ∗ g ∗ , ∪ Cτ ′′ ) ∪ C Σ m ∗ C ∗ , ) ◦ (Φ( e Σ m H ) ∪ C Σ m C , ) ◦ ((1 Σ m C , ) ∪ Cτ ) ∪ C Σ m C , ) ◦ ( e Σ m a , ∪ C Σ m C , ) (by Lemma 2.1(2),(3))= q ′ e Σ m ∗ g ∗ , ◦ (cid:0) (1 Σ m ∗ g ∗ , ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 Σ m C , ∪ Cτ ) ◦ e Σ m a , ∪ C Σ m C , (cid:1) ≃ e Σ m ∗ ω ∗ , ◦ (( e Σ m ∗ a ∗ , ) − ∪ C Σ m ∗ C ∗ , ) ◦ (cid:0) (1 Σ m ∗ g ∗ , ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 Σ m C , ∪ Cτ ) ◦ e Σ m a , ∪ C Σ m C , (cid:1) (by (2.6)) ≃ e Σ m ∗ ω ∗ , ◦ (cid:16)(cid:0) Σ m (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − )) (cid:1) − ∪ C Σ m C , (cid:17) (by Lemma 3.1.2) . Hence ( e Σ m ω , ) − ≃ (Σ m (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − )) ∪ C Σ m C , ) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ )by (3.1.7), and so g , = ( f ∪ C Σ m C , ) ◦ ( e Σ m ω , ) − ≃ ( f ∪ C Σ m C , ) ◦ (Σ m (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − )) ∪ C Σ m C , ) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ )= ( f ∗ ∪ C Σ m ∗ C ∗ , ) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ )= g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ) , that is, g , ≃ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ) . By the last relation, we easily have(3.1.8) e Σ m g , ≃ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [6 , , S ) ∧ (S ) ∧ ) . Take H : g , ≃ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ) arbitrarily and setΦ( H ) = Φ( g , , g ∗ , , Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ , C , ; H ): C , ∪ g , C Σ Σ m [5 , X → C ∗ , ∪ g ∗ , C Σ Σ m ∗ [4 , X ∗ ,a ∗ , = Φ( H ) ◦ a , ◦ (cid:0) X ∪ C Σ m (1 X ∪ C Σ m (1 X ∪ C ( e Σ m a , ) − )) (cid:1) : C ∗ , → C ∗ , ∪ g ∗ , C Σ Σ m ∗ [4 , X ∗ . Let A ∗ be a reduced structure on S ∗ containing a ∗ , as a member.When n = 6, { S ∗ r , f ∗ r , A ∗ r | ≤ r ≤ } is an ¨ s t -presentation of ( f ∗ , . . . , f ∗ ) such that f ◦ e Σ m g , ≃ f ∗ ◦ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ) (by (3.1.8)) . Thus (3.1) holds for n = 6.Fifthly let n ≥ S ∗ = ( e Σ m ∗ S ∗ )( f ∗ , e Σ m ∗ A ∗ ). Then C ∗ ,s = C ,s (1 ≤ s ≤ , C ∗ , = C , , C ∗ , = X ∗ ∪ f ∗ CC ∗ , . Set f ∗ = ( f n = 7 f ◦ Σ m ∗ (1 X ∪ C Σ m ∗ ( · · · ∪ C Σ m ∗ (1 X ∪ C ( e Σ m a , ) − ) · · · )) n ≥ . Proceeding with the above construction, we obtain inductively a desired ¨ s t -presentation of ~ f ∗ . This ends the proof of (3.1).3.2. Proof of (3.2).
Let n ≥
4. It suffices to prove (3.2) when the term on the right handis not empty. Suppose that { f n , . . . , f , [ f , A , Σ m f ] ◦ ( ψ m f ) − , ψ m f ◦ (Σ m f , e Σ m A , Σ m [3 , f ) } (¨ s t )( m n ,...,m , , is not empty for A : f ◦ Σ m f ≃ ∗ and A : f ◦ Σ m f ≃ ∗ . We set f ∗ = ψ m f ◦ (Σ m f , e Σ m A , Σ m [3 , f ) : X ∗ → X ∗ ,f ∗ = [ f , A , Σ m f ] ◦ ( ψ m f ) − : X ∗ → X ∗ , SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 13 f ∗ k = f k +1 : Σ m ∗ k X ∗ k → X ∗ k +1 (3 ≤ k ≤ n − . Note that f ∗ = Σ m ( f , A , Σ m f ) ◦ (1 Σ m [2 , X ∧ τ (S m , S )) by [ OO1 , Lemma 2.4]. Take α ∈ { f ∗ n − , . . . , f ∗ } (¨ s t )( m ∗ n − ,...,m ∗ ) and let { S ∗ r , f ∗ r , A ∗ r | ≤ r ≤ n − } be an ¨ s t -presentation of −→ f ∗ such that α = f ∗ n − ◦ e Σ m ∗ n − g ∗ n − ,n − . We will construct an ¨ s t -presentation { S r , f r , A r | ≤ r ≤ n } of ~ f such that f n ◦ e Σ m n g n,n − ≃ f ∗ n − ◦ e Σ m ∗ n − g ∗ n − ,n − ◦ (1 Σ m [3 , X ∧ τ (S m [ n, , S ) ∧ (S ) ∧ ( n − ) . If this is done, then we obtain (3.2). Set S = (Σ m X ; X , X ∪ f C Σ m X ; f , i f ) , A = { C , } ,f = [ f , A , Σ m f ] ◦ ( ψ m f ) − : Σ m C , → X , S = ( e Σ m S )( f , e Σ m A ) with g , = ( f , A , Σ m f ) . Then f ∗ = e Σ m g , . Take a homotopy equivalence a , ∈ TOP C , ( j , , i g , ). Set A = { C , , a , } which is a reduced structure on S . By definition e Σ m a , = (1 Σ m C , ∪ C (1 Σ m [2 , X ∧ τ (S , S m ))) ◦ ( ψ m g , ) − ◦ Σ m a , : Σ m C , → C ∗ , . By (2.4), we have e Σ m ω , = q ′ e Σ m g , ◦ ( e Σ m a , ∪ C Σ m C , ) = ω ∗ , ◦ ( e Σ m a , ∪ C Σ m C , ): Σ m C , ∪ C Σ m C , → ΣΣΣ m [3 , X and so ( ω ∗ , ) − ≃ ( e Σ m a , ∪ C Σ m C , ) ◦ ( e Σ m ω , ) − . Set f = f ∗ ◦ e Σ m a , : Σ m C , → X which is an extension of f . Set S = ( e Σ m S )( f , e Σ m A ) ,f = ( f ∗ : Σ m C , = Σ m ∗ C ∗ , → X ∗ = X n = 4 f ∗ ◦ Σ m (1 X ∪ C e Σ m a , ) : Σ m C , → X n ≥ . Then C , = C ∗ , , C , = C ∗ , , C , = X ∪ f C Σ m C , . We have g ∗ , = ( f ∗ ∪ C X ∗ ) ◦ ( ω ∗ , ) − ≃ ( f ∗ ∪ C X ∗ ) ◦ ( e Σ m a , ∪ C Σ m C , ) ◦ ( e Σ m ω , ) − = ( f ∪ C Σ m C , ) ◦ ( e Σ m ω , ) − = g , . Take H : g ∗ , ≃ g , arbitrarily and setΦ( H ) = Φ( g ∗ , , g , , Σ Σ m [3 , X , C ∗ , ; H ): C ∗ , ∪ g ∗ , C ΣΣ m ∗ [2 , X ∗ → C , ∪ g , C Σ Σ m [3 , X ,a , = Φ( H ) ◦ a ∗ , ◦ (1 X ∪ C e Σ m a , ) : C , → C , ∪ g , C Σ Σ m [3 , X . Let A be a reduced structure on S containing a , as a member. When n = 4, { S r , f r , A r | ≤ r ≤ } is an ¨ s t -presentation of ( f , . . . , f ) such that f ◦ e Σ m g , = f ◦ Σ m g , ◦ (1 Σ m [3 , X ∧ τ (S m , (S ) )) ≃ f ◦ Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m , (S ) ))= f ∗ ◦ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ S )so that (3.2) holds for n = 4.Let n ≥
5. We have ω , = q ′ g , ◦ ( a , ∪ C C , ) = q ′ g , ◦ (cid:0) Φ( H ) ◦ a ∗ , ◦ (1 X ∪ C e Σ m a , ) ∪ C C , (cid:1) = q ′ g , ◦ (Φ( H ) ∪ C C , ) ◦ ( a ∗ , ∪ C C , ) ◦ (cid:0) (1 X ∪ C e Σ m a , ) ∪ C C , (cid:1) ≃ q ′ g ∗ , ◦ ( a ∗ , ∪ C C , ) ◦ (cid:0) (1 X ∪ C e Σ m a , ) ∪ C C , (cid:1) (by Lemma 2.1(3))= ω ∗ , ◦ (cid:0) (1 X ∪ C e Σ m a , ) ∪ C C , (cid:1) and so e Σ m ω , = (1 Σ m [3 , X ∧ τ ((S ) ∧ , S m )) ◦ Σ m ω , ◦ ψ m j , ≃ (1 Σ m [3 , X ∧ τ ((S ) ∧ , S m )) ◦ Σ m ω ∗ , ◦ Σ m (cid:0) (1 X ∪ C e Σ m a , ) ∪ C C , (cid:1) ◦ ψ m j , = (1 Σ m [3 , X ∧ τ ((S ) ∧ , S m )) ◦ Σ m ω ∗ , ◦ ψ m j ∗ , ◦ (cid:0) Σ m (1 X ∪ C e Σ m a , ) ∪ C C , (cid:1) = (1 Σ m [3 , X ∧ τ ((S ) ∧ , S m )) ◦ (1 ΣΣ m [3 , X ∧ τ (S m ∗ , (S ) ∧ )) ◦ e Σ m ∗ ω ∗ , ◦ (cid:0) Σ m (1 X ∪ C e Σ m a , ) ∪ C C , (cid:1) = (1 Σ m [3 , X ∧ τ (S , S m ) ∧ (S ) ∧ ) ◦ e Σ m ∗ ω ∗ , ◦ (Σ m (1 X ∪ C e Σ m a , ) ∪ C C , (cid:1) . Hence ( e Σ m ω , ) − ≃ (cid:0) Σ m (1 X ∪ C e Σ m a , ) ∪ C Σ m C , (cid:1) − ◦ ( e Σ m ∗ ω ∗ , ) − ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ ) . (3.2.1)Set S = ( e Σ m S )( f , e Σ m A ). Then C ,s = C ∗ ,s (1 ≤ s ≤ , C , = C ∗ , , C , = X ∪ f C Σ m C , . By (3.2.1), we have g , = ( f ∪ C Σ m C , ) ◦ ( e Σ m ω , ) − ≃ ( f ∪ C Σ m C , ) ◦ (Σ m (1 X ∪ C e Σ m a , ) ∪ C Σ m C , ) − ◦ ( e Σ m ∗ ω ∗ , ) − ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ ) ≃ ( f ∗ ∪ C Σ m C , ) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ (1 Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ )= g ∗ , ◦ ((1 Σ m [3 , X ∧ τ (S m , S ) ∧ (S ) ∧ ) , that is,(3.2.2) g ∗ , ≃ g , ◦ ((1 Σ m [3 , X ∧ τ (S , S m ) ∧ (S ) ∧ ) . Take H : g ∗ , ≃ g , ◦ ((1 Σ m [3 , X ∧ τ (S , S m ) ∧ (S ) ∧ ) arbitrarily and set a , = Φ( H ) ◦ a ∗ , ◦ (1 X ∪ C Σ m (1 X ∪ C e Σ m a , )) : C , → C , ∪ g , C Σ Σ m [4 , X . SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 15
Then a , ∈ TOP C , ( j , , i g , ) is a homotopy equivalence in the category TOP C , . Let A be a reduced structure on S containing a , as a member. By (2.2) and (3.2.2), we have(3.2.3) e Σ m g , ≃ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ) . Set f = ( f ∗ : Σ m C , → X n = 5 f ∗ ◦ Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , )) : Σ m C , → X n ≥ f .When n = 5, { S r , f r , A r | ≤ r ≤ } is an ¨ s t -presentation of ( f , . . . , f ) such that f ◦ e Σ m g , = f ∗ ◦ e Σ m g , ≃ f ∗ ◦ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ) (by (3.2.3)) . Hence (3.2) holds for n = 5.Let n ≥
6. Set S = ( e Σ m S )( f , e Σ m A ). Then C ,s = C ∗ ,s (1 ≤ s ≤ , C , = C ∗ , , C , = X ∪ f C Σ m C , . Set f = ( f ∗ : Σ m C , = Σ m ∗ C ∗ , → X n = 6 f ∗ ◦ Σ m (1 X ∪ C Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , ))) : Σ m C , → X n ≥ ,τ = 1 Σ m ΣΣ m [3 , X ∧ τ (S m , S ) ,τ ′ = 1 Σ m [3 , X ∧ τ (S , S m ) ∧ S ∧ S m ,τ ′′ = 1 Σ m [4 , X ∧ τ (S , S m ) , Φ( e Σ m H ) = Φ(Σ m ∗ g ∗ , , Σ m g , , τ ′ , Σ m C , ; e Σ m H ) , Φ ′ ( e Σ m H ) = Φ( i Σ m ∗ g ∗ , , i Σ m g , , Σ m C , , Φ( e Σ m H ); 1 i Σ m g , ) . Note that τ ′′ ◦ τ ′ ◦ τ = 1 Σ m [3 , X ∧ τ (S , S m [5 , ) ∧ S . Here we consider Diagram D ∗ inAppendix B.2. Lemma 3.2.1. e Σ m a , = (1 ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 ∪ Cτ ) ◦ e Σ m ∗ a ∗ , ◦ Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , )) .Proof. We have e Σ m a , = (1 Σ m C , ∪ Cτ ′′ ) ◦ ( ψ m g , ) − ◦ Σ m a , = (1 Σ m C , ∪ Cτ ′′ ) ◦ ( ψ m g , ) − ◦ Σ m Φ( H ) ◦ Σ m a ∗ , ◦ Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , ))= (1 Σ m C , ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ ( ψ m g ∗ , ) − ◦ Σ m a ∗ , ◦ Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , ))= (1 ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 ∪ Cτ ) ◦ (1 ∪ Cτ ) − ◦ ( ψ m g ∗ , ) − ◦ Σ m a ∗ , ◦ Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , ))= (1 ∪ Cτ ′′ ) ◦ Φ( e Σ m H ) ◦ (1 ∪ Cτ ) ◦ e Σ m a ∗ , ◦ Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , )) . This ends the proof. (cid:3)
It follows from Lemma 2.1 that Φ ′ ( e Σ m H ) ≃ Φ( e Σ m H ) ∪ C Σ m C , and Σ τ ′ ◦ q ′ Σ m ∗ g ∗ , ≃ q ′ Σ m g , ◦ Φ ′ ( e Σ m H ) so thatΣ τ ′ ◦ q ′ Σ m ∗ g ∗ , ≃ q ′ Σ m g , ◦ (Φ( e Σ m H ) ∪ C Σ m C , ) . We then haveΣ( τ ′′ ◦ τ ′ ◦ τ ) ◦ e Σ m ∗ ω ∗ , ≃ q ′ e Σ m g , ◦ ((1 ∪ Cτ ′′ ) ∪ C ◦ (Φ( e Σ m H ) ∪ C ◦ ((1 ∪ Cτ ) ∪ C ◦ ( e Σ m ∗ a ∗ , ∪ C ≃ e Σ m ω , ◦ (( e Σ m a , ) − ∪ C ◦ ((1 ∪ Cτ ′′ ) ∪ C ◦ (Φ( e Σ m H ) ∪ C ◦ ((1 ∪ Cτ ) ∪ C ◦ ( e Σ m ∗ a ∗ , ∪ C ≃ e Σ m ω , ◦ (cid:0) Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , )) − ∪ C Σ m C , (cid:1) (by Lemma 3.2.1) . HenceΣ( τ ′′ ◦ τ ′ ◦ τ ) ◦ e Σ m ∗ ω ∗ , ◦ (cid:0) Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , )) ∪ C Σ m ∗ C ∗ , (cid:1) ≃ e Σ m ω , and g , = ( f ∪ C Σ m C , ) ◦ ( e Σ m ω , ) − ≃ ( f ∪ C Σ m C , ) ◦ (Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , )) − ∪ C Σ m ∗ C ∗ , ) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ Σ( τ ′′ ◦ τ ′ ◦ τ ) − ≃ ( f ∗ ∪ C Σ m ∗ C ∗ , ) ◦ ( e Σ m ∗ ω ∗ , ) − ◦ Σ( τ ′′ ◦ τ ′ ◦ τ ) − = g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ )Take H : g ∗ , ≃ g , ◦ (1 Σ m [3 , X ∧ τ (S , S m [5 , ) ∧ (S ) ∧ ) arbitrarily and set a , = Φ( H ) ◦ a ∗ , ◦ (1 X ∪ C Σ m (1 X ∪ C Σ m (1 X ∪ C e Σ m a , ))): C , → C , ∪ g , C Σ Σ m [5 , X . Let A be a reduced structure on S containing a , as a member.When n = 6, { S r , f r , A r | ≤ r ≤ } is an ¨ s t -presentation of ( f , . . . , f ) such that f ◦ e Σ m g , = f ∗ ◦ Σ m g , ◦ (1 Σ m [5 , X ∧ τ (S m , S )) ≃ f ∗ ◦ Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ∧ S m ) ◦ (1 Σ m [5 , X ∧ τ (S m , S ))= f ∗ ◦ e Σ m ∗ g ∗ , ◦ (1 Σ m ∗ [4 , X ∗ ∧ τ (S , S m ∗ )) ◦ (1 Σ m [3 , X ∧ τ (S m [5 , , S ) ∧ (S ) ∧ ∧ S m ) ◦ (1 Σ m [5 , X ∧ τ (S m , S ))= f ∗ ◦ e Σ m ∗ g ∗ , ◦ (1 Σ m [3 , X ∧ τ (S m [6 , , S ) ∧ (S ) ∧ ) . Hence (3.2) holds for n = 6.Let n ≥
7. Proceeding the above process, we obtain inductively an ¨ s t -presentation { S r , f r , A r | ≤ r ≤ n } of ( f n , . . . , f ) such that f n ◦ e Σ m n g n,n − ≃ f ∗ n − ◦ e Σ m ∗ n − g ∗ n − ,n − ◦ (1 Σ m [3 , X ∧ τ (S m [ n, , S ) ∧ (S ) ∧ ( n − ) . This ends the proof of (3.2).
SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 17 Brackets in
TOP ∗ In this section, we work in TOP ∗ .Let n ≥ X n +1 , . . . , X ) a sequence of spaces, ~ m = ( m n , . . . , m ) a sequenceof non-negative integers, and ~ f = ( f n , . . . , f ) a sequence of maps f k : Σ m k X k → X k +1 . Wedefine { ~ f } ~ m inductively as follows: { ~ f } ~ m = [ { [ f , A , Σ m f ] , (Σ m f , e Σ m A , Σ m [3 , f ) } (0 , ( n = 3) , { ~ f } ~ m = [ { f n , . . . , f , [ f , A , Σ m f ] , (Σ m f , e Σ m A , Σ m [3 , f ) } ( m n ,...,m , , ◦ (1 Σ m [3 , X ∧ τ (S m [ n, , S ) ∧ (S ) ∧ ( n − ) ( n ≥ , where the union S is taken over all pairs ( A , A ) of null homotopies A : f ◦ Σ m f ≃ ∗ and A : f ◦ Σ m f ≃ ∗ . If A or A does not exist, then { ~ f } ~ m denotes the empty set. Notice that { ~ f } ~ m is the classical Toda bracket when n = 3 and that { ~ f } ~ m ⊂ [Σ n − Σ m [ n, X , X n +1 ]. Proposition 4.1. If X k is well pointed for all k with ≤ k ≤ n + 1 , then { ~ f } ~ m = { ~ f } (¨ s t ) ~ m .Proof. By the induction on n , this follows easily from Theorem 1.1. (cid:3) Remark . We can prove that { ~ f } ~ m depends only on homotopy classes of f k (1 ≤ k ≤ n )and that if n ≥ { ~ f } ~ m is not empty then f k +1 ◦ Σ m k +1 f k ≃ ∗ for all k with n > k ≥ { f k , . . . , f , f } ( m k ,...,m ) contains 0 for all k with n > k ≥
2, where { f , f } ( m ,m ) denotesthe one point set consisting of the homotopy class of f ◦ Σ m f . Problem . What can we say about uniqueness of a system of unstable higher Toda brackets?
Appendix A. A remark to our previous paper
Lemma A.1.
Given a sequence ( f n , . . . , f ) of maps f k : Σ m k X k → X k +1 (0 ≤ k ≤ n ) , wehave { f n , . . . , f , f ◦ Σ m f } (¨ s t )( m n ,...,m ,m + m ) ⊂ { f n , . . . , f , f ◦ Σ m f , f } (¨ s t )( m n ,...,m ,m + m ,m ) . Proof.
We use the following notations:( X ′ n +1 , . . . , X ′ ) = ( X n +1 , . . . , X , Σ m X ) , ( m ′ n , . . . , m ′ ) = ( m n , . . . , m ) , ( f ′ n , . . . , f ′ ) = ( f n , . . . , f , f ◦ Σ m f ) , ( X ′′ n +1 , . . . , X ′′ ) = ( X n +1 , . . . , X , X ) , ( m ′′ n , . . . , m ′′ ) = ( m n , . . . , m , m + m ) , ( f ′′ n , . . . , f ′′ ) = ( f n , . . . , f , f ◦ Σ m f ) , ( X ∗ n +1 , . . . , X ∗ ) = ( X n +1 , . . . , X , X , X ) , ( m ∗ n , . . . , m ∗ ) = ( m n , . . . , m , m + m , m ) , ( f ∗ n , . . . , f ∗ ) = ( f n , . . . , f , f ◦ Σ m f , f ) . In [
OO3 , (4.3)], we proved { ~ f ′ } (¨ s t ) ~ m ′ ⊂ { ~ f ∗ } (¨ s t ) ~ m ∗ . As is easily seen, the set of ¨ s t -presentationsof ~ f ′ is the same as the set of ¨ s t -presentations of ~ f ′′ . Hence { ~ f ′ } (¨ s t ) ~ m ′ = { ~ f ′′ } (¨ s t ) ~ m ′′ and so thelemma is proved. (cid:3) Corollary A.2.
Under the situation of Lemma A.1, if moreover m = 0 , then { f n , . . . , f , f ◦ f } (¨ s t )( m n ,...,m ,m ) ⊂ { f n , . . . , f , f ◦ Σ m f , f } (¨ s t )( m n ,...,m ,m ) and if moreover f : X → X is a homotopy equivalence, then { f n , . . . , f , f ◦ f } (¨ s t )( m n ,...,m ,m ) = { f n , . . . , f , f ◦ Σ m f , f } (¨ s t )( m n ,...,m ,m ) . Proof.
The first containment is obvious from Lemma A.1. Assume that f is a homotopyequivalence. Let f − be a homotopy inverse of f . We have { f n , . . . , f , f ◦ Σ m f , f } (¨ s t )( m n ,...,m ,m ) = { f n , . . . , f , f ◦ Σ m f , f − ◦ f ◦ f } (¨ s t )( m n ,...,m ,m ) (by [ OO3 , Lemma 6.2]) ⊂ { f n , . . . , f , f ◦ Σ m f ◦ Σ m f − , f ◦ f } (¨ s t )( m n ,...,m ,m ) (by Lemma A.1)= { f n , . . . , f , f , f ◦ f } (¨ s t )( m n ,...,m ,m ) (by [ OO3 , Lemma 6.2]) ⊂ { f n , . . . , f , f ◦ Σ m f , f } (¨ s t )( m n ,...,m ,m ) (by Lemma A.1) . Hence we obtain the desired equality. (cid:3)
Appendix B. Diagrams
B.1.
Diagram D r +1 . The diagram is commutative except for two squares containing τ ′ r +1 or Σ τ ′ r +1 . The exceptional two squares are homotopy commutative. Σ mr +1 C r +1 ,r +1 e Σ mr +1 ar +1 ,r (cid:15) (cid:15) i Σ mr +1 jr +1 ,r / / Σ r − Σ m [ r +1 , X e Σ mr +1 gr +1 ,r / / τr +1 (cid:15) (cid:15) Σ mr +1 C r +1 ,r i / / Σ mr +1 jr +1 ,r ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ Σ mr +1 C r +1 ,r ∪ e Σ mr +1 gr +1 ,r C Σ r − Σ m [ r +1 , X ∪ Cτr +1 (cid:15) (cid:15) i / / Σ mr +1 Σ r − Σ m [ r, X mr +1 gr +1 ,r / / τ ′ r +1 (cid:15) (cid:15) Σ mr +1 C r +1 ,r i / / Σ mr +1 C r +1 ,r ∪ Σ mr +1 gr +1 ,r C Σ mr +1 Σ r − Σ m [ r, X e Σ mr +1 Hr +1) (cid:15) (cid:15) i / / Σ m ∗ r Σ r − Σ m ∗ [ r − , X ∗ m ∗ r g ∗ r,r − / / τ ′′ r +1 (cid:15) (cid:15) Σ m ∗ r C ∗ r,r − i / / Σ m ∗ r C ∗ r,r − ∪ Σ m ∗ r g ∗ r,r − C Σ m ∗ r Σ r − Σ m ∗ [ r − , X ∗ ∪ Cτ ′′ r +1 (cid:15) (cid:15) i / / Σ r − Σ m ∗ [ r, X ∗ e Σ m ∗ r g ∗ r,r − / / Σ m ∗ r C ∗ r,r − i e Σ m ∗ r g ∗ r,r − / / Σ m ∗ r j ∗ r,r − * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Σ m ∗ r C ∗ r,r − ∪ e Σ m ∗ r g ∗ r,r − C Σ r − Σ m ∗ [ r, X ∗ i / / Σ m ∗ r C ∗ r,r e Σ m ∗ r a ∗ r,r − O O i / / SYSTEM OF UNSTABLE HIGHER TODA BRACKETS 19 Σ mr +1 C r +1 ,r +1 ∪ C Σ mr +1 C r +1 ,r e Σ mr +1 ar +1 ,r ∪ C mr +1 Cr +1 ,r (cid:15) (cid:15) e Σ mr +1 ωr +1 ,r , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ (Σ mr +1 C r +1 ,r ∪ e Σ mr +1 gr +1 ,r C Σ r − Σ m [ r +1 , X ) ∪ C Σ mr +1 C r +1 ,rq ′ e Σ mr +1 gr +1 ,r / / (1 ∪ Cτr +1) ∪ C mr +1 Cr +1 ,r (cid:15) (cid:15) ΣΣ r − Σ m [ r +1 , X τr +1 (cid:15) (cid:15) (Σ mr +1 C r +1 ,r ∪ Σ mr +1 gr +1 ,r C Σ mr +1 Σ r − Σ m [ r, X ) ∪ C Σ mr +1 C r +1 ,rq ′ Σ mr +1 gr +1 ,r / / Φ ′ ( e Σ mr +1 Hr +1) (cid:15) (cid:15) ΣΣ mr +1 Σ r − Σ m [ r, X τ ′ r +1 (cid:15) (cid:15) (Σ m ∗ r C ∗ r,r − ∪ Σ m ∗ r g ∗ r,r − C Σ m ∗ r Σ X ∗ ) ∪ C Σ m ∗ r C ∗ r,r − q ′ Σ m ∗ r g ∗ r,r − / / (1 ∪ Cτ ′′ r +1) ∪ C (cid:15) (cid:15) ΣΣ m ∗ r Σ r − Σ m ∗ [ r − , X ∗ τ ′′ r +1 (cid:15) (cid:15) (Σ m ∗ r C ∗ r,r − ∪ e Σ m ∗ r g ∗ r,r − C Σ r − Σ m ∗ r X ∗ ) ∪ C Σ m ∗ r C ∗ r,r − q ′ e Σ m ∗ r g ∗ r,r − / / ΣΣ r − Σ m ∗ [ r, X ∗ Σ m ∗ r C ∗ r,r ∪ C Σ m ∗ r C ∗ r,r − e Σ m ∗ r a ∗ r,r − ∪ C m ∗ r C ∗ r,r − O O e Σ m ∗ r ω ∗ r,r − ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ B.2.
Diagram D ∗ . The diagram is commutative except for two squares containing τ ′ orΣ τ ′ . The exceptional two squares are homotopy commutative.Σ m ∗ C ∗ , e Σ m ∗ a ∗ , (cid:15) (cid:15) i Σ m ∗ j ∗ , / / Σ Σ m ∗ [4 , X ∗ e Σ m ∗ g ∗ , / / τ (cid:15) (cid:15) Σ m ∗ C ∗ , i e Σ m ∗ g ∗ , / / Σ m ∗ j ∗ , ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Σ m ∗ C ∗ , ∪ e Σ m ∗ g ∗ , C Σ Σ m ∗ [4 , X ∗ ∪ Cτ (cid:15) (cid:15) i / / Σ m ∗ Σ Σ m ∗ X ∗ m ∗ g ∗ , / / τ ′ (cid:15) (cid:15) Σ m ∗ C ∗ , i / / Σ m ∗ C ∗ , ∪ Σ m ∗ g ∗ , C Σ m ∗ Σ Σ m ∗ X ∗ e Σ m H ) (cid:15) (cid:15) i / / Σ m Σ Σ m [4 , X m g , / / τ ′′ (cid:15) (cid:15) Σ m C , i / / Σ m C , ∪ Σ m g , C Σ m Σ Σ m [4 , X ∪ Cτ ′′ (cid:15) (cid:15) i / / Σ Σ m [5 , X e Σ m g , / / Σ m C , i / / Σ m j , * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Σ m C , ∪ e Σ m g , C Σ Σ m [5 , X i / / Σ m C , e Σ m a , O O i / / Σ m ∗ C ∗ , ∪ C Σ m ∗ C ∗ , e Σ m ∗ a ∗ , ∪ C Σ m ∗ a ∗ , (cid:15) (cid:15) e Σ m ∗ ω ∗ , + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ (Σ m ∗ C ∗ , ∪ C Σ Σ m ∗ [4 , X ∗ ) ∪ C Σ m ∗ C ∗ , q ′ e Σ m ∗ g ∗ , / / (1 ∪ Cτ ) ∪ C Σ m ∗ C ∗ , (cid:15) (cid:15) Σ Σ m [5 , ΣΣ m [3 , X τ (cid:15) (cid:15) (Σ m ∗ C ∗ , ∪ C Σ m ∗ Σ Σ m ∗ X ∗ ) ∪ C Σ m ∗ C ∗ , q ′ Σ m ∗ g ∗ , / / Φ ′ ( e Σ m H ) (cid:15) (cid:15) ΣΣ m ∗ Σ Σ m ∗ X ∗ τ ′ (cid:15) (cid:15) (Σ m C , ∪ C Σ m Σ Σ m [4 , X ) ∪ C Σ m ∗ C ∗ , q ′ Σ m g , / / (1 ∪ Cτ ′′ ) ∪ C (cid:15) (cid:15) ΣΣ m Σ Σ m [4 , X τ ′′ (cid:15) (cid:15) (Σ m C , ∪ C Σ Σ m [5 , X ) ∪ C Σ m C , q ′ e Σ m g , / / Σ Σ m [5 , X Σ m C , ∪ C Σ m C , e Σ m a , ∪ C Σ m C , O O e Σ m ω , ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ References [ BBG]
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E-mail address : [email protected] (K. ¯Oshima) E-mail address ::