Some formulas for the generalized Hardie-Jansen product and its dual

J. Korean Math. Soc. 36 (1999), No. 3, pp. 527-544

SOME FORMULAS FOR THE GENERALIZED HARDIE-JANSEN PRODUCT AND ITS DUAL

NOBUYUKI ODA AND TOSHIYUKI SHIMIZU

ABSTRACT. The generalized Hardie-Jansen product and its dual are defined and the fundamental results on these products are obtained.

By studying the adjoint maps, we give proofs to them. Moreover we characterize the generalized Hardie-Jansen product making use of the rw-Whitehead product. We also obtain a characterization of the dual of the generalized Hardie-Jansen product using (rW)*- Whitehead product.

Introduction

We work in the category of compactly generated Hausdorff spaces with nondegenerate base point*[8]. LetX /\Y be the smash product of topological spacesX and Y. We denote an element ofX /\Y by x /\y for any x ^E X andy E Y. Letr be a co-Hopf space. We definerX =r/\X.

The space r^X is called the r-suspension space of a space ^X. Let ^XW be the space of base point preserving maps from W to X. We define

r*X = X r . The spacer*X is called ther-loop space of a space X. Let [A, Z] be the set of base point preserving homotopy classes of base point preserving maps from A to Z. For any maps ^0:, {3 : A ^{- 7} Z, we define the following maps; ifA is a co-Hopf space and v :A ^{- 7} A VA is the co-multiplication ofA,'then we define a map

0:

+

where 'Vz :Z VZ - 7 Z is the folding map; or if Z is a Hopf space and Il : Z x Z - 7 Z is the multiplication ofZ, then we define a map

0: t {3= Il0 (0: X {3)o.6.A : A ^{- 7}Z Received July28, 1998.

1991 Mathematics Subject Classification: 55P91, 55Q15.

Key words and phrases: r-Whitehead product, Hardie-Jansen product.

where ~A ^: A ^-+ A x A is the diagonal map. We use the same symbol for a homotopy class and a map which represents the homotopy class.

Now let f be a co-grouplike space, namely an associative co-Hopf space with an inverse. Throughout this paper, f is a co-grouplike space, except otherwise stated explicitly. Then the f-Whitehead product is defined in [6]. It is an element [a,,6]r E [f(XA.Y), Z] for any elements a E [fX,Z] and,6^E [fX,Z]. Iff = sI, then the f-Whitehead product [a,,6]r is the generalized Whitehead product [a,,6] of Arkowitz [1] and Barratt [2]. Let

o:^{[fX A.W;Z] ----+ [fX,ZW]}

be the adjoint isomorphism for a fixed spaceW. The generalized Hardie- Jansen product is defined in[7] as follows, using the f -Whitehead prod- uct and the adjoint isomorphism; for any elements^{a E} [fX A.W, Z] and ,6 E [fY A. W; Z], we define the element [a, ,6]r E [f(XA.Y) A. W, Z]

by O-l([O(a),O(,6)]r). Iff = sI, X = sm-l and Y = sn-I, then this product is the Hardie-Jansen product in[3] and [4]. In§1,we prove the following results of the generalized Hardie-Jansen product.

THEOREM 1.5 Let a E [fX ^A. W;Z] and ,6 E [fY^A. W;Z] be any elements. Then wehave the following results.

(i)10 [a,,6]r = h^{0 a,1 o,6W for} anyI :Z ^-+Z'.

(ii) [a,,6W 0 (f(81\E)1\1_{w )} = [a0 (f8A. 1_w),,60 (fE1\1_w)]r for any 8 : X' -+ X and E :Y' -+ Y.

THEOREM 1.6 Let T: Y A.X -+ X A.Y be the switchingmap. Let a E [fX1\W;Z] and,6 E [fY1\W;Z] beany elements. Then the product [a, ,6]rE [f(X1\Y)1\W;Z] satisfies

(rT1\1w)*([a,,6lf) = _~ [,6, alf·

THEOREM 1.7 If X and Y are co-Hopf spaces,then we have

(i) [a

+

+

'Y E [fY1\W, Z].

(ii) [a,,6

+

+

(3'1 E [fY1\W;Z].

Generalized Hardie-Jansen product and its dual 529

THEOREM 1.8 IfXl, X2^andX3 areco-Hopf spaces, then we have [[al,a2]r, a3W

+

. * W W

+ (fT3l2^{A lw) ([[a3,}arJr ,a2]r )=⁰

for anyal E [fXl AW,Z], a2 E [fX2^AW,Z] and a3 E [fX3^AW,Z], where^{Tijk :} XlAX2^AX3~ XiAXj AXk is the permutation of factors of the smash products.

THEOREM 1.10 Let f l be a co-grouplike space and f² a co-Hopf space. Then we have f2([a,j3]~) = 0 for any elements a E [fIXAW,Z]

and13 E [flYA W, Z].

In§2, we consider dual results. The f* -Whitehead product is defined in [6]. It is an element [a,j3]r' E [A,f*(Xl>Y)] for any elements ^{a E} [A,f*X] and 13 E [A,f*Y]. We define an isomorphism

T: [A,f*(Xw)] ~ [A, (f*X)w] ~ [AA W,f*X].

where 'fJ is an isomorphism induced by a natural homeomorphism and B-1 is- the inverse map ofB. Hardie and Jansen do not formulate the dual product. Using the f*-Whitehead product and the isomorphism^T, we define the dual of the generalized Hardie-Jansen product as follows.

For any elementa E [A, f*(X^W)] and 13 ^E [A, f*(Y^W)], it is an element [a,j3]fY. = T-l([T(a), T(j3)]p) ^E [A, f*((Xl>Y)W)]. In this section, we prove the following results of the dual of the generalized Hardie-Jansen product.

THEOREM 2.6 Leta ^E [A, P(XW)] and13^E [A, f*(YW)] be anyele- ments. Then we have the following results.

(i) [a,j3]fY.^{0 ,} =[a0 I,13⁰ ,]fY. for any ,: A' _~ A.

(ii) (f*((8l>c)W)) ⁰ [a,j3]fY. = [(P(8^{W)) 0} a, (f*(cW)) ⁰ j3]if. for any8 : X _~ X' andc : Y ~Y ' .

THEOREM 2.7 Let T : Xl>Y ~ Y l>X be the switching map be- tween induced fibres. Leta ^E [A,f*(XW)] and 13^E [A,f*(Y^W)] beany elements. Then the product [a,j3]fY. E [A,f*((Xl>Y)W)] satisfies

(f*(Tw))*([a,j3]i!:') = -:- [13,a]i!:' .

THEOREM 2.9 Let r1 be a co-grouplike space and r2 a co-Hopf space. Then we have q([a,,6]f.) = 0 for any a E [A,ri(x^W)] and

1

,6E [A, fi(Y^W)].

In §3, we consider an isomorphism ';wI\X which is closely connected with the generalized Hardie-Jansen product. We define the isomorphism

';w^{I\X :} [rX 1\W; Z] ^---+ [rW1\X, Z]

induced by the switching map tWI\X : W ^1\ X --. X 1\W. Then we have a relation between the generalized Hardie-Jansen product andrw-

Whitehead product in [7]. Itis the following _~elation [a, ,6]r=';w11\(XI\Y) ([,;wl\x(a) , ,;wl\y(,6)]rw)

for any elements a E [rX 1\W;Z] and ,6E [rY 1\W;Z]. Hence the iso- morphism';wI\Xis also useful to investigate the properties of the general- ized Hardie-Jansen product. In this section we prove some fundamental properties of the isomorphism';wI\X .

We also consider an isomorphism(w*x : [A, r*(XW)] --. [A, (rW)*X]

for the characterization of the dual of the generalized Hardie-Jansen product. We prove some fundamental properties of the isomorphism (w*x· And we have a relation between the dual of the generalized Hardie- Jansen product and (rW)*-Whitehead product.

THEOREM 3.7 Let a E [A,r*(X^W)] and ,6 E [A,r*(Y^W)] be any elements. Then we have

[a,,6]~ =(w~(ny)([(w*x(a),(w*Y(,6)](rw)*).

This isomorphism (w*x is also useful to investigate the properties of the dual of the generalized Hardie-Jansen product.

1. Formulas for generalized Hardie-Jansen product Let W be a fixed space. We define an adjoint map

o:^{[X 1\}W;Z] ---+ [X, Zw]

by (O(a)(x))(w) = a(x ^1\w) for any map a : X ^1\W --. Z and any elements x E X and wE W. In the following lemmas, we obtain some formulas for e.

Generalized Hardie-Jansen product and its dual 531 LEMMA 1.1. Let f : X ---+Y be any map. Then there is a commuta- tive diagram

[Y /\ W,Z] ~ ^{[X /\ W,Z]}

91 19

[Y,ZW] r ' [X,ZW].

ThereforewehaveB(o:)of =O(o:o(f/\lw)) for any map 0: :Y/\W ^{- t}z.

Proof. By the definition of B, we have

(0(0:0 (f /\ lw))(x))(w) (0:0 (f /\ lw ))(x /\ w) o:(f(x) /\ w)

= ((B(o:) 0 f)(x))(w) for any elements x E X and w EW.

LEMMA 1.2. Let f: X ---+Y be any map. Then there is a commuta- tive diagram

[A /\ W,X] ~ ^[A/\W,Y]

9

i i

[A,X^W] w • [A,y^W].

(f ).

Therefore we have fW 0 B(o:) ⁼ B(f0 0:) for any 0: : A /\W ---+ X.

Proof By the defini,tion of 0, we have

((fW ⁰ B(o:))(x))(w) (f0 B(o:)(x))(w) f((O(o:)(x))(w)) f(o:(x /\ w))

(f ⁰ 0:)(x /\ w) = (B(f⁰ o:)(x))(w) for any elements x E A and w E W.

LEMMA 1.3. (i) Let X be a co-Hopf space. For any elements 0: and (3in [X /\W,Z]' we have

B(o:

+

+

(n) Let Z be a Hopf space. For any elements a and (3 in [X /\W;Z], we have

O(a t (3) = O(a) t 0«(3) E [X, Zw].

Proof (i) Letvbe a co-multiplication of X. Then the co-multiplication of X /\W is obtained by

v /\ lw :X /\W _~ (X V X) /\W _~ (X /\W) V (X /\W).

We have

O(a+^{(3) -} O(V'zo(aV(3)o(v/\lw)) - O(V'z ⁰ (aV(3» 0 v.

Here, we see O(V'z0(aV(3» 0 jI = O(V'zO(aV(3)0 (jI /\lw)) = O(a) and O(V'z⁰ (aV(3» ⁰j2 = 0«(3) wherejI, j2 :X ^{- t}XVX isinclusions defined by jI(X) = (x,*) and j2(X) = (*,x) for any element x E X. Hence we have

O(a

+

+

(n) Let p, be a multiplication of Z. Then the multiplication of ZW is obtained by

We have

O(a t (3)= O(p,⁰ (a x (3) ⁰ ~xJ\w)

= p,w⁰ O«ax (3)⁰ _~xJ\w),

Here, we seeP'f oO«ax_(3)o~XJ\w) =O(PIo(ax_(3)o~XJ\w) =O(a) and

pr

O(at ⁽³⁾ == p,w⁰ (O(a) x 0«(3» ⁰ ~x ⁼ O(a) t 0«(3).

We notice that 0 is an isomorphism of groups when the domain and . the codomain of0have the group structure. (cf.·Theorem 6.3.28 Maun-

der [5])

We recall the definition of the generalized Hardie-Jansen product in [7]. Itdefines a pairing

[rX /\W, Z] x fry /\W, Z] ~ [r(x /\ Y) /\W; Z]

Generalized Haxdie-Jansen product and its dual 533

by [a, ,B]r = 0-1([0(a),0(,B)]r) E [f(X /\ ¥) /\ W;Z] for any elements a E [rX /\ W;Z] and,BE fry /\ W;Z], where [ " .]r is the f-Whitehead product. Now we prove the following results by making use of the lemmas mentioned above. These results are quoted in [7] without proof.

THEOREM 1.4. If f is a commutative co-Hopf space or ifW is a co-Hopf space orifZ is a Hopf space, then [a,,BlP' = 0 for any a E

[fX /\W, Z] and ,BE try /\ W;Z].

Proof. Using Propositions 1.2 and 1.3 of [6], we have the result.

THEOREM 1.5. Let a E [rX /\ W; Z] and ,B E try /\W;Z] be any elements. Then wehave the following results.

(i) "/0 [a,,BlP' = boa, ,,/o,BlP' for any,,/ :Z^{- t}Z'.

(ii) [a, ,B]r0 (f(8 /\ e) /\1w) = [a0 (f8/\1w),,B⁰ (re /\1w)]r ^{for any} 8: X' ^--t X ande : ¥' ^--t ¥.

Proof. (i) Using Lemma 1.2 and Proposition 1.1 (i) of [6], we have Ob⁰ [a, ,B]r) - "/w ⁰ O([a, ,B]r)

- "/w0 [O(a),O(,B)]r - b^{W 0} O(a),,,/w0 O(,B)]r

= [Ob⁰ a), Ob⁰ ,B)]r

= O(b⁰ a,"/0,B]r).

(ii) Using Lemma 1.1 and Proposition 1.1 (ii) of [6], we have O([a, ,B]r0 (r(8 /\ c) /\1w)) - O([a, ,B]r)0 f(8/\ e)

[O(a), O(,B)]r0 f(8/\ c) [O(a) ⁰ r8,O(,B)⁰ re]r

[O(a0 (r8/\1w)), O(,B0 (re /\ 1w))]r O([a0 (r8 /\ 1w),,B0 (fe /\1w)]r).

THEOREM 1.6. Let T: ¥ /\X --t X /\¥ be the switching map. Let a E [rX /\^W; Z] and,BE try/\^W;Z] be any elements. Then the product [a,,BlP' E [f(X /\¥) /\W;Z] satisfies

(rT /\ 1w )*([a, ,B]p") ⁼ ~ [/3,a]p".

Proof Using Lemma 1.1 and Proposition 1.5 of [6], we have O([a,,8]r⁰ (fT /\1w)) O([a, ,8]r)0 fT

[O(a),O(,8)]r ⁰ fT ..:.. [O(,8),O(a)]r

. [,8 W

- O( ,a]r ) O( ..:.. [,8,a]r)·

THEOREM 1.7. (Bilinearity) If X andY are co-Hopfspaces, then we have

(i) [a

+

+

(ii) [a,,8 +^{I')f = [a,,8]f} + [a,l']f for any a E [rX /\ W;Z] and ,8, I' E [fY /\W;Z].

Proof We only prove (i). Using Lemma 1.3 and Proposition 1.9 of [6], we have

O([a+,8,I')f) = [O(a+,B),o(')')]r - [O(a)

+

= [O(a),O(I')]r

+

= O([a,I'Jr) +^0([,8,I'Jr)

= O([a,l']f

+

THEOREM 1.8. (Jacobi identity) IfXI, X2andX3areco-Hopfspaces, then we have

[[aI, a2]r,a3Jr

+

+^(rT312 /\ lw)*([[a3,al]r,a2]r) = 0

for anyal E [fXI /\W, ZJ, a2 E [fX2 /\W, ZJ and a3 E [fX3 /\ W;Z]' whereTijk : Xl1\X2 /\X3 ----+Xi /\Xj /\Xkis the permutation of factors ofthe smash products.

Generalized Hardie-Jansen product and its dual

Proof. Using Lemma 1.1 and Proposition 1.10 of [6], we have () ([[ab_{a2]~, a3]~}

+

+(fT312 /\¹w)*([[a3,al]~, a2]~)) [()([aI, a2]f) , ()(a3)]r

+

+

[[B(ad,()(a2)]r, ()(a3)]r +^[[()(a2),()(a3)]r, ()(ad]r⁰ fT231

+

This completes the proof.

535

We define an inclusion map i :Z ^--t f*(fZ) byi(z)(t) =t /\ zfor any zE Z and t Ef. Let

f :[X, Z] ^----t [fX, fZ]

be the f -suspension map. The following lemma says that the inclusion mapi and the f-suspension map are essentially equivalent.

LEMMA 1.9. Let K,: [fX,fZ] ^--t [X,f*(fZ)] be theadjoint isomor- phism of f. Then the following diagram is commutative.

[X,Z] _ _r _ . [fX,fZ]

~

[X,f*(fZ)]

Proof. For any map a : X ^--t Z and any elements t E f, x E X, we have (K,(fa)(x))(t) = fa(t /\ x) = t /\ a(x) and (i*(a)(x))(t) =

((i⁰ a)(x))(t) = t /\ a(x). Hence we have the result.

THEOREM 1.10. Let f _l be a co-grouplike space and f ₂ a co-Hopf space. Then wehavef₂([a,,8]}\) = 0 foranyelementsa E [fIX /\W; Z]

and,8E [flY /\W;Z].

Proof. Leti :Z --Q(f2Z) be the inclusion map. By Lemma 1.9, we have

x:(f2[a,_,8]~) = ^{i 0} [a,_,8]~

= [i0 a,i0 ,8]~ in [fl(X1\Y) 1\W;f2^(f2Z)].

Since f;(f₂Z) is a Hopf space, the element [i0 a, i 0 ,8)~ vanishes by Theorem 1.4. Hence, we have the result.

2. Dual product

LettwI\r :W1\f -- f 1\W be the switching map. We define a natural homeomorphism tWl\r : f*(XW) ~ Xrl\w __ XWl\r ~ (f*X)W induced bytWl\r. And we define a map

",: [A,r*(Xw )] ^{- - - t} [A, (f*X)W]

by",(a) = tWAr oafor any mapa: A --f*(XW). We notice that", isa bijection of sets. We now show some formulas for the bijection ",.

LEMMA 2.1. Let a and ,8 be any elements in [A, f*(XW)). Then we have

Proof. We see that the map twI\r is a Ropf map. Then we have

",(a t,8). tWl\r⁰ (a t ,8)

= tWl\r0a t tWl\r0,8

= ",(a) t ",(,8).

PROPOSITION 2.2. Leta :A --f*(XW) be any map. Then we have (i) ",(f*(fW) ⁰ a) =^(f*f)w ⁰ ",(a) for any map f: X -- Y.

(ii) ",(a0 g) =^{",(a)0 9 for} any map9 : A' -- A.

Proof. (i) The following diagram is commutative.

f*(XW)~ ^(f*X)w

r*(fW)

1 1

f*(yW ) tWAf. (f*Y)W

Generalized Hardie-Jansen product and its dual 537 Then we have1](f*(fW)⁰a) = tWAr⁰f*(fW) ⁰a = (f*J)w^{0 tWAr 0}a = (f*J)wo1](a).

(ii) From the definition of1], we see1](a0g) =tWl\r 0 a09=1](a)0 g.

Letr : [A, f*(XW)] - [A/\ W,f*X] be the composition map of1]and (J-1, namely r = (J-1 01].

PROPOSITION 2.3. Letaand(3 be anyelements in[A,f*(XW)]. Then we have

r(a-t (3) = r(a) t r((3).

Proof From Lemmas 1.3 and 2.1, we have r(a t /3) = ((J-1⁰ 1])(a T (3)

(J-1(1](a) t 1]((3))

(J-1(^1](a)) t (J-1(1]((3)) r(a) -t r((3).

The homomorphismr is an isomorphism since1]and (Jare bijections.

We obtain the following formulas for r.

LEMMA 2.4. (i) Letf :X - Y be any map. Then there is a com- mutative diagram

[A, f*(Xw )] (r'(JW)),. [A, f*(Yw)]

Tl I T

[A1\W";f* X] (r*I),' [A /\W";f*Y].

Thereforewe havef*f ⁰ r(a) = r(f*(fW)⁰ a) for anya: A _ f*(XW).

(ii) Let 9 : A' - A be any map. Then there is a commutative diagram

[A, f*(XW)] ^{g' •} [A', f*(XW)]

Tl I T

[A^1\W";f* X] - ( g_A1w~)' [A' /\ W,f*X].

Thereforewe haver(a) ⁰ (g /\ 1w)= r(a⁰ g) for anya: A _ f*(XW).

Proof (i) From Lemma 1.2 and (i) of Proposition 2.2, the result fol- lows. (ii) From Lemma 1.1 and(ii) of Proposition 2.2, the result follows.

We recall the dual of the Hardie-Jansen product in [7]. It defines a pairing

[A, r*(Xw)] x [A, r*(Yw)] ^{- - - t} [A,r*((X~y)W)]

by [a,,B]~ = T-1([T(a),T(,B)]r*) ^E [A,r*((X~Y)W)] for any elements a E [A, r*(XW)]and,BE [A, r*(YW)],where [', . ]r-is the r*-Whitehead product and^X ~Y isthe homotopy fibre of the inclusion mapj :XVY ~

X xY. Now we give proofs to the following propositions. These results are quoted in [7] without proof.

THEOREM 2.5. Hr is a commutative co-Hopf space or if W or A is a co-Hopf space, then [a,,B]~= 0 for anya ^E [A, P(XW)] and ,B ^E

[A, r*(YW)].

Proof. Using Proposition 2.2 of [6], we have the result.

THEOREM 2.6. Let a E [A, r*(XW)] and ,B E [A, f*(YW)] be any elements. Then we have the following results.

(i) [a,,B]~^{0 ,}= [a o",B0 ,]~ for any, :A' ~ A.

(ii) (r*((8bc:)W)) 0 [a,,6}~ = [(r*(8W)) 0 a, (r*(c:W)) 0 ,6}~ for any 8 : X ~ X' and c :Y ~Y ' .

Proof. (i) Using Lemma 2.4(ii) and Proposition 2.1 (i) of [6], we have T([a,,B]~^{0 , )} [T(a) ,T(,B)]r* 0 ( ,tdw)

- [T(a) ⁰ (,1\ 1w),T(,B)⁰ (,1\ 1w)]r*

- [T(a0 , ) ,T(,Bo,)]r*

T([a o",B⁰ _,]~).

(ii) Using Lemma 2.4 (i) and Proposition 2.1 (ii) of [6], we have T((r*((8~c:)W))⁰ [a,,B]~) r*(8~c) ⁰ T([a,,B]~)

r*(8~E) ⁰ [T(a),T(,B)]r*

[r*8⁰ T(a),r*E⁰ T(,B)]r*

[T((r*(8^W))0 a), T((r*(c:w))⁰ ,B)]r*

T([(r*(8W))0 a, (r*(c:W))0 ,B]~).

Generalized Hardie-Jansen product and its dual 539

7((f*(T^W))0[a,(3]~)

THEOREM 2.7. Let T : ^{X DY} ~ ^{Y DX be} the switching map be- tween induced fibres. Leta ^E [A, f*(XW)] and(3^E [A, f*(YW)] beany elements. Then the product [a,(3]fY. E [A,f*((XDy)W)] satisfies

(f*(TW))*([a,(3]fY.) = -:- [{3,a]fY. .

Proof. Using Lemma2.4 (i) and Proposition2.4of [6],we have

- W

f*T0 7([a,(3]r') f*T0^[7(a),7({3)]r'

= -:- [7({3),7(a)]p 7( -:- [(3,a]~).

We define the evaluation map e : r(f*A) ~ A by e(t /\>..) = X(t) for any elements t ^E f and >..^E f*A. Let

f* :[A,X] ----+ [f*A,f* X]

be the f-Ioop map. Then the following lemma says that the evaluation map e and f-Ioop map are essentially equivalent.

LEMMA 2.8. Let ^{K :} [r(f*A),X] _~ [f*A,f* X] be the adjoint iso- morphism of thespacef. Then the followingdiagram is commutative.

[A,X] r' . [f*A,f*X]

~

[r(f*A),X]

Proof. For any mapa: A _~ X and any elements t E f, >..E f*A, we have

(K(e*(a)) (>..))(t) ^{e*(a)(t /\}>..) (a⁰ e)(t /\>..) a(e(t /\>..)) a(>..(t)) ((f*a)(>..))(t).

THEOREM 2.9. Let f ₁ be a co-grouplike space and r2 a co-Hopf space. Then we have q([a, ,8]r.) = 0 for any a ^E [A, fi(XW)J and

1

,8E [A, fi(yW)J.

Proof. Let K-1

: [qA,q(ri(XDy)w))J ^{- 7} [r2(qA) ,fi«XDy)W)J be the inverse map of the adjoint isomorphism ^K. By Lemma 2.8,

K-1(f;([a,,8]r.)) = [a,,8J~ 0 e= [a0e,,80e]r.

1 1 1

in [r2(qA),r;:«XDy)W)J. The element [a0 e,,80 eJr. vanishes by The-

1

orem 2.5, sincer2(r2A) is a co-Hopf space. This completes the proof.

3. Related isomorphisms ~WI\Xand (w*x

LettwI\X : W/\X ^---t X /\Wbe the switching map defined bytwI\X (w /\

x) = x/\wfor any elementsx E X andw EW,wherex/\wis an elements in X /\W. Itis a natural homeomorphism. We define

~wI\X :[rX /\W;Z] ---+ [rW /\X,Z]

by ~wl\x(a) = a ⁰ ftwl\X for ftwl\x : fW /\ X = r 1\W /\ X ^---t f /\X /\W = rx /\W. Then we can define the fW-Whitehead product

[~wl\x(a)'~Wl\y(,8)]rwE [fW /\(X1\Y),ZJ for any elementsa^E [rX1\

W, Z] and ,8 E [fY /\W;Z]. Then we obtain the following equality in Theorem 1.3 [7J;

[a, ,8]f,V= ~W~(Xl\y)([~wl\X(a), ~wl\y(,8)]rw).

Therefore we can use both isomorphisms () and~to study the properties of the product la,,8uY. Now we consider the properties of~wI\X in this section as is shown below.

PROPOSITION 3.1..Let a and,8be any elements in[fX1\W;Z]. Then wehave

~W!IX(a -+- ,8) = ~wl\X(a) -+- ~Wl\x(,8).

Proof. By the definition of~wI\X, we have

~wl\x(a-+- ,8) (a -+- ,8)⁰ rtWI\X a0 ftwl\X -+- ,80 rtwl\x

~wl\x(a)-+- ~Wl\x(,8).

Generalized Hardie-Jansen product and its dual 541 We notice that ~W!IX is an isomorphism of groups. We have the fol- lowing formula for the isomorphism~W!lX'

PROPOSITION 3.2. Let f :X ^{- 7} Y and9 : V ^{- 7} W be any maps.

Then the following diagram is commutative.

[fY /\W;Z] (r(f!lg))*. [fX /\V,Z]

{WAY

1 1

[fW /\1";Z] (r(g!lj)): [fV /\x,Z].

Therefore we have~W!ly(a) ⁰ r(g /\J) = ~v!IX(a⁰ f(f/\g)) forany map a : fY /\W ^{- 7}Z.

Proof We have

~v!IX(a⁰ f(f /\g)) a0 f(f /\g)0 ftv!IX

= a0 ftwJ\Y0 f(g /\J)

~W!ly(a)⁰ f(g /\J).

COROLLARY 3.3. H 9 = lw : W ^{- 7} W in Proposition 3.2, then we have

~W!ly(a) ⁰ fWf =~w!IX(a⁰ (ff /\lw)) for any mapa :fY /\W ^{- 7} Z.

From the results above, we can give other proofs to Theorems 1.5, 1.6, 1.7, 1.8 and 1.10, using the isomorphism ~ instead of O. But we omit them here.

Let<Px : f*(XW) ^{- 7} (fW)*X be the homeomorphism which is defined in Definition 6.2.35 of [5] or (2.9) of Chapter IIIof [8]. Now we define a map

(w*x : [A, f*(X^W)] ~ [A, (fW)*X]

by(w*x(a) =<Px0 a for any elementa E [A, f*(XW)].

PROPOSITION 3.4. Let a andj3be anyelements in [A,f*(XW)]. Then we have

Proof We see that <Px isa Hopf map. Thus we have Cw*x(a t f3) <Px⁰ (a t f3)

= <Px⁰ a t <Px⁰ f3

= ^Cw*x(a) t Cw*x(f3).

Ifwe are given any two maps 1 :^{X ---7} Y and 9 : V ^---7 W, then we define two maps Ig :X^{W ---7} Y v by f9(A) = lOA 0 9 for any element AE X W,and(rg)<*) f : (rW)*X ^--t (rV)*Yby«rg)(*)f)(X) = foNorg for any element X E (rW)* X. Then we have the following result.

PROPOSITION 3.5. Let f : X ^--t Y and 9 : V ^--t W be any maps.

Then the followingdiagram iscommutative.

[A,f*(XW)] (r*u

g

))*. [A,f*(Yv )]

(w*x

1 1

[A,(fW)* X] ()' [A,(fV)*Y]

«rg)^{* 1)*}

Therefore we have«rg)(*) f) ⁰ Cw*x(a) =Cv*y(f*(fg) ⁰ a) for any map

a : A^---7 f*(XW).

Proof The following diagramiscommutative.

f*(Xw )~ ^(fW)*X

f*(fg)

1 1

f*(Y^V) ~y ^I (rv)*Y Hence we have

((fg)(*)f) 0 <Px⁰ a

<Py0 f*(fg) 0 a Cv*y(f*(fg) 0 a).

COROLLARY 3.6. If9 = lw : W --t W in Proposition 3.5, then we have

(rw)* f 0 (w*x(a) =(w*y(f*(fW)0 a) for any mapa: A ^---7 f*(XW).