lifting gottlieb sets and duality - American Mathematical Society

proceedings of the american mathematical society Volume 119, Number 4, December 1993

LIFTING GOTTLIEB SETS AND DUALITY YEON SOO YOON (Communicated by Frederick R. Cohen) Abstract. Let p : Ef —»X be a fibration induced by a map f : X —>Y from the path space fibration e : PY —>Y . Let g : A -* X be cyclic. When does g lift to a map A —► Ef which is cyclic? We give an answer of this question for arbitrary A and Y . Also, we give an answer in the dual situation.

1. Introduction A based map f : A —>X is cyclic [8] if there exists a map cj>: X x A ^ X such that cj)j is homotopic to V(l V/), where j : XvA —► XxA is the inclusion

and V : X V X -►X is the folding map. The Gottlieb set denoted G(A, X) is the set of all homotopy classes of cyclic maps from A to X. Dually, a based map / : X -> A is cocyclic [8] if there exists a map 6 : X ^ X WA such that jd is homotopic to (1 x /)A, where j : X V A —>X x A is the inclusion and A : X ^> X x X is the diagonal map. The dual Gottlieb set DG(X, A) is the set of all homotopy classes of cocyclic maps from X to A. In this paper we consider the following problem and its dual. Let /: X —>Y be a map and PY the space of paths in Y which begin at *. Let e : PY —>Y be the fibration given by e(n) = n(l). Let p : Ef —>X be the fibration induced by / from e, that is, Ef = {(x, n) e X x PY \ f(x) = e(n)} is the pullback of /: X -» 7 and e : PY —► Y. Let g : A —>X be cyclic, that is, there is a map :X x A —>X such that cf>j~ V(l V g), where j : X V A -» X x A is the inclusion. When does g lift to a map A —>Ef which is cyclic? This can be achieved, if there is a map : Ef x A -» Ef such that 4>\Ef~ 1e{ and tj)(p x 1) = /?.In case ^4 = 5" and Y is an Eilenberg-Mac Lane space, Gottlieb [1, Theorem 6.3], has given a necessary and sufficient condition for the existence of this 4> . In case A is arbitrary and Y is a product of Eilenberg-Mac Lane spaces, Halbhavi and Varadarajan [2] have given a necessary and sufficient condition for the existence of such a 4>. In case A is arbitrary and Y is an 77-space, Hoo [4] has also given a necessary and sufficient condition for the existence of such a .We can obtain a necessary and sufficient condition for the existence of such a with arbitrary A and Y, and we show that Hoo's necessary and sufficient condition Received by the editors May 29, 1991. 1991 Mathematics Subject Classification. Primary 55R05, 55P05. Key words and phrases. Cyclic maps, cocyclic maps, principal fibrations, principal cofibrations.

The author was partially supported by KOSEF 891-0103-008-2and TGRC during this research. © 1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page

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YEONSOOYOON

follows from ours. Our method is a modification of Haslam's which was used to study Postnikov systems and (/-spaces. We now consider the dual situation. Let / : X -> Y be a map and cX the reduced cone of X. There is a cofibration

X -U cX —>2ZX, where i(x) = [x, 1]. Let i : Y —>Cf be the cofibration induced by / from i, that is, Cf = cX II Y/i(x) ~ f(x) is the pushout of /: X —>Y and i : X —► cX. Let g : Y —>A be cocyclic, that is, there is a map 9 : Y -» YV.4 such that ;0 ~ (1 x g)A, where ;': YMA-» 7 x.4 is the inclusion. When does g extend to a map Cf —>A which is cocyclic? When there is a map

9 : Cf -> CfV A such that Pxj'9 ~ ley and (iv 1)(9= di, this can be achieved, where / : Cf V A —► Cf x A is the inclusion and px : Cf x A -» Cf is the projection. In case .4 is arbitrary and X is a wedge product of Moore spaces, Halbhavi and Varadarajan [2] have given a necessary and sufficient condition for the existence of this 9 . In case A is arbitrary and X is a co-77-space, Hoo [4] has given a necessary and sufficient condition for the existence of such a 6 . We can obtain a necessary and sufficient condition for the existence of such a 9 with arbitrary A and X, and we show that Hoo's necessary and sufficient condition follows from ours. All our spaces will be homotopy type of connected locally finite CW complexes. We assume also that spaces have nondegenerate base points. All homotopies and maps are to respect base points. The base point as well as the constant map will be denoted by *. Also, we denote by [X, Y] the set of homotopy classes of pointed maps X —>Y. The identity map of space will be denoted by 1 when it is clear from the context. ZX and QX denote the reduced suspension and the loop space of X respectively. The adjoint functor from the group [LX, Y] to the group [X, QY] will be denoted

by t. 2. Lifting

Gottlieb

sets

Let / : X —>Y be a map and PY the space of paths in Y which begin at *. Let e : PY —>Y be the fibration given by evaluating a path at its end point. Let p : Ef —»X be the fibration induced by / from e. That is, Ef is the pullback of / : X -» 7 and e : PY — Y ;

Ef -►

PY

d

-I

X —^

7

where Ef = {(x, n) 6 X x PY \ f(x) = e(n)},p(x, fibration p : Ef —>X is principal. The following two lemmas are standard.

n) = x.

In fact, the

Lemma 2.1. A map g : A -►X can be lifted to a map A -> Ef if and only if

fg~*Lemma 2.2 [3]. Given maps gk : Ak —>Ef, k = 1, 2, and g : Ax x A2 —>Ef satisfying pg\Ak ~ P8k, k = 1,2, then there is a map h : Ax x A2 —>Ef such that ph = pg and hlAk~ gk, k = 1, 2.

Theorem 2.3. Let g : A —>X be cyclic, that is, there is a map 4>'■X x A —>X such that j~ V(l V g), where j:X\/A^XxA

is the inclusion. Then there


LIFTING GOTTLIEBSETS AND DUALITY

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exists a map 4>:Ef x A —► Ef such that \Ef ~ l^y and the diagram

Ef x A ->

Ef

pxl

P

XxA

—£-» X

commutes if and only if ftf>(px 1) ~ *. Proof. If such a tf> exists, we have, from Lemma 2.1, that fcf>(p x 1) ~ *. Conversely, suppose f(px1) ~ * . By Lemma 2.1, there is a map ':EfxA —► Tsy such that /?(/>'= cp(p x 1). Then /70j£ = tf>(px l)\Ef ~ plEf. Thus

we have, from Lemma 2.2, that there is a map q\ : Ef x A -* Ef such that ptj) = p4>' = tb(p x 1), (j>\Ef~ lEf , and ^ ~ ',A . This proves the theorem.

Consider the following diagram where each square homotopy commutes and each column is the Puppe sequences of the cofibration:

Ef x A

1 Ef x cA

EfXcA/EfxA

-^-

XxA

| -!——»

X x cA

—p—> XxcA/XxA

"[

Z{Ef x A)

I

\.

BEflX

I(X x A)

-^

IX -^

17

where p is induced by p x 1. Corollary 2.4 [4, Theorem 1]. Let 7 be an H-space and g : A —>X cyclic, that is, there is a map : X x A —>X such that XxA is the inclusion. Then there exists a map cf>:Ef x A —► Ef

such that 4>\E~ lEf and p§ = 4>(px 1) if and only if L(f)qp ~ *. Proof. From Theorem 2.3, it is sufficient to show that L(f)qp ~ * if and

only if f(p(p x 1) ~ *. If f(px 1) ~ *, then 2Z(f(j))qp ~ 2Z(f(f)(p x l))q ~ * . Suppose ~L(f4>)qp~ *. Since Ef x A —+ Ef x cA —► Ef x cA/Ef x A is a cofibration, there is an exact sequence —>[L(Ef x cA), X7]

[L(Ef x A),1Y] £ [Ef x A/Ef x A, IY] -.

(E(lxi))*

—►

Since q*(l(f(f>(p x 1))) ~

^(f)qP~ * and Ker^r* = Im(Z(l x i))*, there is a map k : L(Ef x cA) -> L7 such that k\x(EfXA)= L(f(px 1)). Taking adjoints, we get a map Ef x cA —> QE7 extending e'ftf)(p x 1), where e' = T(lj;y). Since 7 is an 77-space, there is a map r : QX7 -»• 7 such that re' ~ ly. Hence we have a map

77 : EfxAxI -» 7 satisfying 77( , 0) = f(j>(px1) and 77( , , 1) = fppx ,where px: Ef x A —>Ef is the projection. From Lemma 2.1, 77( , , 1) = fppx ~ * .

Thus f(px 1) ~ * . License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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YEONSOOYOON

We can also obtain the following proposition . The proof of Proposition 2.5 is similar to that of Theorem 2.3. So we will omit the proof.

Proposition 2.5. Let g : A -» Ef be a map such that pg : A —>X is cyclic,

that is, there is a map tf>: X x A —>X such that j~ V(l V pg), where j:X\lA^>XxA

is the inclusion. Then there exists a map 4>: Ef x A —► Ef

such that \A ~ g, and p4>= 4>(px 1) if and only if f(h(p x 1) ~ *. Corollary 2.6. Let [A, X] = 0. Then, for any map f : X —>7, [A, Ef] = G(A,Ef). Proof. Consider the map (px 1) = ppx , where px : Ef x A —»Ef is the projection. That is, (px 1) : Ef x A —>X can be lifted to the map px : Ef x A —>Ef. Thus we know, from

Lemma 2.1, that ftj>(px 1) ~ *. Let [g] e [A, Ef]. Since p*([g]) e [A, X] = 0, pg ~ *. Thus 4>j~ V(l V *) ~ V(l Vpg). From the fact f(px 1) ~ * and Proposition 2.5, [g] e G(A, Ef). Remark 2.7. Consider the map ip : Ef x Q7 —► Ef given by y/((x, n), co) = (x, n + co), where + denotes the usual product of two paths. Then the inclusion /: Q7 —► Ef is cyclic. Thus the induced map /, maps [A, Q7] into G(A, Ef) [8]. Therefore, Corollary 2.6 can also be obtained from the long exact sequence of homotopy sets for the fibration £IY —>Ef —► X.

3. Extending We now consider cX —*LX, where reduced suspension. Cf —► LX induced

dual Gottlieb

sets

the dual situation. There is a well-known cofibration X -^ i(x) = [x, 1], cX is the reduced cone, and LX is the Given a map / : X —>7, consider the cofibration 7 -U by / : X —► 7 from i. That is, Cf is the pushout of

/: X - y and i : X - cX:

X —^

7

cX -►

Cf

where Cf is the mapping cone of /,

the space obtained from cX U 7 by

identifying [x, 1] G cX with f(x), i(y) = y. In fact, the cofibration 7 -U Cf —*LX is a principal cofibration [7]. Thus there is a map ([x,/]) = ((x, 2t), *), 0 < t < 1/2, tj>([x,t]) = (*,[x,2t-l]),l/2:Cf —>LX v Cf

is given by Ak such that gk ~ V(7t vpkjg), k = 1, 2. Let y = (yx V y2)p :LX ^ AXVA2 ,where p : IX -» ZXVlX is the co-77-structure. Consider the map * - V(yVg):C/ -+ ^i VA2. Then /»' = gi follows from the fact /= (1 V i) 1*2 . Moreover, Pkjh = pkJV(y V g)(f>= V(pk Vpk)(j V 7)(y V g)
Cf is the projection.

Proof. If such a 9 exists, we have, from Lemma 3.1, that (i v 1)9f ~ *. Conversely, suppose (i v 1)0/ ~ *. By Lemma 3.1, there is a map 9' : Cf ->

Cy V,4 such that 0'/ = (i V 1)0 . Then pxj'9'i = pxj'(i V 1)0 = p{(i x l)j9 ~ Px(i x g)A - lcfi ■ Thus we have, from Lemma 3.3, that there is a map 0 :

Cf^CfVA

such that Oi = 9'i = (i V 1)0, p,/0 ~ 1C/, and p2/0 ~ PiJ'B'.

This proves the theorem. Consider the following diagram where each square homotopy commutes and


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each column is the Puppe sequence of the fibration: IV 1

CfVA

i-

YVA

1V£

CfWPY

1V£

^—

] F2

YWPA

_ I CfxA

is the inclusion and

Px : Cf x A —»Cf is the projection. Proof. From Theorem 3.4, it is sufficient to show that iq£l(9f) ~ * if and only

if (i V 1)0/ ~ * . If (i V 1)0/ ~ * , then lqQ(9f) ~ qQ((i V 1)0/) ~ * . Now suppose iqQ(9f) ~ * . Since F2 —>CfWPA —* CfVA is a fibration, there is an

exact sequence -» [£2X, £2(CyV TM)] "(-^* [£2*, £2(CyV A)] &>[QX, F2] -»

[£2X,CyV7M] -.

Since g,(fi((iv

1)0/)) = itf£2(0/) ~ * and Ker^ =

Imfi(l V e),, there is a map p : QX -> £2(Cy V TM) such that £2(1 V e)/? ~ £2((/V 1)0/). Since A" is a co-77-space, there is a map Sx '■X —► L£IX such that exsx ~ lx , where ex = ?~x(lax) ■ Let v = t~x(B)s : X —>Cf V PA . Since T-'Off) = e{Cf\fPA)^fi and £2e(C/v^)£2I)3 = P&ex, £2f = £l(e{Cfs/PA)Z0sx)=

pn(exsx) ~ £ • Thus £2((zVl)0/) ~ £2(lVe)jff~ £2((lve)t/) : £2* -* £2(CyV^). Since X is a co-77-space, the function £2 : [X, Cf V A] -> [£2X, £2(Cyv A)], given by / h> £2/, is injective. Thus (/ V 1)0/ ~ (1 V e)u . Let /: Cy V TM -► (Cf V ^)7 be given by l(z, *) = constant path at (z, *) and by /(*, n) = i2n, where i2 : A —*Cf V A is the inclusion. Consider the map * = If : X -» (Cf V A)1. Then * : X -» (Cy V ^)7 gives rise to a map 77 : X x I -» CyV /I

with 77(x, 0 = *(*)(>)• Then H( , 0) = (1 V *)v and 77( , 1) = (1 Mtf)v, where * : PA —*A is the constant map. Thus (i V 1)0/ ~ (1 V e)v ~ (1 v *)v =

i\Pd(l V e)v ~ itPii(» V 1)0/ ~ ii//, where iy: Cf-* CfV A, j : CfV A-* Cf x A are natural inclusions and pi : Cy x ^ —>Cf is the projection. Thus we know, from Lemma 3.1, that (i V 1)0/ ~ * .

We can also obtain the following proposition. The proof of Proposition 3.6 is similar to that of Theorem 3.4. So we will omit the proof.


LIFTING GOTTLIEB SETS AND DUALITY

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Proposition 3.6. Let g : Cf —>A be a map such that gi: 7 —>A is cocyclic, that

is, there is a map 9 : 7 —>7 V A such that j9 ~ (1 x gi)A, where j : 7v A —> 7 x A is the inclusion. Then there exists a map 9 : Cf —>CfV A such that

Pxj'9 ~ ley, P2J'9 ~ g, and 91 = (I V 1)0 if and only if (i V 1)0/ ~ *, where j': CfV A -» Cf x A is the inclusion and px : Cf x A —>Cf, p2 : Cf x A -> A are projections.

Corollary 3.7. Let [7, A] = 0. Then, for any map f : X -» Y,[Cf,A] = DG(Cf,A). Proof. Let 0 : 7 — 7 V A be given by 9(y) = (y, *). Then (j V 1)0 = ixi, where ix : Cf -> CfV A is the inclusion. That is, (j v 1)0 : 7 —>Cy \M can be extended to the map ix : Cf —► Cy V .4. Thus we know, from Lemma 3.1, that

(i V 1)0/ ~ * . Let [g] e [Cf, A]. Since i*([g]) e [7, A] = 0, gi ~ * . Thus 70 ~ (1 x *)A ~ (1 x gi)A and gi is cocyclic. From the fact (i V 1)0/ ~ * and Proposition 3.6, [g] e DG(Cf, A). Remark 3.8. Consider the map : Cf -> LX V Cy given by 4>(y)= (*,y), ([x,t])= ((x,2t),*),0