Associativity in two-cell complexes
Brayton Gray
Dept. of Mathematics, Statistics
and Computer Science (M/C 249)
University of Illinois at Chicago
851 South Morgan Street
Chicago, IL 60607-7045
brayton@uic.edu
1. The purpose of this paper is to examine the smash powers of a two-cell comp*
*lex
P = S2m-1 [ e2n when localized at primes grater than 3. We have two applicat*
*ions
in mind. We intend to introduce Samelson and Whitehead products into the homot*
*opy
groups with coefficient in P given by ssk(X; P ) = [k-2nP; X] for k 2n: Neisen*
*dorfer
did this in the case that n = m [N], and we generalize his results here. This *
*will be
applied in a forthcoming paper that generalizes the splitting theorems of [CMN]*
* for P .
The second purpose is to enhance the spectrum P0 = -2m+1 P = S0 [ e2n-2m+1 with
the structure of a homotopy associative, homotopy commutative ring spectrum. T*
*hus,
in particular, any homotopy associative homotopy commutative ring spectrum can *
*have
"coefficients" introduced by smashing with -2m+1 P and retain its ring structur*
*e. Finally
we will examine the Hurevicz homomorphism in this context.
The essence of these results lies in constructing a favorable splitting:
P ^ P ' 2m P _ 2n+1P ;
that such splittings exist when localized away from 2 has been known for some t*
*ime ([G2],
[K]); here we find that a careful choice of splitting allows us to handle the a*
*ssociativity
and commutativity of the resultant maps. Throughout this paper we will assume t*
*hat all
spaces are localized at a prime p > 2. From 2.5 onwards, we will assume that p *
*> 3.
2. In this section we construct the required splittings. Recall that S1 localiz*
*ed at p is not
a co-H space. However, if P is localized at p, P = S1 ^ P is still p local and *
*a co-H space
where S1 is the unlocalized circle.
Lemma 2.1. P is a co-commutative co-H space.
1
Proof: We will show that for any X the group structure defined on [P; X] is in *
*fact
Abelian. Our argument is strengthening the obvious generalization of [N; 9.7]. *
*We consider
two maps f; g : P ! X and compare their products fg and gf:
P ----! P x P -fxg---!X x X ----! X
P ----! P x P -fxg---!X x X ---o-! X x X ----! X
where o is the twist map. The two composites from P x P to X agree on P _ P . We
take their difference and factor it over P ^ P . It follows that the differenc*
*e between fg
and gf factors:
P ----! P ^ P ----! X:
Our task is complete when we show that : P -! (P ^ P ) is null-homotopic.
By cellular approximation, the map compresses through S2m-1 ^ S2m-1 uniquely *
*if
n > m, and this map factors uniquely over S2n. In other words, there is a uniq*
*ue map
fl : S2n -! S4m-2 so that the diagram:
P ----! P ^ P
? x
j?y ??i^i
S2n --fl--!S2m-1 ^ S2m-1
where i : S2m-1 -! P and j : P ! S2n are the maps from the cofibration sequenc*
*e.
Let o : P ^ P - ! P ^ P be the twist map. Since o = , ofl ~ fl; i.e., (-1) O f*
*l ~ fl.
Consequently fl : S2n+1 -! S4m-1 has order 2 and hence is null-homotopic. The *
*case
n = m is actually easier. The map fl may not be unique; however 2n 4n - 2 onl*
*y if
n = 1. In this case the degree of fl is determined by the cup product structur*
*e in H*P
and this is zero localized away from 2.
We now state the splitting result. The existence of the maps and * goes ba*
*ck to
[G2].
Theorem 2.2. Localized at a prime p > 2, there are unique co-H maps:
: P ^ P ----! 2m P
* : 2n+1P ----! P ^ P
such that
a) O (i ^ 1) ~ 1
b) (j ^ 1) O * ~ 1
c) O ~ *
2
d) O o ~ -
e)o O * ~
where o : P ^ P -! P ^ P is the twisting map.
The proof will reply on constructing self maps of P ^P which are idempotent*
*s as self
maps. This is stronger than knowing that their induced homomorphisms are idempo*
*tent,
but the proof is of the same flavor (see [C]).
Given an idempotent self map e : X -! X, we construct a telescope; i.e., th*
*e homo-
topy colimit
X ---e-! X ---e-! X ----! . . .
which we write as T (e) or T (X; e) and denote byie: : X -! T (e) the inclus*
*ion of the
first factor.
Lemma 2.3. There is a 1-1 correspondence between homotopy classes of maps g : T*
* (e) -!
Y and homotopy classes of maps h : X -! Y such that he ~ h given by composition*
* with
ie.
Proof: We first observe that [T (e); Y ] is the inverse limit of the diagram:
* e*
. . .----! [X; Y ]--e--! [X; Y ]----! [X; Y ];
i.e., there are no phantom maps. This follows since the Mittag-Leffler conditio*
*n holds since
e2 = e (see [G1]). However, given a sequence of maps gi : X -! Y with gie ~ gi-*
*1, we
calculate gi~ gi+1e = gi+1e2 ~ gie ~ gi-1. So gi= g for each i.
Corollary 2.4. Suppose W is another space. Then T (X ^ W; e ^ 1) ' T (X; e) ^ W*
* .
Proof: We use 2.3 and adjointness.
By 2.3, we can construct a unique map je : T (e) - ! X such that jeie ~ e a*
*nd
ieje ~ 1. It is also easy to see that if X is a co-H space and e : X -! X is a *
*co-H map,
then X(e) is a co-H space and ie; je are co-H maps.
Proof of 2.2: Since P is a co-commutative co-H space by 2.1, P ^ P is also. Thus
2 ^ 1 : P ^ P - ! P ^ P is a co-H map and a homotopy equivalence. Let us write
1_ 1_
2 : P ^ P -! P ^ P for the homotopy inverse. 2 is thus a co-H map. Now it is ea*
*sy to
see that the sum and difference of two co-H maps is again a co-H map since P ^ *
*P is co-
associative and co-commutative. Consequently e+ = 1_2O(1+o) and e- = 1_2O(1-o) *
*are
both co-H maps. Thus composition is distributive over addition and we get e+ O *
*e+ = e+ ,
e+ O e- = *, e- O e- = e- , and 1 = e+ + e- . Write X = T (e ), i = ee , j *
*= je .
3
Then j O i ~ e , i O j ~ 1, and i O j ~ * since e O e ~ *. Since 1 = e*
*+ + e- , we
have a splitting P ^ P ' X+ _ X- . Now write ae; oe for the compositions:
2m P -i^1---!P ^ P --i---! X-
X+ --j+--!P ^ P -j^1---!2n+1P
respectively. These are co-H maps since e are co-H maps. They are easily see*
*n to be
homotopy equivalences. So we define = ae-1i- and * = j+ O oe-1 ; the statement*
*s in 2.2
now easy to verify.
From this point forward we will assume that all spaces are localized at a p*
*rime p > 3.
Theorem 2.5. There are homotopy commutative diagrams:
2n*
P ^ P ^ P -^1---! 2m P ^ P 4n+1P ----! 2n+1P ^ P
?? ? ? ?
y1^ ?y2m-1 ?y2n* ?y1^*
2m-1 *^1
2m P ^ P -----! 4m-1 P 2n+1P ^ P ----! P ^ P ^ P
where an obvious shuffle of suspension coordinates is implicit in 1 ^ and 1 ^ *
**.
To prove this we need a further lemma. Suppose e; f : X -! X, ef = fe and e*
*2 = e.
Then by 2.3, f induces a unique map T (e) -f*!T (e) such that the diagram:
T (e)--f*--!T (e)
x? x
? ie ??ie
X ---f-! X
commutes up to homotopy. Furthermore, if f is also an idempotent, so are f* and*
* ef.
Lemma 2.6. There is a homotopy commutative diagram:
X --ief--! T (X; ef)
? x
ie?y ??OE
T (X; e)--if*--!T T (X; e); f*
where OE is a homotopy equivalence.
Proof: We easily construct OE and its inverse using 2,3. For example, there is *
*a unique
map : T (X; e) -! T (X; ef) with ie ~ ief and then . f* ~ by uniqueness. Sim*
*ilarly,
we have
4
Lemma 2.7. There is a homotopy commutative diagram:
T (X; e) --je--! X
x x
jf*?? ??jef
OE
T T (X; e); f*---- T (X; ef)
where OE is a homotopy equivalence.
This follows similarly to 2.6 using
Lemma 2.8. There is a 1-1 correspondence between homotopy classes of maps g : Y*
* !
T (e) and homotopy classes of maps h : Y ! X with eh ~ h given by composition w*
*ith je.
Proof: Let H(g) = jeg and G(h) = ieh. Then H G(h) = eh ~ h an G H(g) = g.
Proof of 2.5. We construct idempotent self maps of P ^ P ^ P by using the actio*
*n of
3. Let a = (12) and b = (123); then ab = (23) and aba = b2. Define e1 = 1_2O (1*
* - a),
e2 = 1_2O (1 - ab), and f = 1_3O (1 + b + b2). Here we interpret the elements o*
*f 3 as self
maps of P ^ P ^ P . As before, these are idempotents as is E = e1f = fe1 = e2f *
*= fe2.
Thus 2.6 applied twice gives a homotopy commutative diagram:
ie2
P ^ P ^ P ----! T (e2)
? ?
ie1?y OE2if*?y
T (e1) ----! T (E)
OE1if*
where OE1 and OE2 are equivalences.
By 2.4, T (e1) ' T (e- )^P ' 2m P ^P and T (e2) ' P ^T (e- ) ' P ^2m P ' 2m*
* P ^
P . Furthermore, under these equivalences ie1 corresponds to ^ 1 and ie2 corre*
*sponds to
1 ^ . To understand if* we must understand the effect of f* on T (e1) and T (e*
*2). On
T (e1), f* is determined by the composition
T (e1) -j1!P ^ P ^ P -f!P ^ P ^ P -i1!T (e1)
if we combine this with the equivalence:
2m P ^ P ' T (e- ) ^ P ' T (e1)
we obtain:
2m P ^ P -i^1-!P ^ P ^ P -e1!P ^ P ^ P -f!P ^ P ^ P -i1!T (e1):
5
However fe1 = -fe1ab = -fe1(23) so we have fe1 O (1 ^ o) = -fe1. Consequently t*
*he
composition 2m P ^ P - ! T (e1) -! T (E) factors through 2m-1 : 2m P ^ P - !
2m-1 X- . The same considerations apply to f* on T (e2) and we obtain a homoto*
*py
commutative diagram:
2m-1
P ^ P ^ P - 1^---!2m P ^ P -----! 2m-1 P
?? ? ?
y^1 ?y ?y2
2m-1
2m P ^ P -----! 4m-1 P ----! T (E)
1
An easy calculation for H* T (E) shows that 1 and 2 are homotopy equivalences.*
* Re-
stricting to S2m-1 ^ P ^ S2m-1 gives 1 ~ 2. Composing with -1 ~ -12gives the
result. The case of * is similar, using 2.7 and replacing e1 and e2 with e01= 1*
*_2O (1 + a),
e02= 1_2O (1 + ab).
3. In this section we will apply the pairings of section 2 to produce the Same*
*lson and
Whitehead products as well as the ring spectra structure. We begin with some no*
*tation.
Let P k= k-2nP for k 2n and Pi = i-2m+1 P for i 2m - 1. We can now define
maps i;jand *k;`as follows. Let i;j= (-1)j+1ci;jwhere ci;jis the composite:
i+j-4m+1
Pi^ Pj = i-2m+1 P ^ j-2m+1 P ' i+j-4m+2 P ^ P --------! i+j-2m+1 P = Pi+j
for i + j 4m - 1. Let *k;`be the composite:
k+`-4n-1*
pk+` = k+`-2nP ---------! k+`-4nP ^ P ' k-2nP ^ `-2nP ' P k^ P `
for k + ` 4n + 1. Write Ii: Si -! Pi and Jk : P k-! Sk for the appropriate sus*
*pensions
of the maps i; j from section 2.
Then we have
Theorem 3.1. Suppose i + j 4m - 1 and k + ` > 4n. Then there are maps
i;j: Pi^ Pj -! Pi+j
*k;`: P k+`-! P k^ P `
such that:
a)i;jO o = (-1)ijj;i
6
b) i;jO (Ii^ 1) ~ 1
c)i;j+kO (1 ^ j;k) ~ i+j;kO (i;j^ 1)
d) o O *k;`~ (-1)k`*`;k
e)(Jk ^ 1) O *k;`~ 1
f)(1 ^ *k;`) O *j;k+`~ (*j;k^ 1) O *j+k;`
Corollary 3.2. Let fflss2k(S0). Then the spectrum S0 [ e2k+1 has a unique ho*
*motopy
commutative homotopy associative ring spectra structure when localized at a pri*
*me p > 3.
Proof: This follows from 3.1 a; b and c.
We now use the maps *k;`to construct Samelson and Whitehead produces in hom*
*o-
topy with coefficients in P . We follow the methods of [N] which we generalize*
* to this
case and make mild improvements due to the uniqueness of our structure maps. I*
*n [N;
section 9], the author defines a space C to be co-abelian if the diagonal map C*
* -! C ^ C
is null-homotopic. This guarantees that [C,G] is an Abelian group whenever G is*
* a group
like space (a homotopy associative H space with homotopy inverse). However, as *
*we have
seen (2.1), it suffices to assume that : C -! C ^ C is null homotopic, and thi*
*s will
suffice for all applications.
Definition 3.3. A space C is quasi-co-Abelian if : C -! C ^ C is null homotopi*
*c.
Following [N], we define external Samelson products as follows. Let ff : C*
*1 -! G,
fi : C2 -! G. Then : C1 ^ C2 -! G is given by the composition
C1 ^ C2 ff^fi---!G ^ G -c!G
where c is an extension over the smash product of the group commutator map (a; *
*b) -!
aba-1b-1 defined in the homotopy category. Then [N, 9.10], in the quasi-co-Abel*
*ian context
yields:
Proposition 3.4. Suppose C1; C2 and C3 are quasi-co-Abelian. Let ff; ff0 2 [C*
*1; G],
fi; fi02 [C2; G], and fl 2 [C3; G]. Then the following formulas hold:
i) = -o***
ii) = +
ff ff ff
iii)ff; + b* fi; + (b2)* fl; = 0
7
where b = (123).
In particular, this applies when Ci = P kfor k 2n. We now apply this to co*
*nstruct
internal Samelson products. Let ssk(X; P ) = [P k; X]. This is an Abelian group*
* if k > 2n
or X is a group like space. If G is a group like space, ff 2 ssk(G; P ), fi 2 s*
*s`(G; P ) we define
[ff; fi] to be the composition:
*k;` k `
P k+`--! P ^ P ---! G:
Strictly speaking, this is not defined if k = ` = 2n. The suspension of the com*
*position is
defined, and we can recover [ff; fi] using the retraction G ! G given by the H *
*space
structure and adjointness. Thus we have the analogue of [N;10.1].
Proposition 3.5. Let ff; ff02 ssk(G; P ), fi; fi02 ss`(G; P ) and fl 2 ssq(G; P*
* ). Then:
i)[ff; fi] = (-1)k`+1[fi; ff]
ii)[ff + ff0; fi] = [ff; fi] + [ff0; fi]
iii)ff; [fi; fl] = [ff; fi]; fl + (-1)k` fi; [ff; fl]
Whitehead products can be defined, as usual, using adjointness. The reader *
*can easily
construct the definition and state the analogous result to 3.5 for Whitehead pr*
*oducts.
The map Jk : P k-! Sk induces a homomorphism of Lie algebras:
ss*(G) -! ss*(G; P ):
Consequently, there is a Samelson product:
ssk(G) ss`(G; P ) -! ssk+`(G; P )
expressing the Lie module structure of ss*(G; P ) over ss*(G). However, this S*
*amelson
product can be defined for all k 1, ` 2n directly and one easily proves
Proposition 3.6. Suppose ff 2 ssk(G), fi 2 ss`(G), fl 2 ssp(G; P ) and ffi 2 ss*
*q(G; P ) with
k; ` 1 and p; q 2n. Then the following identities hold:
a) ff; [fi; fl] = [ff; fi]; fl + (-1)k` fi; [ff; fl]
b) fi; [fl; ffi] = [fi; fl]; ffi + (-1)`p fl; [fi; ffi] .
4. In this section we will discuss the Bockstein homomorphism defined in homoto*
*py with
coefficients in P . Let oe = 2n - 2m + 1. Then we construct
fi : ssk(X; P ) -! ssk-oe(X; P )
which is defined when k 2n + oe = 4n - 2n + 1 and is a homomorphism when k >
4n - 2m + 1 or X is group like. Let Bk = Ik-oeJk-oe: P k-oe-! P k. Then fi is d*
*efined by
fi(f) = (-1)kf O Bk for f : P k-! X.
8
Lemma 4.1. *B4n+1 ' (i ^ 1) + (1 ^ i) O o : S2m ^ P -! P ^ P .
Proof: The right hand side is invariant under o, so it factors as * O OE where*
* OE :
P 2n+2m -! P 4m+1; applying (i ^ 1) to the equation gives OE ~ B4n+1.
Proposition 4.2. If u 2 ssk(X; P ) and v 2 ss`(X; P ) fi**__ = + (-1*
*)k____.
Proof: Suspending and keeping track of signs, we get *k;`O Bk+` = (1 ^ I`) O o *
*+ (-1)`O
Ik ^ 1 : P k+`-oe-! P k^ P `and hence
*k;`O Bk+` = (1 ^ B`) O *k;`-oe+ (-1)`(Bk ^ 1) O *k-oe;`
from which the result follows.
5. Let E be a commutative associative ring spectrum. Let P0 = 1-2m P = S0 [ *
*eoe.
Then by 3.2, EP = E ^ P0 is also a commutative, associative ring spectrum. We d*
*efine
EPr(X) = ssr(X ^ EP ) = eEr+2m-1(P ^ X). In this section we discuss the relati*
*onship
between ssr(X; P ) and gEPr(X).
Lemma 5.1. There is a long exact sequence:
. .-.!eEr-oe+1(X) -*!Eer(X) -i!gEPr(X) -@!eEr-oe(X) -! . . .
where *(x) = 0x where 02 Eoe-1is the composition Soe-1-! S0 -! E.
Corollary 5.2. Suppose 0= *(1) = 0 2 Eoe-1. Then EP*(X) ~=E*(X)^(eoe) as E*(X)
modules where ^(eoe) is the p-local exterior algebra on one generator eoeof dim*
*ension oe.
Proof: Since 0= 0, * = 0 and we have a short exact sequence:
0 -! eEr(X) -i!gEPr(X) -@!eEr-oe(X) -! 0
let eoebe such that @eoe= 1. Since EP* is a module over E*, we can define a rig*
*ht inverse
to @ by the action of eE*(X) on eoe. Consequently gEPr(X) ~=eEr(X) eoeeEr-oe(X*
*). Since
oe is odd, e2oe= 0 and we are done.
We now define a Hurewicz homomorphism. OE : ssr(X; P ) -! gEPr(X) for r 2n*
*. Let
fflr be the composition:
*2n;oe r
Sr+2m-1 -Ir+2m-1----!P r+2n---! P ^ P :
Let er = oer+2m-1 (1) 2 Eer+2m-1(Sr+2m-1 ), and ir = fflr*(er) 2 Eer+2m-1(P ^ P*
* r) =
gEPr(P r). We then define OE by OE(f) = f*(ir) for f : P r-! X. Clearly OE is w*
*ell defined
and a homomorphism for r > 2n.
Define fi : gEPk(X) -! gEPk-oe(X) to be the composition i@ from 5.1.
9
Proposition 5.3. fi OE(f) = -OE fi(f) for f : P k-! X with k 2n + oe.
Proof: This is a simple diagram chase using the definitions here and in section*
* 4.
It now follows that OE is a homomorphism of Bockstein spectral sequences. N*
*ote that if
n > m, OE is nilpotent so both the E-homology and homotopy Bockstein spectral s*
*equences
converge after a finite number of items (depending only on OE).
In case 0= *(1) = 0, we can introduce a Hurewicz homomorphism:
h : ssk(X; P ) -! eEk(X)
as follows. By 5.2 OE(f) = h(f) + eoeh0(f). Since fieoe= 1, we get h0(f) = fi*
*OE(f) =
-OE fi(f) = -h fi(f) = eoeh0 fi(f) . Consequently h0(f) = -h fi(f) and we ha*
*ve:
OE(f) = h(f) - eoeh fi(f) :
This definition depends on the choice of eoewith @eoe= 1. However in the case o*
*f ordinary
Z(p)homology, eoeis unique.
We now examine the multiplicative structure. We first define an external p*
*roduct
pairing in homotopy:
^_: ssk(X; P ) ss`(Y ; P ) -! ssk+`(X ^ Y : P )
by defining f ^_g to be the homotopy class of the composite:
*k;` k ` f^g
P k+`--! P ^ P --! X ^ Y:
With this definition, the Samelson product is given by = c*(ff ' fi) w*
*here c :
G ^ G -! G. is the commutator map for section 3.
Clearly E ^ P0 is a commutative and associative ring spectrum. consequently*
* there is
an external pairing in homology:
^_: gEPk(X) gEP`(Y ) -! gEPk+`(X ^ Y ):
Recalling the definition in [G3; section 2.3], we observe that this pairing is *
*represented by
(-1)r times the composition:
gEP)k(X) gEP`(Y ) =Eek+2m-1(X ^ P ) eE`+2m-1(Y ^ P )
_^_
-! eEk+`+4m-2(X ^ P ^ Y ^ P )
oe-!eE 1
k+`+4m-1(X ^ P ^ Y ^ P ^ S )
o-!E
k+`+4m-1 X ^ Y ^ (P ^ P )
(1^)*----!eE 2n+2m-1
k+`+4m-1(X ^ Y ^ P )
~=eEk+`+2m-1(X ^ Y ^ P ) = eEPk+`(X ^ Y ):
Proposition 5.4. OE(ff ^_fi) = OE(ff) ^_OEfi)
Proof: This is a straightforward diagram chase, using two applications of the f*
*ollowing.
10
Lemma 5.6. *a+b-oe;cO a;b+c' (a;b^ 1) O (1 ^ *b;c) as homotopy classes in [*
*P a^
P b+c; P a+b-oe^ P c].
Proof: It suffices to show that both composites are homotopic when precompos*
*ed with
both (Ia ^ 1) and *a;b+c. In the later case both composites are actually nul*
*l homotopic.
Theorem 5.7. In a group like space OE = OE(ff); OE(fi)
Proof:
OE = OE c*(ff^_fi)
= c*OE(ff^_fi)
= c* OE(ff)^_OE(fi) :
Now although E*(X) is not a coalgebra in general, it is still possible t*
*o define primitive
elements - elements 2 E*(X) such that *() = (i)*() + (i2)*() where ; i1; i2*
* : X -!
X x X are the diagonal and the two axial injections respectively. Clearly ev*
*ery element
in E*(P k) is primitive, and hence OE(ff) and OE(fi) are primitive. Now OE(*
*ff) x_OE(fi) 2
Ek+`(G x G). Let : G x G -! G ^ G. So * OE(ff) x_OE(fi) = OE(ff) ^_OE(fi).*
* Consequently
c* OE(ff) ^_OE(fi) = c** OEOE(ff) x_OE(fi) C is represented by the composite
G x G -x--! G x G x G x G -1xox1---!G x G x G x G -1x1xx-----!G x G x G x *
*G -! G
where is the inverse map. If is primitive, *() = - so one obtains c** OE(f*
*f) x_OE(fi) =
OE(ff)OE(fi) - (-1)|ff| |fi|OE(fi)OE(ff) = OE(ff); OE(fi) .
One can also easily check that if 0= 0, h = h(ff); h(fi) .
References
[C]F. Cohen, Splitting certain spaces via self maps, Ill. J. Math. 20 (1975*
*), 336-347.
[CMN] F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, Torsion in homotopy group*
*s, Ann of
Math. 109 (1979), 121-168.
[G1] B. Gray, Operations and a problem of Heller, thesis, University of Chica*
*go (1965).
[G2] B. Gray, Operations on two-cell complexes. Proceedings of the Conference*
* on Algebraic
Topology, UICC (1968), 61-68.
[G3] B. Gray, Homotopy Theory: An introduction to algebraic topology, Academi*
*c Press,
1975.
[K]W. Komornicki, Multiplication in two-cell complexes, thesis UIC, 1979.
[N]J.A. Neisendorfer, Primary Homotopy Theory Memoirs AMS #232 (1980).
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