On the homotopy groups of 2-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
0. The homotopy groups of CW complexes are, in general, much more mysterious th*
*an
the stable homotopy groups. A notable exception is the case of spheres or cases*
* when the
unstable Adams spectral sequence can be utilized. The problem is clearest in th*
*e case of
a 2-cell complex Sk [ en. Very little knowledge of such spaces was known before*
* the work
of Cohen, Moore, and Neisendorfer who analyzed the case of the Moore space Sk [*
*prek+1
for p an odd prime ([CMN]). Their work gave a clear understanding of the kinds *
*of things
that can occur, and the depth of their analysis was demonstrated by their deter*
*mination
of the exponent of the homotopy groups of Moore spaces when p > 2. Our purpose*
* is
to discuss the homotopy groups of two cell complexes in case the attaching map *
*is an
arbitrary element in an even stem. In some cases we will have results as strong*
* as those
in [CMN].
Throughout this paper all spaces will be localized at a prime p > 2. Let :*
* S2n-1 -!
S2m-1 and write P = S2m-1 [ e2n, P r= Sr-2nP for r 2n, and oe = 2n - 2m + 1.
In section 1, we deal with quite a general decomposition theorem for spaces*
* of the
form S(X [OEe2n-1). There are two applications
Corollary 1.3. Suppose S2n+1 6~ *. Then there is a homotopy equivalence:
+
P 2n+2' FS2 x (P 4n+3^ S2m+1 )
where FS2 is the fiber of S2 : S2n+1 -! S2m+1 .
This generalizes [CMN;1.1]. Another application gives
Corollary 1.4. Suppose p > 3. Let V (1)2n be a space stably homotopy equivalent*
* to the
Smith-Toda complex S2nV (1), and let V (3=4)2n be the 2n + 2p - 1 skeleton of S*
*2nV (1).
Then
V (1)2n ' F x S2n+2p-1V (1)2n ^ SV (3=4)+2n
1
where F is the fiber of the attaching map of the top cell of V (1)2n:
S2n+2p-1 -! V (3=4)2n:
This has application to the program in [G6].
Section 2 is preparatory for the case of P 2n+1. In this section we establi*
*sh a Bock-
stein homomorphism in H*(P 2n+1; Z(p)) corresponding to the homotopy Bockstein *
*in
ss*( ; P ).
In section 3 we prove
W
Theorem 3.2. Suppose = p' and p > 3. Then P 2n+1' T x W where W = P nff
ff
with nff 2m + 2n and T belongs to a fibration sequence:
Y i
2S2n+1 -! S2m-1 x S2np -oe{p} -! T -! S2n+1:
i1
We also prove
Theorem 3.3. Suppose S2n(p-3)+4m-2 ~ *. Then there is a fibration sequence
2S2n+1 ---ss-!S2m-1 ----! T1 ----! S2n+1
fi
where ssfiS2n-1~Q and E2ss ~ 2S2 : 2S2n+1 -!W2S2m+1 . Furthermore, P 2n+1'
T1 x W x (S2npi-oex S2npi-2oe+1) with W ' P nffwhere nff 2n + 2m.
i1 ff
Finally in section 4 we show that the decomposition in 3.2 does not hold in*
* general -
in particular, in case = fi1.
1. In this section we will give a splitting theorem for SY where Y ^Y has a*
* splitting
property. This is a property enjoyed by the Spanier-Whitehead duals of ring spe*
*ctra and
applies to two interesting cases: P 2n+2and V (1)2n+2p (the 2n + 2p dimensional*
* version of
the Smith-Toda comples V (1)).
Definition 1.1. A map p : Y - ! Sk will be called a co-unit if p ^ 1 : Y ^ Y - *
*! Sk ^ Y
has a right homotopy inverse q.
2
Theorem 1.2. Suppose S2n(X [OEe2n-1) is p-complete and atomic, and the projecti*
*on
p : X [OEe2n-1 -! S2n-1
is a co-unit. Then there are fibration sequences:
FSOE ----! S2n-1 --SOE--!SX
S2n-1 ----! SX _ S2n(X [OEe2n-1) ----! S(X [OEe2n-1)
FSOE --*--! S2n(X [OEe2n-1) ^ SX+ --ad--!S(X [OEe2n-1)
inducing a homotopy equivalence:
2n 2n-1 +
S(X [OEe2n-1) ' F x S (X [OEe ) ^ SX
1W
where ad = adi : S2n(X [OEe2n-1) ^ X(i)-! S(X [OEe2n-1) with adi = [; adi-1]*
* and
i=0
ad0 = [; ] O Sq where q is the right homotopy inverse to p ^ 1.
Corollary 1.3. Suppose : S2n-1 -! S2m-1 is such that S2n+1 6~ *. Then there i*
*s a
homotopy equivalence:
+
P 2n+2' FS2 x (P 4n+3^ S2m+1 ):
This generalizes the first decomposition theorem in [CMN].
Let us write V (1)2n = S2n [p e2n+1 [ff1e2n+2p-1 [p e2n+2p for the space wh*
*ose
suspension spectrum is S2nV (1) for n 1. Likewise, write V (3=4)2n for the 2n *
*+ 2p - 1
skeleton of V (1)2n. Then if p > 3, the projection V (1)2n -! S2n+2p is a co-un*
*it, so we
get
Corollary 1.4. Suppose p > 3. Then there is a homotopy equivalence:
2n+2p-1 +
V (1)2n ' F x S V (1)2n ^ SV (3=4)n
where F is the fiber of the attaching map:
S2n+2p-1 -! V (3=4)2n:
This result has an application of the program developed in [G6].
3
The proof of these results depends on constructing the fibration sequences *
*in 1.2.
To this end we recall the clutching constructions of [G5].fiGiven a Hurewicz fi*
*bration
F -! E -ss!X, a subspace A X, and a trivialization of ssfiA, there is a quasi*
* fibering:
F -! E -ss!X [ CA
where a ffi
E = E F x CA (f; a; 0) ~ (f; a):
fi
Here : F x A -! E is a trivialization of ssfiA. A classical example when X = C*
*A and
A = F ' E is the Hopf construction defined by : A x A -! A when A is an H-space
(see [S; 1.3, 1.4]).
Localized away from 2, there is a multiplication
n : S2n-1 x S2n-1 -! S2n-1
giving a well known fibration
S2n-1 ----! S4n-1 --hn--!S2n:
Lemma 1.5. 2hn ~ [n; n].
Proof: The multiplication n is the restriction of the multiplication on 2S2n+1 *
*to S2n-1.
Consequently there is a commutative diagram of Hopf constructions:
S4n-1 ----! 2S2n+1 * 2S2n+1
?? ?
yhn ?yh0
S2n ----! S2S2n+1
however h0 is the restruction of the classifying space construction for 2S2n+1,*
* so
0
2S2n+1 * 2S2n+1 --h--! S2S2n+1 ----! S2n+1
is null homotopic. It follows that
S4n-1 --hn--!S2n ----! S2n+1
is nullhomotopic, so hn = k[n; n]. Applying the second Hopf invariant we get k *
*= 1=2.
More generally, we may use the map OE : S2n-1 -! X to construct:
: S2n-1 x S2n-1 -! X x S2n-1
by (a; b) = OE(a); n(a; b) . is a trivialization of the trivial bundle X x S2*
*n-1 -! X
pulled back by OE, and we get
4
Proposition 1.6. There is a quasi fibering:
S2n-1 - ---! E ----! X [OEe2n-1
with E = X x S2n-1 [ B2n x S2n-1 where S2n-1 x S2n-1 B2n x S2n-1 is identifi*
*ed
with it's image under : S2n-1 x S2n-1 -! X x S2n-1.
Moreover this fibering is natural in X and is the Hopf construction in case*
* X = *.
Proposition 1.7. There is a cofibration sequence:
X -! E -j!S2n-1 ^ (X [OEe2n)
which is natural in X.
Proof: E is the push out of the diagram:
X x S2n-1
x?
?
S2n-1 x S2n-1 ----! B2n x S2n-1:
We will show that this is equivalent to another push out diagram. Let : S2n-1x*
*S2n-1 -!
S2n-1 x S2n-1 be the homotopy equivalence given by (a; b) = a; n(a; b) . Then *
*there
is a homotopy commutative diagram:
X x S2n-1 --F-- S2n-1 x S2n-1 ----! B2n x S2n-1
?? ? ?
y= ?y ?yss2
X x S2n-1 -OEx1---S2n-1 x S2n-1---ff-! S2n-1
where ff is the second component of -1. It follows that E is homotopy equivale*
*nt to the
push out of the diagram:
X x S2n-1
x?
?OEx1
S2n-1 x S2n-1 --ff--!S2n-1:
Clearly ff has degrees -1 and +1 on the axes. Therefore there is a cofibration *
*sequence of
push out diagrams:
5
X _ S2n-1
x?
?OE_1
S2n-1 _ S2n-1 --1_1---!S2n-1
??
??
y
X x S2n-1
x?
?OEx1
S2n-1 x S2n-1 ---ff-!S2n-1
??
??
y
X ^ X2n-1
x?
?OE^1
S2n-1 ^ S2n-1 ----! *
This gives a cofibration sequence of the pushouts, which is the conclusion of 1*
*.7.
Proposition 1.8. Suppose X = SX0 and OE = sOE0 where OE0 : S2n-2 -! X0. Then th*
*ere
is a map w : S(X0[ e2n-1) ^ (X0[ e2n-1) -! E so that:
ss Ow ~ [; ] : S(X0[e2n-1)^(X0[e2n-1) -! S(X0[e2n-1) and S2n-1pOjOw ~ 2S(p^p)
Proof: Since the Whitehead product is natural, there is a commutative diagram:
S(X0[ e2n-1) ^ (X0[ e2n-1) --[;]--!S(X0[ e2n-1)
?? ?
y S(p^p) ?ySp
S4n-1 --[;]--! S2n:
On the other hand, there is a pull back diagram:
2n-1pOj
E S------! S4n-1
?? ?
yss ?yhn
S(X0[ e2n-1) --Sp--! S2n
so we may lift the Whitehead product map on S(X0[ e2n-1) through ss to E .
6
Proof of 1.2: Replacing X0 with X and OE0 with OE, we show that the cofibration*
* sequence
(2.4):
SX -! E -! S2n(X [OEe2n-1)
splits and E ' SX _ S(X [OEe2n-1). To obtain the splitting, we consider the co*
*mposite
: S2n ^ (X [OEe2n-1) -Sq!S(X [ e2n-1) ^ (X [ e2n-1) -w!E
where q is the right inverse to p ^ 1. We have
S2n ^ (X [OEe2n-1)----! E
?? ?
y2S2np j?y
S4n-1 ---- S2n(X [OEe2n-1):
S2np
Now since p is a count, p* : H2n-1(S2n-1; Z=p) ! H2n-1(X [OEe2n-1; Z=p) is a mo*
*nomor-
phism; it follows that the self map j : S2n(X [ e2n-1) ! S2n(X [ e2n-1) is not *
*topologi-
cally nilpotent. Since S2n(X [ e2n-1) is atomic it follows from the results in *
*[AK] that j
is an equivalence. Thus we have a homotopy equivalence:
SX _ S2n(X [OEe2n-1) -_-! E :
We now construct the diagram of fibrations:
S(X [OEe2n-1) ----! F - ---! S2n(X [OEe2n-1) ^ SX+ ----! S(X [OEe2n-*
*1)
?? ? ? ?
y1 ?y ?y ?y1
S(X [OEe2n-1) ----! S2n-1 - ---! SX _ S2n(X [OEe2n-1) ----! S(X [OEe2n-*
*1)
?? ? ? ?
y* ?y ?y ?y*
* ----! SX - ---! SX ----! *:
Clearly there is a right homotopy inverse for S(X [OEe2n-1) ! F , so S(X [ OEe2*
*n-1) '
F x S2n(X [OEe2n-1) ^ SX+ . Here we use the fibering [G2]
1_
SB ^ A(i)' SB ^ SA+ -f!SA _ SB -! SA
i=0
where f = _fi and SB ^ A(i)fi-!SA _ SB is adi(A)(B).
7
2. In order to decompose SP , we need to construct a Bockstein homomorphism*
* in
homology which is compatible with the homotopy Bockstein
fi : ssr(X; P ) -! ssr-oe(X; P )
under the Hurewicz homomorphism. Such a homology operation cannot be defined a*
*nd
natural for all spaces.
W
Definition 2.1. A space X is P -free if there is a homotopy equivalence X ' P*
* nff. A
ff
space is stably P -free if it has the stable homotopy type of a P -free space.
Theorem 2.2. There is a unique Bockstein operation:
bp: eHr(X; Z=p) -! eHr-oe(X; Z=p)
defined when X is stably P -free and natural under continuous maps such that
a)fbi: eHr(P ; Z=p) -! eHr-oe(P ; Z=p) is a isomorphism for all r
b) fbis = sbfiwhere oe is the homology suspension
c)kerfbi= imbfi
d) if r - oe 2n, there is a commutative diagram:
ssr(X; P )---fi-! ssr-oe(X; P )
?? ?
yh ?yh
eHr(X : Z=p)---bfi-!Hr-oe(X; Z=p):
Proof: It is clear what the intention is. The problem is with naturality, and i*
*nWparticular,
to show that the definition does not depend on the choice of an equivalence X '*
* P nff.
*
* ff
Let u 2 H2m-1 (P ; Z(p)) and v 2 H2n(P ; Z(p)) be homology generators. Then*
* define
uk = Sk-2m+1 u 2 Hk(P k+oe; Z(p)) and vk = Sk-2nv 2 Hk(P k; Z(p)). With this we*
* define
Hurewicz homomorphisms when r 2n:
h : ssr(X; P-)! Hr(X; Z=p)
h0: ssr(X; P-)! Hr-oe(X; Z=p)
by h(f) = f*(vr), h0(f) = f*(ur-oe) for f : P r-! X (see [G7]). If r - oe 2n, *
*h0 = hfi
where
fi : ssr(X; P ) -! ssr-oe(X; P )
is the homotopy Bockstein.
8
Lemma 2.3. Suppose X is P -free. Then
a)h is onto
b) kerh kerh0.
Proof: Clearly h : ssr(P nff; P ) -! eHr(P nff; Z=p) is onto if r = nff; it is *
*also onto in case
r = nff- oe if r 2n. Consequently, if X is P -free, a) follows. Now suppose *
*h(f) = 0,
where f : P r-! X. Since P ris compact, f(P r) P ni_ . ._.P nk. Then
X
h0(f) = aiuni-oe+ bivni:
If bj 6= 0 for some j, the composite
P r-! P n1_ . ._.P nk-! Snj
would be non zero in homology and r - oe = nj; it follows that bj = 0, so bj 0*
*(mod p).
Now suppose that aj 6= 0 for some j. Then the composite
P r-! P n1_ . ._.P nk-! P nj
is non zero in homology in dimension r - oe = nj - oe. However since h(f) = 0, *
*the degree
on the top cell is divisible by p. It follows that the degree on the bottom cel*
*l, aj, is also
divisible by p.
Proof of 2.2. We first deal with the case that X is P -free. In this case Lemma*
* 2.3 implies
that we can define bfisuch that h0= bfih. Since h is onto, bfiis unique, natura*
*l, and commutes
with s. Condition d) follows since if r -oe 2n, h0(f) = h(fif). Condition a) i*
*s immediate.
To show that bfi2= 0, note that bfi= 0 if r < 2n. Suppose u 2 eHr(X; Z=p) and r*
* - oe 2n.
Let u = h(f); then bfi(u) = h0(f) = h fi(f) . Thus bfi2(u) = h0 fi(f) = 0. S*
*ince the
homology of eH*(X; Z=p) under bfiis the direct sum of the homology of eHr(P nff*
*; Z=p), it is
zero and c) holds. In case X is stably P -free, we define bfias the composite:
Her(X; Z=p) -si!eH (SiX; Z=p) -bfi!eH (SiX; Z=p) -si eH (X; Z=p):
~= r+i r+i-oe ~= r-oe
Clearly this does not depend on i and is compatible under the Hurewicz homomorp*
*hism
with the homotopy Bockstein. The other conditions are immediate.
Proposition 2.4. The category of stably P -free spaces is closed under Cartesi*
*an and
smash products and retracts. If X is P -free, SX is stably P -free.
9
Proof: The only difficult part is to show that a retract of a P -free space is *
*P -free. This
follows from
Lemma 2.5. Suppose for a given space X, h is onto, kerh kerh0, and bfi= h0h-1
satisfies kerbfi= im bfi. Then X is P -free.
Proof: Choose a basis {vff1uff} for bH*(X; Z=p) with bfi(vff) = uff. Choose fff*
*: P nff-! X
with h(fff) = vff. Define _
f : P nff-! X
ff
fi
by ffiPnff= fff. Then f* is an isomorphism.
Proposition 2.6. Suppose X and Y are stably P -free and u v 2 bH*(X x Y ; Z=p)*
*. Then
bfi(u v) = bfi(u) v + (-1)|u|u bfi(v).
Proof: It suffice to prove this formula in case uv 2 eH*(X ^Y : Z=p). We may as*
*sume that
X and Y are P -free. In this case h is onto so the formula follows from the cor*
*responding
formula in homotopy with coefficients in P .
3. In this section we will discuss the decomposition of SP . Our main tool *
*will be the
method of [CMN].fWeiwill use the results of [G7] to obtain an identical splitti*
*ng to that
in [CMN] when pfi. We will also obtain a splitting generalizing [G4] when is u*
*nstable
and discuss the obstructions in the general case.
Recall from [G7] that if G is a group like space there is a Samelson produc*
*t pairing
[ ; ] : ssk(G; P ) ss`(G; P ) -! ssk+`(G; P )
when k; ` 2n satisfying the usual conditions for a graded Lie algebra except t*
*hat for the
Jacobi identity we require p > 3. Furthermore ss*(G; P ) is a Lie module over s*
*s*(G). There
is a Bockstein homomorphism:
fi : ssk(X; P ) -! ssk-oe(X; P )
defined when k - oe 2n which is a derivation with respect to the Samelson prod*
*uct.
Finally, there is a Hurewicz homomorphism
h : ssk(X; P ) -! eHk(X; Z(p))
and h [x; y] = h(x); h(y) , the graded commutator in the ring H*(X; Z(p)).
The following result is an adaption of the results of [CMN] using the resul*
*ts of section
2 and the above.
10
Theorem 3.1. There is a commutative diagram of fibrations:
2Sn+1 ---@-! V ----! T2m-1 - ---! S2n+1
?? ? ? ?
y ?y *?y ?y
* ----! W --1--! W - ---! *
?? ? ? ?
y ?y ?yf ?y
S2n+1 ----! F --r--! SP - ---! S2n+1
with the following properties:
(a)f has a left homotopy inverse, so
SP ' W x T2m-1
F ' W x V
(b)H*(F ; Z(p)) ' T (xi; 2n(i - 1) + 2m - 1; i 1) where T is a tensor alg*
*ebra on
the generators xi and r*(xi) = adi-1(v)(u).
N V N
(c)H*(V ; Z(p)) ' (p)(oi; 2npi- oe) Z(p)(oei; 2npi- 2oe)
i0 i1
V
as coalgebras where (p)and Z(p)denote exterior and polynomial algebras respec*
*tively and
oi corresponds to xpi while oei corresponds to
i-1
1__pX pi [x ; x i- j]
2p j=1 j j p
under the equivalence in (a).
Proof: As in [CMN; section 12], write L(u; v) for the free Lie algebra over Z(p*
*)generated
by u and v with dim u = 2m - 1 and dim v = 2n. Let L0 be the Lie algebra kernel
of the natural projection: L ! . Then L0 is the free Lie algebra generated*
* by xi,
i 1, and H*(F ; Z(p)) ' UL0, H*(SP ; Z(p)) ~= UL__. This is the same as in
[CMN] except dim xi = 2n(i - 1) + 2m - 1 in this case while m = n in [CMN]. In
[CMN; 12.3] specific free sub-Lie algebras L(k)of L(0)are constructed,Tand we c*
*onsider
the same sub-Lie algebras regraded as appropriate. Let L(1) = L(k). Since SP
is stably P -free, we introduce Bockstein bfi: Hr(SP ; Z=p) -! Hr-oe(SP ; Z=p) *
*from
section 2. Clearly bfi(v) = u and bfiis a derivation. Consequently apart from*
* grading,
L____; bfiis isomorphic to the differential Lie algebra L in [CMN], and L(1)*
* Z=p is
acyclic. Choose a free basis uff; vfffor L(1) with bfi(vff) uff(mod p). Let u*
* = h() and
11
v = h() with 2 ss2m-1 (SP ) and 2 ss2n(SP ; P ). Using the Samelson product
and the action of ss*(SP ) on ss*(SP ; P ), we see that each element of L(u; v)*
* Z=p
except possibly [u; u] is in the image of h : ss*(SP ; P ) - ! H*(SP ; Z=p). C*
*hoose
fff: P nff-! SP so that h(fff)W= vff. Then (fff)*(vnff) = vffandf(fff)*(unff-o*
*e)i
uff(mod p). Construct W 0= P nffand f0 : W 0-! SP with f0fiPnff= fff. Then*
* in
ff
(f0)* : H*(W 0; Z(p)) -! H*(SP ; Z(p)) is contained in L(1) and with Z=p coeffi*
*cients
(f0)* is an isomorphism onto L(1) Z=p. It follows that (f0)* is an isomorphis*
*m onto
L(1) with Z(p)coefficients. Let W = SW 0and f : W -! SP be the adjoint of f0. T*
*hen
(f)* : H*(W ; Z(p)) -! H*(SP ; Z(p)) is the inclusion of UL(1) into UL____. S*
*ince
each fffis an iterated Samelson product and S2n+1 is homotopy commutative, f fa*
*ctors
through F , and we have constructed the diagram of the theorem. As in the case *
*of [CMN],
Sf has a left homotopy inverse, so f does as well and we have proven (a). Part*
* (b)
follows exactly as in [CMN] as does (c) where we write oei and oi in H*(V ; Z(p*
*)) for the
images of oei; oi in H*(F ; Z(p)).
We now write Sk{p} for the fiber of the map p : Sk -! Sk-oe.
Theorem 3.2. Suppose = pOE and p > 3. Then
Y i
V ' S2m-1 x S2np -oe{p}:
i1
Proof: Using the extended ideal structure of ss*(F ; P ) in ss*(SP ; P ) we con*
*struct classes
eo; eoein ss*(F ; P ) whose Hurewicz images are oi and oei respectively. Since*
* p > 3, we
may use the Jacobi identity and apply [CMN, 4.4] to conclude that fieoi= peoe. *
* Now let
fli : P 2npi-2oe(p) -! P 2npi-2oebe a map which is degree 1 on the top cell and*
* OE on
the bottom cell where P 2npi-2oe(p) is the mod p Moore space of that dimension*
*. Now
fieoiO fli= peoeiO fli= eoeO fliO p = 0. Now fieoiis the composition of eowith
i-oe 2npi-2oe 2npi-2oe 2npi-oe
B2np : P -! S - ! P
so eoiextends over the mapping cone of B2npi-oeO fli. This mapping cone is P 2n*
*pi-oe(p).
This extension has homology image generated by oei and oi. Using the H space st*
*ructure
we extend and consider the composite
i-oe 2npi-oe+1
S2np {p} -! P -! F -! V:
V
Clearly the image of this map in homology is (p)(oi; 2npi- oe) Z(p)(oei; 2np*
*i- 2oe). Now
multiply these maps together and with the inclusion of S2m-1 in F to obtain a h*
*omotopy
equivalence: Y i
S2m-1 x S2np -oe{p} -! F -! V:
i1
12
Theorem 3.3. Suppose* E2n(p-3)+4m-2 ~ *. Then
Y i i
V ' S2m-1 x S2np -oex S2np -2oe+1
i1
Q i i
and T ' T1 x S2np -oex S2np -2oe+1where T1 fits into a fibration sequence:
i1
2S2n+1 ---ss-!S2m-1 ----! T1 ----! S2n+1
fi
where ssfiS2n-1~ and E2 O ss ~. 2S2 : 2S2n+1 -! 2S2m+1
Proof: The condition on implies that P k' Sk _ Sk-oewhen k 2np - 2oe. Consequ*
*ently
the classes eoeiand eoimay be constructed and we conclude that oei, oi 2 H*(F ;*
* Z(p)) are
spherical. As in the case of 3.2, we easily construct a composite:
Y i i
S2m-1 x S2np -oex S2np -2oe+1-! F -! V
i1
which is a homotopy equivalence. Since the composition V - ! T -! SP is a homol*
*ogy
monomorphism and SSP is a wedge of spheres and copies of P kwith k < 2np - 2oe,
there are maps ffi : SSP -! S2npi-oe+1, fii : SSP -! S2npi-2oe+1carrying oi an*
*d oei
in homology. From this we construct a map
Y i i
SP -! S2np -oex S2np -2oe+1
i1
which is a left inverse to the inclusion
Y i i
S2np -oex S2np -2oe+1-! F -! SP:
i1
It follows that this product factors off of F , SP , V and T . Finally, con*
*struct the
diagram:
2S2n+1 ---ss-! F ----! SP ----! S2n+1
?? ? ? ?
y 2S2 ?yO ?y ?yS2
2S2m+1 ----! 2S2m+1 ----! * ----! S2m+1
Q i i
O is null homotopic on W and S2np -oex S2np -2oe+1since it is an H map and
21
each piece is mapped in by a Samelson product which is null in 2S2m+1 . Consequ*
*ently,
the composite factors though S2m-1 which is included in 2S2m+1 as the bottom c*
*ell.
An example of this result in the Toda fibration S2n-1 -! S2n -! S2np-1 crea*
*ted
from the class wn 2 ss2np-3(S2n-1).
_________________________
*If p > 3 this is equivalent to begin stably trivial.
13
4. The object of this section is to study the space V , and in particular t*
*o show that
it does not always split as in the case that = pOE. In fact we prove
Proposition 4.1. The 2np - 2oe skeleton of V is S2m-1 [wm p-2 e2np-2oewhere w*
*m :
S2mp-3 -! S2m-1 is the first element in the kernel of the double suspension.
Proposition 4.2. Let fi1 : S2m+pq-3 - ! S2m-1 for m p be a desuspension of th*
*e class
fi1 in the stable pq - 2 stem. Suppose m 6 0(mod p); then wm fip-216~ *.
The proof of 4.1 will require the construction of a generalization of the T*
*oda-Hopf
invariant to this situation. Let us recall the relative James construction [G1*
*]. This is a
reduced product space construction for the fiber of a pinch map. Consider a fib*
*ration:
F -! X [ CA -! SA:
Then there is a model (X; A)1 consisting of all words in X1 with the property*
* that all
letters after the first letter are required to lie in A, and a weak homotopy eq*
*uivalence:
ae1 : (X; A)1 -!F:
There is a natural James-Hopf invariant:
(X; A)1 Hk--!(XA(k-1))1
which agrees with the classical one in case A = X. We now generalize the Toda-*
*Hopf
invariant given in [G4]. In [G4], a map ` was constructed fitting into a diagra*
*m:
(X [ CA) - -`--! (X [ CA) ^ A 1
x? x
?ae1 ??i
(X; A)1 - H2---! (X ^ A)1 :
We filter (X; A)1 by (X; A)k which consists of words of length at most k. Then*
* we have
(SX; SA)k-1 = (SX; SA)k-2 [flkC(Sk-2XA(k-2))
where flk : Sk-2XA(k-2)- ! (SX; SA)k-2 is natural in (X; A). Combining this wit*
*h the
map ` and evaluation yields a map
k-2 (k-2) k-1 (2) (k-2)
(SX; SA)k-1 -! (SX; SA)k-1 ^ S XA 1 -! (S X A )1 :
Now consider the case that SX = S2m , SA = S2n, and the inclusion corresponds *
*to S.
Applying the natural retraction yields
14
Proposition 4.3. Localized away from 2 there is a Toda-Hopf invariant:
H0k: (S2m ; S2n)k-1 -! S2nk-2oe+1
which is natural for maps of pairs (S2m-1 ; S2n-1) and is null homotopic when r*
*estricted
to (S2m ; S2n)k-2.
In particular, there is a commutative diagram:
0
S2nk-1 --Hk--! S2mk-1
?? ?
y ?yk-2
0
(S2m ; S2n)k-1 --Hk--!S2nk-2oe+1
?? ?
y ?y2
0
S2mk-1 --Hk--! S2nk-1
where the upper and lower Hopf invariants are the Toda-Hopf invariants of [G4].
Proposition 4.4 H0p(eoe1) : P 2np-2oe-! S2np-2oe+1is the adjoint of the project*
*ion
j : P 2np-2oe+1-! S2np-2oe+1.
Proof:
p-1Xp
oe1 = _1_2p [exi; exp-i]
i=1 i
where exi= adi-1()() is a mod P homotopy class with h(exi) = xi. Now
H*((S2m ; S2n)i; Z(p)) = T (x1; : :;:xi);
the tensor algebra generated by x1; : :;:xi. Hence the pair (S2m ; S2n)1 ; (S2*
*m ; S2n)i
is 2m + 2in - 1 connected and exiis in the image of the inclusion
2m 2n 2m 2n
ss2m+2n(i-1)-1 (S ; S )i; P -! ss2m+2n(i-1)-1 (S ; S 1 ; P :
In particular, if 1 < i < p - 1, [exi; exp-i] is in the image of ss* (S2m ; S2n*
*)p-2; P and
hence H0p[exi; exp-i] = 0 in this case. Thus H0p(eoe1) = H0p[ex1; exp-1] .
Now consider the homotopy groups of the fibering:
2m 2n 2m+2n(p-2)-1 2m 2n 2m+2n(p-2)
(S ; S )p-2; S 1 -! (S ; S )p-1 -! S
[ex1; exp-1] factors through the left hand space and projects to x1xp-1+xp-1x1 *
*in the homol-
ogy groups in the middle. It factors through (S2m ; S2n)p-2; S2m+2n(p-2)-1 2 f*
*or dimen-
sional reasons. It does not lie in the image of H* S2m ; S2n)p-2; Z(p) ' T (x1;*
* : :;:xp-2).
15
However the 2np - 2oe skeleton of (S2m ; S2n)p-2; S2m+2n(p-2)-1 2 is containe*
*d in
(S2m ; S2n)p-2 [ e2np-2oe+1. This is the first cell that maps nontrivially un*
*der the
relative Hopf invariant
2m 2n 2m+2n(p-2)-1 H2 2m+2n(p-2)-1 2m 2n
(S ; S )p-2; S 2 --! S ^ (S ; S )p-2:
It follows from the definition of H0pthat H0p[ex1; exp-2] induces an isomorphi*
*sm in homol-
ogy in dimension 2np - 2oe.
Proof of 4.1: Let V 2np-2oebe the 2np - 2oe skeleton of V . then eH*(V 2np-2oe;*
* Z(p)) is freely
generated by x1 and oe1. Since foe1factors through V 2np-2oeand V 2np-2oefactor*
*s through
(S2m ; S2n)p-1 for dimensional reasons, we have a composite
P 2np-2oeeoe1-!V 2np-2oe-!(S2m ; S2n)p-1 -! S2np-2oe+1
which is a homology isomorphism in dimension 2np - 2oe. Now consider the diagra*
*m:
V 2np-2oe= S2m-1 [g e2np-2oe----! (S2m ; S2n)p-1 --H;p--!S2np-2oe+1
?? ? ?
y ?y ?yp-2
H0p
(S2m [wm e2mp-1 ) ----! S2mp-1 ----! S2mp-1
where the left hand vertical map exists since S2mp-1is a retract of (S2m [wm e2*
*mp-2 ).
From this we see that g ~ wm p-2.
Proof of 4.2. According to [G3; Theorem 12], if m 6 0(mod p), H0(wm ) = ff1.*
* Con-
sequently H0(wm fip-21) = ff1fip-21which according to [O] is non-zero (stably) *
*and not
divisible by p. Hence wm fip-216= 0 in ss*(S2m-1 ).
Analogously to 4.1, one can easily derive the following result using the Ja*
*mes-Hopf
invariant ([G1]) instead of the Toda-Hopf invariant.
Proposition 4.6. Suppose p ~ *. Then there is a map S2m-1 [ e2np-oe! V which
induces a homology monomorphism where
S2np-oe-1 ----! S2m-1
?? ?
yp-1 ?yE
3S2mp+1 --P--! Sb2m:
In particular, H() = (1 - m)ff1 O p-1 : S2np-oe-1! S2(m-1)p+1 and 6~ * in case*
* = fi1
and m 6 1(mod p).
16
Theorem 4.7. If m 6 0; 1(mod p) there is no extension B : 2S2m+pq-1 - ! S2m*
*-1 of
fi1 : S2m+pq-3 - ! S2m-1 .
Proof: According to [G4], wm fip1= fi1wm+p(p-1)-1. It follows that if an ext*
*ension exists,
wm fip1= 0. We will show that this is impossible in case m 6 0; 1(mod p). We*
* will base this
on the fact that the Toda Brocket {fip1; ff1; ff1} 6= 0 (see [O]). The eleme*
*nt wm is obtained
as the composition:
S2mp-3 --am--!S2m-1 B(m-1)q ----! S2m-1 :
where am is the 2m-1 fold suspension of the attaching map of the cell of dim*
*ension mq -1
in Bmq-1 and is the Kahn-Priddy map (see [G3]). We examine the composite:
fip1 2mp-3 am 2m-1 (m-1)q
S2mp+(pq-2)p-3 ----! S ----! S B
stably. It is easily seen to factor through S2m-1 B(m-2)q-1 and its projecti*
*on onto the top
cell S(m-2)2p+2 belongs to the bracket {ff1; ff1; fip1} since
P 2: H(m-2)q-1(Bmq ; Z=p) -! Hmq-1 (Bmq ; Z=p)
is non-zero when m 6 0; 1. In other words, we have a commutative diagram:
fip1 2mp-3
S2mp+(pq-3)p-3 ----! S
?? ?
y ?yam
S2m-1 B(m-2)q-1 ----! S2m-1 B(m-1)q
??
y
S2p(m-2)+2
where the left hand composite, ffl0, is an element of {ff1; ff1; fip1}. This*
* implies that wm fip1
desuspends to an element em 2 ss2mp+(pq-2)p-4(Sb2(m-2)) with H0(em ) = ffl0*
*. Such an
element is, of course, non-zero, so wm fip16= 0.
Bibliography
[AK] J.F. Adams and N.J. Kuhn, Atomic Spaces and Spectra, Proc. Edin Matt. So*
*c. 32
(1989), 473-481.
[CMN] F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, Torsion in Homotopy Group*
*s, Ann.
of Math. 109 (1979), 121-168.
17
[G1] B. Gray, On the homotopy groups of mapping cones, Proc. London Math. Soc.*
* (3)
26 (1973), 497-520.
[G2] B. Gray, A note on the Hilton-Milnor Theorem, Topology 10 (1971), 199-201.
[G3] B. Gray, Unstable Families related to the image of J, Math. Proc. Comb. P*
*hil. Soc.
96 (1984), 95-113.
[G4] B. Gray, On Toda's Fibration, Math. Proc. Comb. Phil. Soc. 97 (1985), 289*
*-298.
[G5] B. Gray On the Iterated Suspension, Topology 27 (1988), 301-310.
[G6] B. Gray EHP Spectra and Periodicity I: Geometric Constructions, TAMS 340,*
* no. 2
(1993), 595-616.
[G7] B. Gray, Associativity in two-cell complexes, 11 pp. ms., submitted to Pr*
*oc. Aarhus.
Conf. on alg. top, 1998.
[HM] J. Harper and H.R. Miller, On the double suspension homomorphism at odd p*
*rimes,
TAMS 273, no. 1 (1982), 319-331.
[O]S. Oka, The Stable Homotopy groups of sphere II, Hiroshima Math. J. 2 (1*
*972),
99-161.
[S]J. Stasheff, H spaces from a homotopy point of view, Lecture notes in Mat*
*hematics,
vol. 161, Springer Verlag (1970).
18
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