A HOMOTOPYTHEORETIC PROOF OF WILLIAMS'S
METASTABLE POINCARE EMBEDDING THEOREM
WILLIAM RICHTER
0. Introduction
Recall the notion due to W. Browder of a Poincare embedding [Br1 , Br3 , Br2 ,
Br4, Wa , Ra, Wi1 , Wi2 ], which is the homotopy analogue of a smooth embedding*
* of
manifolds. Let (M; A) be a simply connected mdimensional (finite) Poincare pai*
*r.
A Poincare embedding of (M; A) in the sphere Sm will mean a finite CWcomplex
W and a map f :A ! W , such that the homotopy pushout M [ A x [0; 1] [f W is
homotopy equivalent to Sm . Given such a Poincare embedding, B. Williams [Wi1 ]
defines the unstable normal invariant ae: Sm ! M=A by collapsing the subspace *
*W .
Williams defines two Poincare embeddings f :A ! W and f0: A ! W 0of (M; A) to
be concordant if there exists a homotopy equivalence ff :W ! W 0so that the ma*
*ps
ff . f; f0: A ! W 0are homotopic. In this paper we give a homotopy theoretic p*
*roof
of Williams's metastable Poincare embedding theorem [Wi1 ], which was originally
proven geometrically.
Theorem 0.1 (Williams). Let (M; A) be a qconnected, mdimensional, finite or*
*i
ented Poincare pair, with m 6 and ss1(A) ~=ss1(M) ~=0. If m 3q, any degree one
collapse ae: Sm ! M=A is induced by a Poincare embedding f :A ! W . If m < 3q,
the Poincare embedding is unique up to concordance.
We complete a program of Williams [Wi2 ], by proving Theorem 0.1 via a system*
*atic
application of elementary unstable homotopy theory, including
(a) the metastable equivalence between desuspensions and coH space structure*
*s,
due to Berstein, Hilton and Ganea [BH , Ga2 ],
(b) Boardman and Steer's Cartan formula for Hopf invariants [BS ], and Barra*
*tt's
equivalence between JamesHopf and HiltonHopf invariants [Ba1 , BS],
(c)the BarrattGaneaToda relative Hopf invariant and relative EHP sequence *
*[Ba2 ,
Ga3 , To], including a relative Cartan formula (Theorem 2.5 below),
(d) the dual of the BarcusBarratt theorem, and James and Thomas's [JT ] rel*
*ated
obstruction theory on mapping into 2stage towers.
___________
Research supported by an NSF postdoctoral fellowship.
1
2 W. RICHTER
Theorem 0.1 has a corollary [Ri]: knots with ss1 ~=Z are determined by their co*
*m
plements in one dimension better than Farber's range [Fa].
Browder [Br2 ] reduced the construction of Poincare embeddings to a desupensi*
*on
question as follows. Given a degree one collapes ae: Sm ! M=A as in Theorem 0*
*.1,
let X be the homotopy cofiber of ae, giving a homotopy cofibration
ae h
Sm ! M=A ! X; (1)
and let @ :M=A ! A be the boundary map of the pair (M; A). We must desuspend
X, by a homotopy equivalence :W '!X, and find a map f :A ! W so that the
h is homotopic to the composite . (f) . @ :M=A ! X. Then (Theorem 1.7 below)
the map f :A ! W is a Poincare embedding of M with normal invariant ae.
Williams [Wi2 ] used Sduality and part of (a) to show that the cofiber X des*
*us
pends, say X = W , and that the cofibering map h: M=A ! X factors through
the suspension of A by a map :A ! W . It would be enough to desuspend ;
Williams claims that thishis equivalentito vanishing of Boardman and Steer's se*
*cond
Hopf invariant 2() 2 2A; (W )[2].
Our proof uses (a) and (b) to show that there is a unique extension :A ! X
and a unique coH structure on X for which 2() = 0. If m < 3q, the EHP sequence
implies that desuspends. When m = 3q we are outside the metastable range, but
Sduality and (c) give the desuspension of . For uniqueness we also need (d).
In x1 we review Williams's Sduality, and prove the existence of Poincare emb*
*ed
dings for m < 3q. In x2 we give a careful exposition of the elementary unstable
homotopy theory we need to proceed further. We shortened this section considera*
*bly
after seeing Moore and Neisendorfer's [MN ] "algebraic" point of view. Only x*
*2.1
is needed for x1. In x3 we prove existence in case m = 3q, and in x4 we prove *
*the
uniqueness; with similar problems when m = 3q  1.
I would like to thank Bruce Williams for interesting me in this problem, expl*
*aining
to me the beautiful work of Farber, and exposing me to Boardman and Steer's wor*
*k.
I would like to thank Mike Hopkins, who first introduced me to the work of Gane*
*a,
at a time when I was trying to understand Quinn's enigmatic papers [Qu2 , Qu1 ].
I would also like to thank Michael Barratt, William Browder, Sylvain Capell, Bi*
*ll
Dwyer, Mark Mahowald, Frank Quinn, Andrew Ranicki, Chris Stover, Jeff Smith
and JanAlve Svensson for many inspiring conversations. Many thanks to Kathleen
Kordesh, who typed the first draft with me, and to the University of Chicago fo*
*r their
hospitality. Finally thanks to Paul Burchard for developing the latex commutati*
*ve
diagram package diagram.sty, and Michael Spivak for his lamstex fonts.
1. Existence in the case m < 3q
In this section we prove the existence part of Williams's Theorem 0.1, modulo*
* the
proof of Theorem 1.6, which requires the relative Hopf invariant of x3.
METASTABLE POINCARE EMBEDDING 3
Let n = m  q  1. Note by Poincare duality that M is ndimensional as a
CW complex. We first review and extend Williams's SDuality; cf. [Wi2 , diagram
(3.1.1)]. Let A = A0[ Dm1 , where A0 is a (m  2)dimensional CWcomplex. Then
ae "
the composite D :Sm ! M=A ! M ^ M=A0 is an SDuality map, where " is the
reduced diagonal map
" . m1 m1
M=A ! M x M (M x A0 [ D x M) = M=D ^ M=A0;
~= m1
which gives the relative Poincare duality isomorphism H*(M; A0) ! Hm* (M; D *
* ).
Since the composite M=A0 ! M=A ! X is a homotopy equivalence, the cofi
bration (1) splits, and M=A ' Sm _ M=A0. We have the SDuality isomorphism
~= s
D*: {M; M=A0 } ! ssm (M ^ M=A0 ). We need the homotopy commutative diagram
"
M=A0 ____________M=Awss__________Mw^ M=A0
  A A
  AA (2)
  A ^ss
u u AAD
M=A0 ^ M=A0 _____M=Aw^sM=As^ss
which proves the naturality of relative cup products. By Sduality, diagram (2)*
*, and
the fact that the diagonal map of Sm is nullhomotopic, the map : M ! M=A0 is
stably nullhomotopic. By the Freudenthal suspension theorem is nullhomotopic,
since n = dim(M) < 2 conn(M=A0 ) + 1 = 2q + 1.
Therefore the map h: M=A ! X factors through A up to homotopy, by the
BarrattPuppe sequence of the derived cofibration M ! M=A ! A, which further
shows that A splits as a wedge M=A _ M. Thus the group [M ; X] acts freely
and transitively on the set extensions "h2 [A; X] of h.
We now recall the equivalence, in the metastable range, between cospaces and
suspension structures [BH ] [Ga2 , prop. 3.6]). Recall that X is a coH space *
*if there is
a map ff :X ! X _X, called the coH space structure, so that .ff ' : X ! X xX.
For a given coH space X, a suspension structure will be a homotopy equivalence
:W ! X which is a coH map, i.e. ff . ' 1 . + 2 . :W ! X _ X.
Theorem 1.1 (Hilton, BersteinGanea). Let X be a qconnected, mdimensional
finite CWcomplex, and assume that m 3q. If X is a coH space, then there exis*
*ts
a suspension structure :W '!X. Furthermore X will be a coH space, and hence
a suspension, if the reduced diagonal map : X ! X ^ X is nullhomotopic.
Since ' *: M ! M=A0 , diagram (2)shows that the reduced diagonal map of
M=A0 is nullhomotopic, and thus the space X ' M=A0 is a suspension.
For any suspension structure :W '!X, the suspension map E :[M; W ] !
[M ; W ] is surjective, by the Fruedenthal suspension theorem, since n < 2q,*
* since
4 W. RICHTER
m 3q. Thus, the extensions of h: M=A ! X are parametrized by the group
[M ; W ], which is mapped onto by the set [M; W ] via the suspension map.
By suspending the 2nd James Hopf invariant J(W )=W ! J(W ^ W ) Boardman
and Steer [BS ] define the Hopf invariant 2: [A ; W ] ! [2A; W ^ W ], a*
*nd
prove the Cartan formula [BS , thm. 3.15, def. 2.1]
Theorem 1.2 (Boardman and Steer). For f; g :A ! W , the Hopf invariant
of the sum satisfies the Cartan formula 2(f + g) = 2f + f ^ g + 2g.
We discuss Barratt's equivalence between JamesHopf and HiltonHopf invariant*
*s.
The 2nd HiltonHopf invariant H2: [A ; W ] ! [A ; W ^ W ] is uniquely defined
by the equation [BS , (4.11)]
X
(x + y) = x . + y . + [y; x] . H2() + ! . H!() 2 [A ; W _ W ];(3)
!>2
where x; y :W ! W _ W are the inclusions of the 1stand 2nd summands, and
! 2 {[[y; x]; x]; [[y; x]; y]; : :}:is a Whitehead product in the standard Hall*
* basis of
the free Lie algebra on {x; y}. Then [BS , thm. 4.17(a)] 2() = E (H2()) , and
Theorem 1.3. Suppose W is (q  1)connected, q > 1, and dim A < 3q. Then for
any map :A !W , 2() = 0 if and only if f :A ! W is a coH map.
Proof.In the metastable range, the higher Hopf invariants will all vanish, since
[A ; W [k]] = 0 for k > 2. By this and the HiltonMilnor theorem [Wh , XIx6
7] [BS , thm. 4.7], is a coH map iff H2() = 0. But in our range the suspensi*
*on
map E :[A ; W [2]] ! [2A; (W )[2]], which takes H2() to 2(), is injective.*
* 
Notice that A splits into three pieces; Sm _ M=A0 _ M '! A : the first*
* two
summands map into A naturally by the composites ae: Sm ! A and M=A0 !
M=A @!A ; M is mapped in by any splitting of the map (): A ! M .
Theorem 1.4. There is a unique extension "h:A ! X of the map h: M=A ! X,
and a unique coH space structure ff on X such that "his a coH map. Furthermore
if :W '!X is a suspension structure on (X; ff), then 2() = 0 for the map
:A ! W defined by the homotopy commutative diagram
M=A _______wXh
" [] u
 h[ [ 
@ [ 
u[ 
A _______wW:
METASTABLE POINCARE EMBEDDING 5
Proof.The second statement follows from the first by Theorem 1.3. Choose an ex
tension "hand a coH space structure ff on X. "his a coH map if and only if t*
*he
diagram
"h
Sm _ X _ M '! A ! X
?? ??
(x+y)?y ?yff (4)
"h_"h
A _ A ! X _ X
commutes up to homotopy when pulled back to the wedge M _ X _ Sm ; this
gives us an equation for each summand. The equation for M is vacuous, since
[M ; X _ X] ae [M ; X x X], since n + 1 < 2q + 1. The equation for X is exa*
*ctly
that ff is homotopic to the composite
(x+y) "h_"h
ff :X ! A ! A _ A  ! X _ X: (5)
Thus the coH space structure ff is determined by the extension "h. The equatio*
*n for
Sm is that the composite
ae x."h+y."h
Sm ! A ! X _ X (6)
is nullhomotopic. Notice that this condition is independent of the coH space s*
*tucture
ff. Thus we will be done if we can show there is a unique extension "h2 [A ; *
*X]
satisfying (6).
Let :W '!X be any suspension structure. Then equation (6)becomes
0 = ((x + y) .  [y; x] . H2()). ae = ((x + y) _ [y; x]). (x .  y . H2())*
*.(ae:7)
Let f = x . ; g = y . H2(): A ! W _ W . Using the HiltonHopf expan
sion [Wh ], which follows by naturality from equation (3), we have
(f + g) . ae = g . ae + [g; f] . H2(ae) + [[g; f]; f] . H3(ae) +(.8.*
*:.)
By metastability, H2() desuspends to an element i 2 [A; W [2]], so by naturality
[g; f] = [y; x . ] . (i ^ id): A ^ A ! W [2]^ A ! W _ W [2]:
By a simplified form a theorem of Barcus and Barratt [BB , cor. 7.4] about Whi*
*tehead
products, we have
[g; f] . H2(ae) = [y; x] . i ^ . H2(ae) (9)
To see this, notice that [y; x . ] = [y; x . R] . (id^^), where R :J(W ) ! W *
* is the
evaluation map, and ^ :A ! J(W ) is the adjoint of . But the two maps
[y; x . R]; [y; x] . id^R :W [2]^ J(W ) ! W _ W [2]
6 W. RICHTER
are equalized by the map (id^E): W [2]^W ! W [2]^J(W ), and thus are indis
tinguishable from the point of view of Sm , by dimensional reasons. This proves*
* (9).
Since all the higher Whitehead products in (8) contain [g; f], we can use the s*
*ame
argument to show that all the higher terms vanish, and (8)reduces to (f +g).ae *
*= g.ae,
and (7)becomes 0 = [y; x] . (H2()) . ae. By the HiltonMilnor the composite (*
*6)is
nullhomotopic if and only if H2() . ae = 0. By metastability this is equivale*
*nt to
2() . ae = 0.
The group [M ; W ] acts bijectively on the set of extensions of h: to each*
* d 2
[M ; W ], we associate the extension "h0= . 0, where 0 = + . d . . As we
noted above, the map d desuspends; d = e for some e: M ! W . By Theorem 1.2,
2(0 ) = 2() + ^ (e . )
2(0 ) . ae= 2() . ae + ( ^ (e . )) . ae: (10)
The second term of the right hand side of (10)is the composite
ae ______ _________________^(e.)
Sm+1 ________w2A Aw ^ A Ww ^ W :
446
id^ 4 4^d
A ^ M
But we have the following diagram, which commutes
" h^id
M=A ____________________wM=A0 ^ M _____Xw^'Mu
[[]   
ae[   
[ @ @^id ' ^id
[   
[ ae u id^ u ^id 
Sm __________wA ________wA ^ A ______Aw ^ M _____Ww ^ M
up to homotopy. In homology the middle rectangle is Browder's compatibility of *
*cup
products and boundary maps [Br5 , xI]. Thus the bottom row, the composite
ae @ ^
Sm ! M=A ! A ! A ^ A ! W ^ M; (11)
is an Sduality map, and the term ( ^ (e . )) . ae of equation (10) is the Sdu*
*al
~= s
of d, under the duality isomorphism D :{M ; W } ! ssm+1 (W ^ W ). Thus, *
*by
equation (10), there is a unique extension "h= . :A !X so that 2() . ae = *
*0.
This completes the proof of Theorem 1.4. 
Now we recall the EHP sequence of G. Whitehead [Wh , thm. XII(2.2)]. We have
Theorem 1.5. Let A be a finite complex of dimension dim A 3q  1, and let W be
a (q  1)connected finite complex. Then we have the following exact sequence o*
*f sets
h i
[A; W ] E![A ; W ] 2! 2A; (W )[2] :
METASTABLE POINCARE EMBEDDING 7
That is, an element :A !W desuspends to an element f :A ! W if and only
if the second Hopf invariant 2(): 2A ! (W )[2]is nullhomotopic.
Let :A !W be the extension guaranteed by Theorem 1.4 with 2() = 0. If
m < 3q then desuspends by Theorem 1.5 to a map f :A ! W . In x3 we prove
Theorem 1.6. The extension :A ! W of Theorem 1.4 with 2() = 0 desus
pends to a map f :A ! W when m = 3q.
Theorem 1.7. The map f :A ! W gives a Poincare embedding of (M; A) in Sm ,
whose unstable normal invariant Sm '!M [fW ! M=A is homotopic to our original
degree one collapse map ae: Sm ! M=A.
ae h
Proof.The split cofibration Sm ! M=A ! X and Theorem 1.4 give a bijection
~=
((f) . @)*: ssdegreem1(M=A) ! ssm (W );
which sends ae to the 0 element. Let ae0:Sm ' M [ A x I [f W ! M=A be the
degree 1 collapse given by the Poincare embedding (M; A; f :A ! W ), obtained *
*by
collapsing the subspace W . Then the composite
ae0 @ f
Sm ! M=A ! A  ! W
is nullhomotopic, which implies that ae and ae0are homotopic. 
We have now completed the proof of the existence part of Williams's Theorem 0*
*.1,
modulo the proof of Theorem 1.6.
2. A Review of Elementary Unstable Homotopy Theory
We would like to rewrite the literature we survey here from the point of view*
* of
Whitehead [Wh ], James [Ja1, Prop. 6.42] and Moore and Neisendorfer [MN ], of
working in the category of compactly generated spaces and NDR pairs, and using
the result of Strom [St] [Wh , thm. I.7.14], that the pullback under a fibrati*
*on of an
NDR pair is an NDR pair in the total space. We contend that only with such care
for the point set topology is it possible to write down explicit homotopies. Ho*
*wever
in this work we have been able to dispense with explicit homotopies, so we avoid
such technical discussions. We shall assume that all spaces have the homotopy t*
*ype
of CW complexes, and have nondegenerate basepoints.
8 W. RICHTER
2.1. Suspensions, loops, adjoint functors and cup products. We will carefully
follow Boardman and Steer [BS ] in our conventions involving suspensions. We d*
*efine
the suspension X by smashing on the right with the circle S1 = I={0; 1}, so that
X := X ^ S1, for any space X. Defining the ksphere to be Sk = Ik=@(Ik) =
S1 ^ . .^.S1, we have a canonical identification between kX = . .X.and
X ^ Sk. For any two spaces X and Y we will identity (X ^ Y )with X ^ Y ,
using the strict associativity of the smash product. On the other hand we will*
* be
careful with shuffle maps 2(X ^ Y ) ! (X) ^ (Y ) , which sends ff ^ fi ^ s ^ *
*t to
(ff ^ s) ^ (fi ^ t). Following Boardman and Steer [BS ], the cup product f ^ g*
* of the
maps f :nA ! X and g :nA ! Y is defined to be the composite
n+m() n+m shufflen m f^g
f ^ g :n+m A ! A ^ A ! ( A) ^ ( A)  ! X ^ Y: (12)
X will always mean the measured loops space [Wh ] of pairs (r; fl(t)) such t*
*hat
fl :[0; r] ! X with fl(0) = fl(1) = *. This will simplify the discussion in x2*
*.4. The
measured loop functor is adjoint up to homotopy to suspension using the suspens*
*ion
map E :X ! X and the evaluation map oe :X ! X which sends (r; fl) ^ t to
fl(rt). Under the adjoint functor isomorphism [X; Y ]~= [X; Y ]between suspensi*
*on
and loops, we write the adjoint of a map f as f^ , so that f^ ^= f.
2.2. The Moore splitting of the suspension of a product. We recall the Moore
splitting of (A x B), which Moore [Mo , Hu ] used to give the first proof of Ja*
*mes
splitting of (X) for a connected space X [Ja3].
Following Husemoller [Hu , app.,Prop. 3.1], we use the suspension coordinate *
*to
"add up" the projections, giving a homology equivalence
= (1 . ss1)+ (3 . ss12)+ (2 . ss2):(A x B) ! A _ B _ (A ^ B);
where ss1, ss12, and ss2 are the three projections of the product onto A, A ^ B*
*, and
B respectively. By the five Lemma, we see that is a homology isomorphism, so
for connected CW complexes A and B, is a homotopy equivalence. Since is a
homotopy equivalence, the projection ss12:(A x B) ! (A ^ B) is a surjection in
the homotopy category, which means that two maps out of (A ^ B) are homotopic
if their pullbacks to (A x B) to (A ^ B) under ss12 are homotopic.
We then define the universal Hopf construction to be the unique homotopy class
Hid:(A ^ B) ! S(A x B) such that
. Hid= 3 2 [(A ^ B); A _ B _ (A ^ B)]: (13)
Given a map f :A x B ! X, we define the Hopf construction Hf: (A ^ B) ! X
of f to be the composite
Hid f
Hf: (A ^ B) ! (A x B) ! X:
We have some easy consequences of the definitions.
METASTABLE POINCARE EMBEDDING 9
Lemma 2.1. For spaces A, B and X we have
(1) The composite (ss1 + ss2) . Hid2 [(A ^ B); A _ B] is the zero element.
H oe
(2) For any space B, the composite (B ^ B) ! B ! B is nullhomo
topic, where : B x B ! B is the loop multiplication.
(3) For any map f :A x B ! X we have
f = (fA . ss1)+ Hf . (ss12)+ (fB . ss2)2 [ (A x B); X];
where fA :A ! X and fB :B ! X are the restrictions of f.
(4) The universal Hopf construction Hidis uniquely determined by the equation
Hid. (ss12)=  (ss1)+ 1  (ss2)2 [(A x B); (A x B)]:
Proof.(1)follows composing the defining equation (13)with the projection to A _
B. (2)follows from (1)and the fact that oe. ' oe.(ss1 + ss2) : BxB ! B.
(3)follows from the definition (13). (4)follows from (3)by letting f = id:A x B*
* !
A x B. 
Using the Hopf construction twice, we can split (A ^ B ^ C) off (A x B x C),
in two different ways. We have the two composites
id^Hid Hid
Hid. (1A ^ Hid): (A ^ B ^ C) ! A ^ (B x C) ! (A x B x C) ;
Hid^id
Hid. (Hid^ 1C ): (A ^ B ^ C)shuffle! (A ^ B)^ C ! (A x B) ^ C
shuffle! [(A x B)^ C] Hid! (A x B x C) :
We note that we used the following result (without proof) in the proof of Mahow*
*ald's
Xk splitting of SU(3) [MR , p. 604].
Lemma 2.2. For spaces A, B and C, the two composites
Hid. (1A ^ Hid); Hid. (Hid^ 1C )2 [ (A ^ B ^ C); (A x B x C)]
become equal after one suspension in the group [2(A ^ B ^ C) ; 2(A x B x C) ] .
Let B ! F ! X be a principal fibration with action : B x F ! F . Then the
two maps
(H . 1B ^ H ); (H . H ^ 1F:)2B ^ B ^ F ! 2F
are homotopic.
Proof.To prove the first statement, we use property (4) above and the fact that
the track group [2X; Y ] is abelian. By prefixing the two maps by the homotopy
surjection (A x B x C) ! (A ^ B ^ C) we see that both composites can be
rearranged, after a suspension, to the sum 1  ssAB  ssAC  ssBC + ssA + ssB +*
* ssC . The
second statement follows from the first and the associativity of the action . *
* 
10 W. RICHTER
2.3. The relative Hopf invariant and the relative EHP sequence. Let W !
X h!B be a cofibration sequence, meaning that (X; W ) is an NDR pair [Wh ,
thm. I(5.1)]), and B = X [ CW is the homotopy cofiber of the inclusion map . Let
F be the homopy fiber of the the map h: X ! B, that is, the pullback of the pa*
*th
p p
fibration B ! P B ! B to X B. We have a principal fibration B ! F ! X,
with action map : B x F ! F . There is a natural lifting e: W ! F of the
inclusion : W ! X, since the composite W ! X ! B is canonically nullhomo
topic. Recall Ganea's [Ga3 , thm. 1.1] stable splitting of the fiber F , which *
*extends
the MooreJames splitting of SX [Wh ], which was first proved in homological f*
*orm
by James [Ja2, thm. (2.1)].
Theorem 2.3 (Ganea). Let W ! X h!B = X [ CW a cofibration sequence, and
let F be the homotopy fiber of the map h: X ! B. Then the composite
(1^e) H h
(B ^ W ) ! (B ^ F ) ! (F ) ! (F [eCW )
is a homotopy equivalence, and furthermore we have a homotopy equivalence
e _ H . (1 ^ e): W _ (B ^ W ) ! F:
Let : F ! B ^W and J :F ! W be the homotopy classes uniquely defined
so that the diagram
J _______________________________
W u_____________F_u'* Bw^uW
'  
'  
' '  
' ' e_H .(1^e) oe^1W (14)
ss1 '  
'  
 
W _ (B ^ W ) ___w(Bs^sW2) ______w(B)s^hWuffle
is homotopy commutative. Clearly the following two maps are homotopic:
J . e ' E :W ! W: (15)
We then define the relative Hopf invariant h: F ! 2(B ^ A ) , following Ganea*
* [Ga3 ],
to be the composite
(E) 2
F ^! (B ^ W ) ! (B ^ W ) : (16)
The relative Hopf invariant h gives rise to a natural transformation
h 2
[P; F ] ! [P; 2(B ^ A ) ] !~[ P; B ^ W ];
=
for any space P , which we will also refer to as h.
METASTABLE POINCARE EMBEDDING 11
__
Notice that h factors through a map h:F [eCW ! 2(B ^ W ), which is adjoint
to a homotopy equivalence. From this we deduce the relative EHP sequence, varia*
*nts
of which were proved by Barratt [Ba2 ], Ganea [Ga1 , Ga3 ] and James [Ja2, Thm.*
* 1.4].
Theorem 2.4 (Barratt, Ganea)._ Suppose that conn W = (n  1), conn(X; W ) =
m, with n 2, m 1. Then h: F [e CW ! 2(B ^ W ) is (2m + n)connected.
Thus_if A is a finite CWcomplex with dim A 2m + n, then the induced map
h*:[A; F [e CW ] ! [A; 2(B ^ W )] is bijective. Furthermore we have an exact
sequence
h 2
[A; W ] e![A; F ] ! [ A; B ^ W ];
if dim A < min(2m + n; m + 2n  1).
For any space A, let *: [A; B] x [A; F ] ! [A; F ] denote the product on the
homotopy sets given by the action of B on the fiber F , so that ! * ff 2 [A; F*
* ] is
the composite
! * ff :A ! A x A !xff!B x F ! F;
for ! 2 [A; B] and ff 2 [A; F ].
Following Boardman and Steer [BS , thm. 5.6] and James [Ja2, cor. (1.7)] we *
*prove
a "Cartan formula" for the relative Hopf invariant.
Theorem 2.5. For any space A we have
(1) The suspension of the composite . H : (B ^ F ) ! B ^ W is homotopic
to the composite
2(B ^ F ) shuffle!(B) ^ (F ) oe^J!B ^ W : (17)
(2) For ! 2 [A; B] and ff 2 [A; F ], we have
h(! * ff) = h(*!) + !^ ^ J . ff + h(ff) 2 [A; 2(B ^ W ) ]:
Proof.The first statement follows from the second by Lemma 2.1, part (3). To pr*
*ove
the first part we use Theorem 2.3. It is enough to show that the two maps in qu*
*estion
are equalized by the suspensions of the two maps
(1^e) 1B ^H .(1^e)
(B ^ W ) ! (B ^ F )  (B ^ B ^ W ) :
Notice that the composite (17)is homotopic to the suspension of the composite
(B ^ F ) = B ^ (F ) 1^J!B ^ (W ) shuffle!(B)^ F oe^1!B ^ W: (18)
One sees this for instance by noticing that the twist map T of S2 = S1^S1 has d*
*egree
1. By this observation and the definition (14)of , 2(1^e) equalizes the two ma*
*ps.
12 W. RICHTER
By definition (14)of J, the map 1B ^ H . (1 ^ e) annihilates composite (18). *
*It
remains to show that the suspension of the composite
1B ^H .(1^e) H
(B ^ B ^ W ) ! (B ^ F ) ! F ! B ^ W
is nullhomotopic. But this follows from Lemma 2.2 and Lemma 2.1, part (2). 
One difficulty in applying the Cartan formula is understanding the composite *
*term
A ff!F J!W . The following lemma simplifies the computation. Our argument
is motivated by the obstruction theory of James and Thomas [JT , Ja2].
()
Lemma 2.6. The composition F J!W  ! X is homotopic to the suspension
of the fibering map p: F ! X.
Proof.This follows immediately from the definition (14) of J, the fact that p .*
* =
p . ss2: B x F ! X, and Lemma 2.1, part (1). 
Recall the James construction J(W ), which has a natural homotopy equivalence
oW :J(W ) ! W for a connected space W . Recall the JamesHopf invariants [W*
*h ]
Hk: J(W ) ! J(W [k]), which extend to give maps Hk :J(W )=W ! J(W [k]) for
k > 1. Recall we have maps E :W ! J(W ) and oe :J(W ) ! W . Consider the
relative EHP sequence for the cofibration E :W ! J(W ), so that B = J(W ) [E
CW ' J(W )=W . Letting F be the fiber of the map h: J(W ) ! J(W )=W we
have a relative Hopf invariant h: F ! 2(J(W )=W ^ W ) . Suppose W is (q  1)
connected. Then the map : W [2]= J2(W )=W ae J(W )=W is (3q  1)connected.
Note that Hk ^:J(W )=W ! W [2]is a left inverse of theisuspensionjof : W [2]ae
J(W )=W , and is therefore 3qconnected. Let j: F ! 2 W [3] be the composite
of h with the double loop of the composite
Hk^^1 i [2]j shuffle[3]
J(W )=W ^ W shuffle!(J(W )=W^)W  ! W ^ W ! W ;
which is 4qconnected. Thus we can substitute j for h in our relative EHP seque*
*nce.
By Theorem 2.4, Theorem 2.5 and Lemma 2.6, we have
Theorem 2.7. Let A be a space with a map :A ! W , with adjoint^ : A !
^ h
J(W ). Suppose that the composite A ! J(W ) ! J(W )=W is nullhomotopic, so
METASTABLE POINCARE EMBEDDING 13
that^ lifts to a map ": A ! F , so that the diagram
______ ____2ij [3]j
W [2] Fw w W

aeaeoij
e " i p
ae i 
ae i 
ae u
i ___h
W ______J(Ww)i wJ(W )=W
i AC
i aeaeoA A A
i ae AA
iaeA A *
aeAAi
A
is homotopy commutative. Then we have an exact sequence
[A; W ] e![A; F ] j![2A; W [3]]; if dim A < 4q  2.
i j
Using the action W [2]x F ae (J(W )=W )x F ! F , we have
j(!^ * ") = j(*!^) + ! ^ + j(") 2 [2A; W [3]]; for ! :A ! W [2]:
2.4. The dual of the BarcusBarratt theorem and stable decompositions of
James and Thomas. Recall that the BarcusBarratt theorem [BB , Ru , Ba4, Ba3]
computes maps out of a homotopy cofiber X [f CW , in the case that f is a map
of suspensions. The stabilizers of the action of the group [W; M] on the set [X*
* [f
CW; M] are computed as the image of the group [X; M], by a homomorphism
which is a sum which includes (f)* and also higher order terms involving the Ho*
*pf
invariants of f.
We need a dual result; we will consider maps into the homotopy fiber F of a m*
*ap
h: X ! B of loop spaces. Various versions of this theorem have been published
by James and Thomas, Rutter and Baues [Ru , Ba4 , Ba3 ] but not in the form we
need, Theorem 2.11 below, so we have given our own proof. We were motivated by
the application of James [Ja2, Thm. 3.1]. We will need this result for uniquen*
*ess
of Poincare embeddings. At the tip of our range we will also need a further res*
*ult,
basically due to James and Thomas, about mapping into a two stage tower.
Let h: X ! B be a pointed map and let F be the homotopy fiber of the map h,
so that p : F ! X is a principal fibration. Suppose in addition that h() = **
* for
some point 2 E. Let " 2 F be the canonical lift of to F . Recall the long exa*
*ct
homotopy sequence bijection
i jp*
ss1(X; ) h*!ss1(B; *)! ss0 F ; "! ss0(X; ) : (19)
14 W. RICHTER
Now let X = X and B = B be loop spaces, h as above. We define the dual
Hopf invariant (h): X ^ X ! B of h to be the composite (h) := h^ . H of the
adjoint of h and the Hopf construction of the loop multiplication of X.
Lemma 2.8. Let h: X ! B be a map of loop spaces as above. Then there is a
bijection between the cokernel of the homomorphism h*: ss1(X;i) !jss1(B; *) a*
*nd
the cokernel of the homomorphism ss1(X; *) ! ss1(B; *)~= ss2 B; * given by
i 7 ! h^*(i) + (p) . (i ^ "):
Proof.Loop multiplication on the right gives an isomorphism ]:ss1(X; *) ! ss1(*
*X; ) .
FollowinghJamesiand Thomas [JT , p. 101] we note that the natural map ss1(B; **
*)!
S1 +; B is an injection since B is a loop space, normally we must mod out by co*
*nju
i +j
gation. Let H :S2 = D2=S1 ! S1 be the map H(t . z) = z+ ^ t, which splits *
*off
the top cell. The following composite is easily seen to be the adjunction isomo*
*rphism
h + i ~=h i +j i H* i j
ss1(B; *)ae S1 ; B ! S1 ; B ! ss2 B; * : (20)
The cokernel of h*: ss1(X; ) ! ss1(B;i*)jis then bijective with the cokernel *
*of the
homomorphism (h; ): ss1(X; *) ! ss2 B; * which is the composition of (20)and
] h* 1
the composite ss1(X; *) ! ss1(X; ) ! ss1(B; *). This sends a map i :S ! X
i +j (ix") H u h^
to composite S2 H! S1 ! (X x X)  m!X ! B . Lemma 2.8 then
follows from Lemma 2.1 part (3). 
Suppose further that we have a map j: F ! C from the homotopy fiber F to a
loop space C = C , and let T be the homotopy fiber of the map j, so that we have
q j
a homotopy fibration T ! F ! C. Suppose that " lifts to T by a map . Followi*
*ng
James and Thomas [JT ], suppose that the composite j^.H : (B^F ) ! C factors
through a map c: (B ^ X) j!C, so that the diagram
H
(B ^ F ) ! F
?? ??
(1^p)?y ?yj^
(B ^ X) c! C
commutes up to homotopy. In the language of James and Thomas, (q; p) is a stable
decomposition of the nonprincipal fibration p . q :T ! X.
Lemma 2.9. Under the aboveihypotheses,jthere is a surjection from the cokerne*
*l of
the map ss1(B; *) ! ss2 C; *, which sends i 7 ! j^*(i) + c . (i ^ "), to the
i jq* i j
kernel of the map ss1 T ; ! ss1 F ; "3 ".
METASTABLE POINCARE EMBEDDING 15
Proof.The proof is similar to the proof of Lemma 2.8. There is an added complic*
*ation
that we are only assuming that j(") ' *, but we still have an isomorphism
i j h + i ~=h i +j i H* i j
ss1 C; j(")ae S1 ; C ! S1 ; C ! ss2 C; * :
similar to the isomorphism (20). Precomposing with the homomorphism
]" i jj* i j
ss1(B; *) ! ss1 F ; "! ss1 C; j(");
we argue as before. 
Now take a finite CW complex A and consider the induced map hA :XA ! BA of
function spaces. We first have the analogue of (19).
Lemma 2.10. Let h: X ! B be a pointed map and let A be a finite CW complex,
with a fixed map :A ! X such that h . = "*. Then we have a short exact seque*
*nce
i j hA i j p*
ss1 XA ; !* ss1 BA ;"*! [A; F ]![A; X]3
We now have the dual of the BarcusBarratt theorem, which follows from Lemma *
*2.8.
Let ":A ! F be the canonical lift of .
Theorem 2.11. Assume in Lemma 2.10 that X = X and B = B are loop spaces.
Then we have an exact sequence
(h;) h2 i p*
[A; X]  ! A; B ! [A; F ]! [A; X]3 ;
where the homomorphism (h; ) is defined by
(h; )(i) = h^*(i) + (h)*(i ^ ) :
We apply Theorem 2.11 to the cofibrationiWjae J(W ) ! J(W )=W . iLet F 0bej
the fiber of the map 2: W ! 2 (W )[2]. Call o0W:J(W )=W ! 2 (W )[2]
the 3qconnected composite
H2 i [2]joW[2] i [2]jshuffle2i [2]j
o0W:J(W )=W ! J W ! W  ! (W ) :
Then we have a (3q  1)connected map OE: F ! F 0making the diagram
i j
F 0! W 2! 2 (W )[2]
x? x x
?? oW???' ???o0W (21)
p
F ! J(W ) ! J(W )=W
commute up to homotopy. Let Consider the composite e0= OE . e: W ! F 0. Applyi*
*ng
Theorem 2.11 to F 0, and using the Cartan formula [BS ] for 2 we have
16 W. RICHTER
Corollary 2.12. Let A be a space with a map f :A ! W , where dim A 3q  2
and W is (q  1)connected. Then we have an exact sequence
h i (h;(e0.f))h i p*
2A; W ! 3A; (W )[2] ! [A; F 0]![A; W ]3 (e0. f);
where (h; e0:f)(i) = 2(i) + (i ^ f).
We now extend Lemma 2.9 as well to mapping spaces.
q p
Theorem 2.13. Suppose the two principal fibrations T ! F ! X form a stable
decomposition as in Lemma 2.9, and let :A ! T be a lifting of the map ":A ! *
*F .
Then we have an exact sequence
Cok ((h; j; ))! [A; T ]![A; F ]3 ";
h i
where the homomorphism (h; j; ): [A; B] ! 2A; C is defined by
(h; j; )(i) = (j . )^*(i) + c*(i ^ ) :
Consider the cofibration W ae J(W ) ! iJ(W )=Wj. Let T be the fiber of the
modified relative Hopf invariant j: F ! 2 W [3] . Using Theorems 2.13 and 2.7,
we obtain
Theorem 2.14. Let A be a space with a map f :A ! W , where dim A < 4q  3 and
W is (q  1)connected. Then the following sequence
h i ji (h;j;f)h i e
A; W [2] ! 3A; W [3] ! [A; W ] *![A; F ] 3 e . f
is exact at [A; W ], where (h; j; )(i) = j(*i) + (i^^ f).
3. Existence in the case m = 3q; the proof of Theorem 1.6
By Theorem 2.7 there is an additional obstruction to desuspending :A !W
beyond 2() = 0, which arises from the relative Hopf invariant of the cofibrati*
*on
W ae J(W ) h!J(W )=W .
^ h
We claim the composite A ! J(W ) ! J(W )=W is nullhomotopic.iThis followsj
from 2() = 0 and diagram 21, since the map o0W: J(W )=W ! 2 (W )[2]is
3qconnected. Thus the map^ : A ! J(W ) lifts to the fiber F . Call a lifti*
*ng
":A ! F . We can vary the lift " by any map ! :A ! (W [2]). As in x2.4 we
h i
write "]: A; W [2] ! [A; F ]for the map sending ! to ! * ". By Theorem 2.7, t*
*he
composite (not a homomorphism)
h i * h i "] j (@ae)* i j
M; W [2] ! A; W [2] ! [A; F ]![2A; W [3]] !~ssSm+1 W [3] ;
=
METASTABLE POINCARE EMBEDDING 17
n o ~= i j
is the sum of j(") and the Sduality isomorphism D : M; W [2] ! ssSm+1 W [3]
induced by our Sduality map (11). Thus there is a unique class !0 2 [M; W [2]]*
* so
that relative Hopf invariant j((*!0) * ") vanishes. By Theorem 2.7 the lift (*!*
*0) * "
of^ : A ! J(W ) factors through e: W ! F , by a map f :A ! W , which is then*
* a
desuspension of . 
4. Uniqueness of the Poincare Embedding for m < 3q
f f0 0
Given two Poincare embeddings A ! W and A ! W such that their unstable
normal invariants are homotopic, N(f) ' N(f0): Sm ! M=A, we need to construct
a map ff :W ! W 0so that ff . f ' f0 2 [A; W 0]. Note that such a map ff will
automatically be a homotopy equivalence.
Suppose N(f) = N(f0) 2 ssdegreem1(M=A). Call ae = N(f), and ae0= N(f0). There
is then an induced map (unique up to homotopy) of cofibers :W 0 ! W , so that
the diagram
ae @ f
Sm ! M=A ! A ! W
flfl fl x
flfl flflfl ???
ae0 @ f0 0
Sm ! M=A ! A ! W :
is homotopy commutative. That is, the two maps f; . f0: A ! W are
homotopic when prefixed by the map @ :M=A ! A . Thus there exists a map
e: M ! W so that f ' . f0 + e . : A ! W . But the suspension
map is onto, since [M; W [2]] = 0, since n < 2q, since m < 3q + 1. So e desuspe*
*nds;
i.e. e = E(d) for some d 2 [M; W ]. Thus f ' . f0 + (d . ), so by the Cartan
formula, Theorem 1.2, we have
2(f) = 2( . f0) + . f0^ (d . )
0 = 2() . 2f0+ (1 ^ d) . (( . f0) ^ ):
Prefixing this equation with ae yields f0. ae ' f0. ae0' 0, so that
0 = (1 ^ d) . (( . f0) ^ M ) . ae: Sm+1 ! W ^ M :
By Sduality d is stably nullhomotopic, which implies d is nullhomotopic by the
Freudenthal theorem, since n < 2q1, since m < 3q. Thus .f0 ' f :A ! W .
p0 2 2i [2]j
Recall the fibration F 0! W ! (W ) , and the modified relative sus
pension e0:W ! F 0of x2. By the functoriality of 2 we have
0 = 2(f) = 2( . f0) = 2() . 2f0;
18 W. RICHTER
which implies that 2() is nullhomotopic, since f0: A ! W 0 is a homotopy
retraction. Thus ^ :W 0! W lifts to a map ":W 0! F 0, so that
i j i j
p . (e0. f)= (f)^ ' ( . f0)^ = ^ . f0 = p . ". f0 = p . ". f0 :A ! W :
By the BarrattPuppehsequenceithejtwoimapsh(e0. f);i(" . f0): A ! F 0differ by
an element fi 2 A; 3 (W )[2] ~= 3A; (W )[2], i.e. e0. f ' fi * (" . f). *
* First
i j
notice that [A; 3 (W )[2]] ~= sssm(W [2]) [3W; (W )[2]], since [M ; W [*
*2]] = 0,
since n + 1 < 2q, since m < 3q. Thus we can write fi uniquely up to homotopy as
fi = fi1 . pinch+fi2 . f0,hforifi1:jSm+3i ! (W )[2]and fi2: 3W 0! (W )[2]*
*.i Thus j
e0. f ' (fi1. pinch) * fi02* ". f0, by the associativity of the action of 2 *
*(W )[2]
on the principal fiber space F 0. We thereforeiexchangejthe lift fi02* " of ^ *
*for ",
and we now have e0. f ' (fi1 . pinch)* ". f0. Recall that W 0is at most (m  3*
*)
dimensional, since M is 1connected. Since e0 is (3q  2)connected " lifts uni*
*quely
to a map ff :W 0! W , so that " = e0. ff, and therefore = ff, and e0. f '
(fi1 . pinch)* e0. (ff . f0):A ! F 0.
We now show that e0. f ' e0. (ff . f0):A ! F 0, by using Corollary 2.12. We *
*see
that the composite
* E h2 i (h;(e0.f))h3 [2]i2(@ae)*h m+2 *
*[2]i
[M; W ] ! [A; W ] ! A; W ! A; (W ) ! S ; (W )
is the isomorphism, induced by our Sduality map (11), which sends a class i0: *
*M !
@.ae () ^f i0^1
W to the double suspension of the composite Sm  ! A  ! (M) ^ W  !
W [2]: note that the composite 2. E is trivial. Thus the difference element fi1*
*. pinch
lies in the stabilizer of the action, and we have e0. f ' e0. (ff . f0):A ! F *
*0.
In the case m < 3q  1 we are done, since e0is (3q  2)connected, which impl*
*ies
that f ' ff . f0: A ! W . At the tip of our range m = 3q  1 there is an addit*
*ional
obstruction. We use Theorem 2.14 to show the obstruction group is zero. Since t*
*he
map F ! F 0is (3q1)connected and dim A = m1, we have e.f ' e.(ff . f0):A !
F . The differencehbetweenithejmapsif; ff . f0: A ! W is measured in a coset o*
*f the
(@ae)* s i [3]j
obstruction group A; 3 W [3] !~ ssm+3 W . But by Theorem 2.14 this
=
coset is a quotient of the cokernel of the composite
h i ji h i ji (h;j;f)h i (@ae)* i j
M; W [2] ! A; W [2] ! 3A; W [3] !~ sssm+3 W [3] ;
=
i j
which is the Sduality isomorphism sending i0: M ! W [2] to the suspen
(@ae) 2 (2) ^f 2 i0^1 [3]
sionhofithejcompositeiSm+2 ! A ! ( M) ^ W  ! W : note that
M; 3 W [3] = 0 by dimensional reasons. Thus the obstruction coset is zero; *
*the
homomorphism (h; j; f) of Theorem 2.14 is surjective, proving Theorem 0.1. 
METASTABLE POINCARE EMBEDDING 19
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20 W. RICHTER
Department of Mathematics, MIT, Cambridge MA 02139
Current address: Mathematics Department, Northwestern University, Evanston IL*
* 60208
Email address: richter@math.nwu.edu