HIGH DIMENSIONAL KNOTS WITH ss1 ~=Z ARE DETERMINED
BY THEIR COMPLEMENTS IN ONE MORE DIMENSION THAN
FARBER'S RANGE
WILLIAM RICHTER
Abstract.The surgery theory of Browder, Lashof and Shaneson reduces the
study of high-dimensional smooth knots n ,! Sn+2 with ss1 ~=Z to homotopy
theory. We apply Williams's Poincare embedding theorem to the unstable
normal invariant ae:Sn+2 -!(M=@M) of a Seifert surface Mn+1 ,! Sn+2.
Then a knot is determined by its complement if the Z-cover of the comple*
*ment
is [(n + 2)=3]-connected; we improve Farber's work by one dimension.
0.Introduction
A high-dimensional n-knot will mean a smooth, oriented, codimension two em-
bedding n ,! Sn+2 of an exotic sphere, with n 5. See the survey of Kervaire and
Weber [K-W ] for more details. For our purposes, two knots ni,! Sn+2 are said to
~=
be equivalent if there is diffeomorphism_OE:_Sn+2_-!_Sn+2 such that OE(n1) = n2.
A knot n ,! Sn+2 has a complement X = Sn+2 - n x D2 , and is determined by
its complement if it is equivalent to any other knot with diffeomorphic complem*
*ent.
The orientation of n and the trivialization~of the normal bundle neighborhood
give a preferred diffeomorphism fi :n x S1 =-!@X. The Poincare conjecture gives
an oriented homeomorphism $ :Sn -! n. We will call the composite
ff: Sn x S1 $xid---!n x S1 fi-!@X -! X
the attaching map of the knot. Let o :Sn x S1 -'! Sn x S1 be the homotopy
equivalence (diffeomorphism) given by the generator of ss1(SO(n + 1)) ~=Z=2.
A knot n ,! Sn+2 is called r-simple [Ke2 , Fa1] if the Z-cover eXof the com-
plement is r-connected. Levine and Browder's [Le3, B-L, Le2] work shows that
[(n+1)=2]-simple n-knots are trivial. Following Kervaire-Milnor [K-M ], Levine *
*[Le1]
used ambient surgery on the Seifert surface to show that PL [(n - 1)=2]-simple *
*n-
knots were determined by their complements for n odd. Levine's work was partial*
*ly
extended to the case n even by Kearton [Ke1 ] and Kojima [Ko ]. Using Wall's th*
*ick-
ening theory [Wa3 ], Farber [Fa2] showed this for ([n=3] + 1)-simple n-knots.
Theorem A . For n + 3 3q and n 5, (q - 1)-simple smooth knots n ,! Sn+2
are determined by their complements.
____________
1991 Mathematics Subject Classification. Primary 57Q45, 57N65, 55P40, 57R65.
Key words and phrases. knots with ss1 ~=Z, Poincare embeddings, unstable nor*
*mal invariant.
1
2 W. RICHTER
Theorem B . Let n ,! Sn+2 be a knot with complement X and attaching map
ff: Sn x S1 -!X. There exists a homotopy equivalence i :X -'!X so the diagram
Sn x S1 ---ff-!X
? ?
o?y ?yi (1)
Sn x S1 ---ff-!X
commutes up to homotopy, if the knot is (q - 1)-simple, n + 3 3q and n 5.
We prove Theorem B via Williams's Theorem 1.7. In our metastable range the
Poincare embedding M xI ,! Sn+2 of a Seifert surface is determined by its unsta*
*ble
normal invariant ae 2 ssn+2 ((M=@M)) . We construct another Poincare embedding
M x I ,! Sn+2 suggested by ff . o (Lemma 1.4), with the same unstable normal
invariant ae (Theorem 1.6). Theorem 1.7 implies the two Poincare embeddings are
concordant. We use this concordance to construct the homotopy automorphism i.
By the work of Browder, Lashof, Shaneson and Gluck [Br1, L-S, Gl], it is well
known that Theorem B implies Theorem A for piecewise linear (PL) knots. In x2
we extend their work to smooth knots.
Our proof requires Levine's result [Le3], that there exists highly connected *
*Seifert
surfaces. Together with Barratt [B-R ], circa `82, we have a purely homotopy pr*
*oof
which uses Z-equivariant Hopf invariants and Ranicki's [Ra ] equivariant S-dual*
*ity.
We conjecture that high-dimensional knots with ss1 ~=Z are determined by their
complements. If ss1 AE Z, there exist counterexamples due to Cappell and Shanes*
*on,
Gordon, and Suciu [C-S, Go , Su]. Theorem 1.6, which is true outside our range
n + 3 3q, and the Appendix provide some evidence for this conjecture.
We have a homotopy theoretic proof [Ri] of Theorem 1.7, completing a program
of Williams's [Wi2 ], to prove the result using Browder-Quinn Poincare surgery *
*[Br4,
Qu]. The present paper (except x3) is independent of [Ri], but not [Wi2 ].
Given a subspace A X and a map f :A -!Y , we will write the identification
space X [f Y as lim-!(X- A f-!Y ), the colimit or pushout of the diagram.
I want to thank William Browder and Andrew Ranicki for sparking my interest
in Poincare embeddings and knots with ss1 ~=Z, and Sylvain Cappell for many
enlightening conversations about whether such knots are determined by their com-
plements. I want to thank Bruce Williams for explaining Farber's work to me, and
encouraging me to finish his program [Wi2 , Ri]. I want to thank Michael Barrat*
*t,
Neal Stolzfus, Fred Cohen, Derek Hacon and Nigel Ray for helpful conversations
about Farber's work and Hopf invariants. The proof of Theorem 1.6 is largely due
to Jeff Smith. I want to thank to the referee for helpful comments about organi*
*z-
ing the paper. Thanks to Paul Burchard for developing the LATEX commutative
diagram package diagram.sty, and Michael Spivak for his LAMS-TEX fonts.
1. Poincare embeddings and the proof of Theorem B
In Theorem B we can eliminate the condition that i be a homotopy equivalence.
Lemma 1.1. Any selfmap i of the knot complement X making diagram (1)com-
mute up to homotopy is a homotopy equivalence.
KNOTS DETERMINED BY THEIR COMPLEMENTS 3
Proof.Diagram (1)implies that i is a selfmap of the pair Poincare (X; @X), and
i*[X] = [X] 2 Hn+2(X; @X). Furthermore i induces the identity on ss1(X) ~=Z.
By naturality of Z-equivariant Poincare duality [Le2, Wa2 ], the composite
i ji* i j[X]\. i j i i j
Hn+2-* eX; @Xe; -! Hn+2-* eX; @Xe; ---! H* eX -!*H* eX
is the cap product isomorphism [X] \ .. Hence i* is surjective. Since the group*
* ring
= Z[Z] is Noetherian and X is a finite complex, i*: H*(Xe) -!H*(Xe) must be an
isomorphism, hence i is a homotopy equivalence by the Whitehead theorem. __|_|
For a smooth knot n ,! Sn+2 with complement (X; @X)n+2, Alexander duality
and relative transversality give a map h: X -! S1 which is transverse to the po*
*int
1 2 S1, with inverse image h-1(1) = (M; @M), and @M = n. Mn+1 is called a
Seifert surface for the knot. By the relative tubular neighborhood theorem there
is a codimension zero embedding M x I X extending the embedding n x I
@X = n xS1_=_n_xI_[n_xI, where I is the interval [-1; 1]. Let A = @(M xI).
Let W = X - M x I be the Seifert surface complement. The knot complement
X is then obtained by glueing together M x [-1; 1] and W along their common
boundary M x {-1; 1}. By writing X as the union of the Seifert surface and its
complement, we obtain the decomposition Sn+2 = M xI [W [n xD2. Let ^Wbe
the manifold with corners ^W= W [ n x D2, so that A = @W^. Let ffl: ^Wi W be
the deformation retraction which maps n x D2 onto n x I, using a map D2 i I.
Let f :A -! W be the composite of the inclusion A = @(M x I) = @W^ W^
and the retraction ffl: ^W-! W . Note that f is a cofibration, since ffl restri*
*cts to a
homeomorphism ffl: @W^ -! @W .
Williams [Wi1 ] studies codimension zero Poincare embeddings [Br1, Br2, Br3,
Wa1 ] of an m-dimensional oriented finite Poincare pair (Y; @Y ) in the sphere *
*Sm ,
which consist of a complement Z along with an attaching map f :@Y -! Z, such
that the pushout Y [f Z is homotopy equivalent to Sm . Williams [Wi1 ] defines
two Poincare embeddings (Y; @Y ) ,! Sn+2 with attaching maps f1: @Y -! Z1 and
f2: @Y -! Z2 to be concordant if there exists a homotopy equivalence :Z1 -!Z2
so that f2 ' . f1: @Y -! Z2.
For the above Seifert surface embedding, (M; @M) is an (n + 1)-dimensional
oriented Poincare pair, with an oriented homeomorphism $ :Sn -! @M given
by the orientation of the knot. (M x I; A) is an (n + 2)-dimensional Poincare
pair, with A = M x {1} [ Sn x I, and the attaching map f :A -! W gives a
Poincare embedding (M x I; A) ,! Sn+2, by the homotopy equivalence Sn+2 =
M x I [ ^W 1[ffl--!M x I [f W . Let ! :Sn -! M be the composite of $ and the
inclusion : @M -! M. Let v :M -! W be the restriction of f to M x 1 A.
Let :M -! A be the inclusions M x {1} A. For any map g :A -! W we
denote by g the restrictions g+ : M x{1} A f-!W and g- : Sn xI A f-!W .
~=
Definition 1.2.Define ffi :Sn x I -! Sn x I by ffi(x; t) = eiss(t+1). x;,tusin*
*g the
standard action of S1 = SO(2) SO(n+1) on Sn. We define the diffeomorphism o
of Sn xS1 = lim-!(Sn xI- Sn x{1} -!Sn xI) to be the identity on the left Sn xI
and ffi on the right Sn x I. We define the selfmap fl of A = lim-!(M x {1} -!xi*
*d--
Sn x {1} -!Sn x I) to be the identity on M x {1} and ffi on Sn x I.
4 W. RICHTER
Lemma 1.3. Let Sn x S1 -coaction----!Sn x S1 _ Sn+1 and A -coaction----!A _ Sn*
*+1 be
the coaction maps [B-B , Ga, Ba] onto the top cells. Then
(1) The selfmap o of Sn x S1 is homotopic to the composite
o0:Sn x S1 coaction-----!Sn x S1 _ Sn+1 -id_j--!Sn x S1 _ Sn -id_1--!Sn *
*x S1:
(2) The selfmap fl of A is homotopic to the composite
A coaction-----!A _ Sn+1 -id_j--!A _ Sn -id_!--!A _ M -id_+.---!A:
Proof.By the Barratt-Puppe sequence of the CW-complex SnxS1 = Sn_S1[en+1,
the selfmap o0 is characterized up to homotopy by the property that: o0 induces*
*0is
the identity in homology; and the Hopf construction of the composite Sn x S1 o-!
Sn x S1 -ss1!Sn is the generator j 2 ssn+2(Sn+1) ~=Z=2. But o clearly satisfies
both of these properties; hence (1). Now consider the rel boundary coaction map
Sn x I -coaction----!Sn x I _ Sn+1
of Sn xI. We can define the coaction map of A = M xI [Sn xI onto its top cell by
glueing the identity map on the left half M x I to the above rel boundary coact*
*ion
map. Furthermore the selfmap ffi of Sn x I is the identity on Sn x {1} [ N x I
for some point N. Therefore ffi is homotopic, rel boundary, to the composite
Sn x I -coaction----!Sn x I _ Sn+1 -id_g--!Sn x I _ Sn -id_1--!Sn x I
for some map g 2 ssn+1 (Sn). By part (1), we see that g = j 2 ssn+1 (Sn). By
glueing in this rel boundary homotopy, the second assertion follows. __|_|
Lemma 1.4. Let f :A -!W be the attaching map of a Poincare embedding (M x
I; A) ,! Sn+2. Then the composite A -fl!A -f!W is also the attaching map of a
Poincare embedding (M xI; A) ,! Sn+2. Furthermore (f .fl)+ = f+ :M x{1} -!
W and (f . fl)- = f- . ffi :Sn x I -! W .
Consider the general case of a Poincare embedding (M; A) ,! Sm of an oriented,
finite m-dimensional Poincare pair (M; A), with complement W and attaching map
f :A -!W . Let :Sm -'!M [f W be the homotopy equivalence so the composite
ae: Sm -! M [f W -! M=A
is orientation preserving. Williams [Wi1 ] calls ae 2 ssm (M=A) the unstable no*
*rmal
invariant of (M; A) ,! Sm . Williams [Wi2 ] shows that Browder's cofibration [B*
*r3]
Sm -ae!M=A f.@---!W
is split by the degree one map M=A pinch---!Sm , that the composite
M=A @-!A (f).@_pinch--------!W _ Sm (2)
is a homotopy equivalence. From this we deduce
Lemma 1.5. Let ae; ae02 ssm (M=A) be the unstable normal invariants of the Poi*
*ncare
embeddings (M; A) ,! Sm with attaching maps f :A -!W and f0:A -!W 0. Then
KNOTS DETERMINED BY THEIR COMPLEMENTS 5
0 f.@
(1) ae = ae0iff the composite Sm -ae!M=A ---! W is nullhomotopic.
(2) Suppose W 0= W , and assume the suspensions of the attaching maps
f; f0:A -! W are homotopic; f = f0 2 [A; W ]. Then the unsta-
ble normal invariants are equal; ae = ae02 ssm (M=A).
Now consider our Seifert surface Poincare embedding (M x I; A) ,! Sn+2, with
attaching map f :A -!W , and unstable normal invariant ae 2 ssn+2((M x I)=A).
Theorem 1.6. The Poincare embeddings (M xI; A) ,! Sn+2 with attaching maps
f . fl; f :A -!W have equal unstable normal invariant.
Proof.This follows from the codimension one framed embedding of the Seifert
surface, which implies the vanishing of ! 2 ssn+1(M). The cofibration sequence
Sn+1 = Sn -!-!M -! (M=Sn) @--!2Sn = Sn+2
splits; we have a homotopy equivalence M _ Sn+2 -_ae--!(M=Sn). Hence
: M -! (M=Sn) has a left homotopy inverse, which implies that Sn+1 -!-!
M is nullhomotopic. By Lemma 1.3 (2), fl ' id:A -! A. Hence the maps
f; (f . fl): A -!W are homotopic. The result follows from Lemma 1.5. __|_|
We recall the uniqueness part of Williams's [Wi1 ] Poincare embedding theorem.
Theorem 1.7. Let (M; A) be an oriented, finite, m-dimensional Poincare pair,
with ss1(A) = ss1(M) = 0 and m 6. Suppose M is n-dimensional as a CW-
complex, and let q = m - n - 1. If m < 3q, then any two Poincare embeddings of
(M; A) in Sm whose unstable normal invariants are equal are concordant.
Proof of Theorem B.Let n ,! Sn+2 be a (q-1)-simple knot, n+3 3q, with knot
complement Xn+2, and attaching map ff: Sn x S1 -! X. Let Mn+1 be a Seifert
surface with resulting Poincare embedding (M xI; A) ,! Sn+2, with attaching map
f :A -! W . By a theorem of Levine [Le3] we can assume that Mn+1 is (q - 1)-
connected. By Poincare duality of the pair (M; @M)n+1, M is then (n + 1 - q)-
dimensional. Theorem 1.6 and Theorem 1.7 imply the Poincare embeddings with
attaching maps f .fl; f :A -!W are concordant. Let :W -'!W be a concordance,
so .f ' f .fl :A -!W . Since the geometric map f is a cofibration, we may assu*
*me
that . f = f . fl :A -!W . By Lemma 1.4 we have . f+ = f+ and . f- = f- . ff*
*i.
The knot complement X is the pushout X = lim-!(M x I- M x {-1; 1} f+-!W ),
so we can define a selfmap i :X -! X to be the identity on M x I and on W .
But the attaching map ff: Sn x S1 -! X and the composite ff . o :Sn x S1 -! X
are the induced maps of colimits of the following strictly commutative diagrams
Mxx I----- M xx{1} --f+---!Wx Mxx I----- M xx{1} --f+---!Wx
??!xid !xid?? f-?? ??!xid !xid?? f-.ffi??
Sn x I----- Sn x {1} -----!Sn x I Sn x I----- Sn x {1} -----!Sn x I:
Using Definition (1.2), we have i . ff = ff . o :Sn x S1 -! X, and thus diagram*
* (1)
commutes. By Lemma 1.1 i :X -! X is a homotopy equivalence. __|_|
6 W. RICHTER
2. Proof of Theorem A ; smooth knots and surgery
Lashof and Shaneson [L-S, Thm. 2.1] show that any self homotopy equivalence
of a knot complement pair (X; @X)n+2 is homotopic to a diffeomorphism, if n 4
and ss1(X) ~=Z. This follows from the Sullivan-Wall exact sequence [Wa2 , x10]
i id j
0 = Ln+3 Z[Z] -! Z[Z]-! SO (X) -![X; G=O] = 0:
~=
Let OE: X1 -! X2 be a diffeomorphism between the knot complements of two smooth
(q -1)-simple knots ni,! Sn+2 with n+3 3q, n 5, for i = 1; 2. The homotopy
equivalence~i :X1 -'!X1 of Theorem B is thus homotopic to a diffeomorphism
:X1 =-!X1. Following Browder [Br1, Cor. 2], we have we have an exact sequence
n+1 n+2 -!Diff(n1x S1) F-!E(Sn x S1) ~=Z=2 Z=2 Z=2; (3)
involving the pseudo-isotopy and homotopy automorphism groups. The first two
Z=2 summands are given by the degree -1 maps of Sn and S1, and the third
summand is given by the selfmap o, which is detected by the Hopf construction
(cf. Lemma 1.3). We have to modify Browder's argument slightly, since F will
not be surjective if the exotic sphere n1does not possess an orientation-preser*
*ving
diffeomorphism (cf. [K-M ]).
~=
Let fii:nix S1 -! @Xi be the preferred diffeomorphisms, for i = 1; 2. The
~=
restriction of OE to the boundary gives a diffeomorphism @OE: n1xS1 -! n2xS1. If
the Hopf construction of the composite ss2.@OE: n1xS1 -!n2is j 2 ssn+2 Sn+1 ~=
Z=2, then replace the diffeomorphism OE by the composite OE . . By Browder's
application of the Browder-Levine fibering theorem [Br1, Lem. 2], we can assume
~=
that @OE restricts to a diffeomorphism OE0: n1-! n2. Let ffl = 1 be the degree *
*of @OE
on the S1 factor. Consider the diffeomorphism = (OE0xffl)-1 .@OE 2 Diff(n1xS1*
*),
which induces the identity in homology. Since its Hopf construction is zero, *
*' id,
so F( ) = id, and (3)shows that @OE = (OE0xffl). is pseudoisotopic to the comp*
*osite
n1x S1 e-!n1x S1 dx1--!n1x S1 OE0xffl---!n2x S1;
where d 2 n+1 is a diffeomorphism of n1, and e 2 n+2 is obtained from the
identity map of n1x S1 by "connecting sum" with a diffeomorphism of an (n + 1)-
~=
disk. We claim that @OE extends to a diffeomorphism of f@OE:n1x D2 -! n2x D2.
Certainly (OE0.d)xffl extends. But e must be pseudoisotopic to the identity, ot*
*herwise
we could glue in the tubular neighborhoods to get a diffeomorphism between the
standard sphere Sn+2 and the exotic sphere represented by e 2 n+2. By glueing
together OE with this extension f@OEwe have an equivalence of the two knots. _*
*_|_|
3.Appendix
Farber [Fa2] shows that PL (q - 1)-simple knots with n + 3 < 3q are determined
by the homotopy class v+ 2 [M; W ], which is stable by the Freudenthal suspensi*
*on
theorem. Let ae0 = @ .ae: Sm -! A be the composite. Using the S-duality map [Ri]
D: Sn+2 -ae0!A --! A ^ A (f^)----!W ^ M
KNOTS DETERMINED BY THEIR COMPLEMENTS 7
we have a bijection [M; W ]D\.--!sssn+2W [2] in our range. We note that Farber
uses a dual S-duality map M ^W -! Sn+1. We show that Farber's stable homotopy
invariant is essentially the second Hopf invariant 2(ae) of our unstable normal
invariant ae: Sn+2 -! (M x I)=A = (M=@M).
Theorem 3.1. The 2nd Hopf invariant of the unstable normal invariant ae: Sn+2 *
*-!
(M=@M) is the S-dual of the map v+ :M -! W :
-1 [2] s i [2]j
2(ae) = . (v- - v+ ) (id^v+ ) . D 2 ssn+3 ((M=@M)) ;
using the isomorphisms
i j [.(v--v+)-1] [2] i j
v+ 2 [M; W ]D\.---!~sssn+3 (W )[2] --------------! sssn+3 ((M=@M))[2] :
= ~=
Consider the general case of a Poincare embedding (M; A) ,! Sm , with comple-
ment W and attaching map f :A -! W as in [Ri]. The boundary map @ :M=A -!
A is defined to be the homotopy class making the diagram
M [ CA _______Awwpinch
| [[] CA = A x [0; 1]=A x 0;
pinc|h' [
| [ @ A = CA=A = A ^ ([0; 1]={0; 1})
|uu[
M=A
commute up to homotopy. Extending the splitting (2), from Williams's work [Wi2 ]
we have a homotopy equivalence
= x . f + y . pinch+z . : A '-!W _ Sm _ M;
where x, y and z are the inclusions of the three factors, and pinch:A -! Sm is
the unique degree one homotopy class. The two maps ; y :A -!W _ Sm _ M
are equalized up to homotopy by the collapse map M [Ax1 A x [0; 1] [f W -! A:
the first and third maps x . f and z . can be nullhomotoped when restricted
to M [Ax1 A x [0; 1] [f W since the ends M and W are "free" (as in the proof
of the equivalence of Whitehead products and Samelson products [Wh ]). Thus the
diagram
M=Au u________________Smae__________wWy_ Sm _uM
u| | ae0 ||
pinc|h ' | ' |
| |u |
M [ CA u_____Mp[Ax1iAnxc[0;h1] [f W_______Awwpinch
is homotopy commutative. Now apply Boardman and Steer's Cartan formula and
composition formula [B-S] to the equation . ae0 ' y :Sm -! W _ Sm _ M. We
obtain ^ . 2(ae0) + 2() . ae0 = 0 and 2() = xf ^ z, which implies
i j
^ . 2(ae0) = - (xf ^ z) . ae0 2 ssm+1 (W _ Sm _ M)[2] :
8 W. RICHTER
Proof of Theorem 3.1.Now consider a Seifert surface (M x I; A) ,! Sn+2, with
complement W and attaching map f :A -! W . Let h: A -! M=@M be defined
by collapsing the subspace M x 1 [ @M x I. Then we see that h is a homotopy
retraction of @ :(M=@M) -! A. Thus ae ' h . ae0: Sm -! (M=@M), and
2ae = (h)[2]. 2ae0.
We factor h: A -! (M=@M) through the homotopy equivalence , by a
map ff _ fi _ fl :W _ Sn+2 _ M -! (M=@M). The homotopy commutative
diagram
M _ M _ Sn+2
| N N N N (xv++z)_(xv-+z)_y
|| ' N NN NP
*__ae ||+_-_ae 4 A _____wW'_ Sn+2 _ M
| h 4 4 fflflflfll
|u 447fflflfllfliff_fi_fl
(M=@M)
yields ff = .(v- -v+ )-1 2 [W; (M=@M)] , fl = -.(v- -v+ )-1.v+ 2
[M; (M=@M)] , and fi = ae. Recall that [2].2(ae0) = - (xf ^ z) .ae0, which
is the composite of -x ^ z with the suspension of : Sn+2 -! W ^ M. Thus
-1 -1
h ' . (v- - v+ ) _ ae _ - . (v- - v+ ) . v+ . ;
-1 [2] s i [2]j
(h)[2]. 2(ae0)= . (v- - v+ ) (id^v+ ) . 2 ssn+3 ((M=@M)) :
__|_|
The reason that the knot can be determined by both the unstable homotopy class
ae and it's Hopf invariant 2(ae) is that Williams's relation of concordance is *
*stricter
than the usual relation of isotopy, where one also allows diffeomorphisms of M *
*x I
arising from diffeomorphisms of M. There is an EHP sequence interpretation of t*
*his
fact, but we will not give it here. Theorem 3.1 thus provides some further evid*
*ence
for our conjecture: the two Seifert surface Poincare embeddings we consider are
equivalent under a stronger equivalence relation than Farber's relation of isot*
*opy.
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Department of Mathematics, MIT, Cambridge MA 02139
Current address: Mathematics Department, Northwestern University, Evanston IL*
* 60208
E-mail address: richter@math.nwu.edu