HOUSTON JOURNAL OF MATHEMATICS,
THOROUGHLY
Volume 3. No. 2, 1977.
KNOTTED
Martin
HOMOLOGY
Scharlemann
SPHERES
*
For Hn a homologyn-sphere,considerthe problemof classifying
locallyflat
imbeddings
Hn•'-•Sn+2upto isotopy.
Since
anyimbedding
maybealtered
byadding
knotsSn• Sn+2, the classification
problemis at leastascomplexasthe isotopy
classificationof knots. Elsewhere [8] we show that there is a natural correspondence
between
knottheory
andtheclassification
ofthose
imbeddings
Hn• Sn+2which
satisfya certain conditionon fundamentalgroup. Thoseimbeddingswhich do not
satisfythe fundamentalgroupconditionwill be calledthoroughlyknotted.The goal
of thepresent
paperis theconstruction,
foralln •>3, ofhornology
spheres
Hn andof
PLlocally
fiatimbeddings
Hn%Sn+2which
arethoroughly
knotted.
A corollary
will
be that, for thesenomologyspheres,the codimensiontwo classification
problemis
measurablymore complex than knot theory.
The outline is as follows: In õ1 we define thoroughly knotted imbeddings and
present,
for any homology
n-sphere
Hn, necessary
conditions
for rrl(H) to be the
fundamental group of a thoroughly knotted homology n-sphere.These conditions are
shown to be sufficient if n •> 5. Also, for n •> 3, the conditions are shown to be
sufficient
toproduce
athoroughly
knotted
imbeddings
H # H• Sn+2.
In õ2 we review enough of Milnor's K-theory and of Steinberg's results on
Chevalley groups to produce in õ3 an example of a group (the, binary icosahedral
group) satisfying the algebraicconditions of õ t, and, therefore, providing thoroughly
knotted homology n-spheresfor all n •> 3.
Finally, in õ4 we use the same example to show that for n = 3 the algebraic
conditions of õ 1 are not sufficient to produce thoroughly knotted imbeddingsof H
itself, but only of H # H.
I would like to thank Jon Carlson Ibr discussionsuseful in preparing õ2 and
* Supported in part by an N.S.F. grant.
271
272
MARTIN
SCHARLEMANN
Michael Stein for pointing out the connection between K-theory and the calculation
of H2(SL(p,q)).
õ1. The algebraicproblem.Supposea hornologyn-sphereHn is locallyflatly
imbedded
in Sn+2 andlet i: Hnx D2(-• Sn+2 be an imbedding
of a tubular
neighborhood.
Letrr1(H)=Kand
letGbethecommutator
subgroup
ofrr1(sn+2
- H).
Since
Hi(S
n+2- H)•_Z,rrl(Sn+2
- H)may
bewritten
asasemidirect
product
G*r Z.
Inother
words,
there
isanautomorphism
r: G--*Gsuch
thatrrl(Sn+2
- H)consists
of
ordered
pairs(g,t),gC G, t C Z withgroup
multiplication
(g,t)-(g',s)
= (g-rt(g'),t+ s).
Van Kampen's theorem gives the following push-out diagram of fundamental
groups, induced by inclusion:
•rl(HX0D2)• K• Z
• G©rZ
•2
K
•
•2
DEFINITION.
Theimbedding
i: H• Sn+2isthoroughly
knotted
if SOl(K©
{0}) 4: 0.
The importance of this phenomenon will be explored in [8].
Notethat,since
Sn+2-i(HX •21 iscompact,
Ger Z isfinitely
generated
(though
Gneed
notbe).Furthermore,
Hi(Gßr A)• Hi(S
n+2-H)• Z and,
bythe
Hopf theorem [3 ],
.•,
•
H2(sn+2
- H)
H•(Ger
Z) P•r2(sn+2
- H)• 0•
wherep is the Hurewiczhomomorphism.
Insum,
if i' H --*Sn+2isathoroughly
knotted
imbedding
wehave
(a)apush-out
diagram
K©Z Søl ) G©rZ
K
•2 > 1
THO ROUGHLY
KNOTTED
HOMOLOGY
SPHERES
27 3
such that
(b) K•rrl(H)
(c) •i(K• (0})4:0
(d) G *r Z isfinitelypresented,
HI(G*r Z) = Z, H2(G*r Z) = 0.
(e) •2 isprojection
and•1 commutes
withprojection
to Z.
The following is a partial converse. Let H_ denote H with an open n-disk
removed.
1.1 PROPOSITION. Supposethere is a pttsh-outdiagram satisf•'ing(a) -(e).
Thenthereisa compact
contractible
(nq-2)-manifold
N withaN"' O(H_X 1)2) xuch
that
(i)
the natural inclusion OH_ X (0} c-• ON exte•Ms to an imbeddi•g
(H_,OH_)X D2 • N,ON
(ii) the diagram induced by inclusions
rrl(H_X0D2)
rrl(H_X D2)
, rrl(N-(H_X (0}))
• rrl(N)
is the given push-out diagram.
We begin the proof with a preliminary
1.2 LEMMA. Thereis a closedm-manijbld
M. m >•5, suchthat•rl(M) • G •r Z
and
Hi(M)"-Hi(S
1 XSin-l).
PROOF.Let {x 1....,Xs;r
1.....rt} be a presentation
for G*r Z. Identifyin the
natural
way
rrl(s•(S1
XS4))
with
thefree
group
generated
byXl,...,x
s.Perform
surgery
oncircles
oq:S1• #(S1• S4) i= 1.....t representing
r1....,rt. Bystandard
arguments,
framings
maybechosen
forneighborhoods
of theoq(S
1) sothatthe
s
resulting manifold M' is almost parallelizable. From the Mayer-Vietoris sequence,
H2(M')isfree,andfromVanKampen's
theorem
rrl(M')= G *r Z.
According
to Hopf [3], 0 = H2(G•r Z) = H2(M')/Prr2(M').
Thusa basisfor
H2(M')
isrepresentable
byafinite
collection
ofimbeddable
spheres
7i(S2).
Since
M'is
almost
parallelizable,
each
7i(S
2)has
trivial
normal
bundle.
Now
perform
surgery
on
274
MARTIN
SCHARLEMANN
each
7i(S
2)toobtain
aclosed
manifold
M"withrrl(M"
) • Gßr Z andH2(M")=
0.
ByPoincare
dualityH3(M") = H2(M"): 0 andH4(M") = Hi(M")= Hi(G ßr Z) = Z.
This completes the construction if m = 5.
If m > 5 removean opentubularneighborhood
in M" representing
a generatorof
Hi(M"),andtakethecross-product
of thismanifold
withDm-5.Theresultant
manifold
isahornology
S1X Dm-1withfundamental
group
Gßr Z andboundary
=
O(S
1 X D4 X Dm-5)= S1 X Sm-2.Attach
S1 X Dm-Ibyidentifying
theboundaries.
It
is easilyverifiedthat the resultingmanifoldM hasrrl(M)• G ßr Z and H,(M)-
H,(S1 X Sin-l).Thisproves
1.2.
PROOF OF 1.1. Take m = n + 2 in 1.2 and consider the parallelizable manifold
(H_XS1X I)#M withfundamental
group
(KoZ), (Go
r A).ForYl....,Yq
generators of some presentation for K ß Z, perform framed surgery on q circles in
(H_XS1X I)#M representing
thewords
yi1.el(yi)
G(KøZ)*(Gor Z)and
call
theresulting
parallelizable
manifold
M'. Aneasycalculation
shows
rrl(M') "- G ßr Z,
H2(M') -- Hn(M')is freeof rankq - 1 andHi(M')= 0, i • 1,2,n.Furthermore,
aM'•
O(H_XS1X I) andtheinclusion
H_X S1X {0}->M'induces
themapel on
fundamental groups.
As in Lemma1.2, sinceH2(Gor Z)= 0 it is possible
to do surgeryon q- 1
2-spheres
inM'toobtain
N'ahornology
S1X Dn-Iwith
rrl(N')
_•Gßr ZandON'
•
O(H_X S1 X I).
Finally,
let N bethemanifold
obtained
fromN' by attaching
H_ X D2 bya
diffeomorphism
H_XaD2•_H_XS1X {0}.Then
aN= a(H_x D2),Hi(N)=0,i •>
1. Van Kampen's
theoremimpliesthatrrl(N) is the push-out
of themaps•1 and•2,
hence,
byhypothesis
(a),rr1(N)istrivial.
Thus
theimbedding
H_XD2• Nsatisfies
the requirements of 1.1.
Before trying to improve 1.1 we note the following corollary, which is the central
result of this section.
1.3 COROLLARY. If there is a push-outdiagramsatisfying(a) - (e) above,then
there
isa thoroughly
knotted
imbedding
ofthedouble
H # (-H)ofH intoSn4-2.
PROOF. The double along aN of the manifold N of 1.1 is a homotopy
n+2-sphere,
hence
Sn+2,andcontains
thedouble
H # (-H)ofH.Thecomplement
ofa
THOROUGHLY
KNOTTED
HOMOLOGY
SPHERES
275
tubular
neighborhood
of H # (-H)in Sn+2isthedouble
of themanifold
N' appearing
h•theproof
of1.1along
H_X S1X (1}C aN'.Hence
rtl(Sn+2-(H#H))
isthefree
product
of rtl(N') withitselfamalgamated
alongs01(K
mZ) C G•r Z • rtl(N').In
particular
theinclusion
N'-• sn+2-(H#H)
induces
aninjection
rt1(N')-->
rt1(sn+2-(H#
H)).Since
s0t(K
X {0}) isnon-trivial
in rtl(N'),itsimage
inrtl(Sn+2-(H
#H))is
non-trivial. Hence H # H is thoroughly knotted.
The strongest sufficiency theorem possible for constructing thoroughly knotted
homology sphereswould be a version of 1.1 in which we replace H_ by H and N by
Sn+2 Issuch
a theorem
possible?
In õ4a counterexample
isgiven
incase
n = 3,but
for n >• 5 we prove the following slightly weaker version'
1.4 PROPOSITION. Suppose there is a push-out diagram satisfying (a) - (e).
Thenfor anym• 5 thereisa hornology
tn-sphere
H' andanimbedding
H' X D2 •
Sm-t-2
suchthatthediagram
induced
byinclusions
•rl(H'XOD
2)
•rl(H'X D2)
, rtl(sm+2
- (H'X
) rrl(Sm+2
)
is the given push-out diagram.
Note that we are free to choose m 4= n, yet even when m = n we do not know
that H' = H.
PROOF OF 1.4. In [5] Kervaire shows that, if a group K is the fundamental
groupof a homology
sphere,
thenHi(K) = 0 for i = 1,2, and,conversely,
anyfinitely
presented
groupK with Hi(K) = 0, i = 1,2, is the fundamental
groupof a hornology
m-sphere for each m >• 5. In fact, the latter construction creates the hornology
m-sphere
H' withrrl(H') mK bybeginning
withSTM
andsuccessively
doingsurgery
on
i-spheresfor i = 0,1,2. It is not difficult to see from his construction that, if W is the
traceof thesurgeries
(beginning
withtheO-handle
Dm+lwithboundary
sm),thenW
is a hornology disk with boundary 3W = H, and the inclusion H-> W induces an
isomorphism of fundamental groups.
It willbeconvenient
to viewH' "• 3WasH_'t-J
o DTM.
Consider
themanifold
N',
constructed
asin 1.1,whose
boundary
is3(H__'
X S1X I). Attach
WX S1 to H'_X
276
MARTIN
SCHARLEMANN
S1 X {1}C 3N'along
H'_X S1C WX S1.Thismanifold
hasboundary
(H'_X S1 X
{0})CJ
3 (DmX S1)"•H'X S1.Standard
calculations
again
show
thatif H'X D2 is
attached
to thisH' X S1 theresulting
manifold
isSm+2andthat,since
i' I-t • W
induces
anisomorphism
onfundamental
groups,
thisimbedding
of H' X D2 intoSm+2
has a push-out diagram asin (ii) of 1.1 and hence as required to prove 1.3.
{}2. K-theory and Chevalley groups. It is relatively easy to produce examples of
push-out
diagrams
satisfying
all the hypotheses
of 1.1 otherthanH2(G©r Z)= 0.
Unfortunately, this last condition is tedious to verily and difficult to produce. In this
section we review the algebra (associated with
K-theory)
which shows that
H2(SL(n,p))= 0. HereSL(n,p)is the speciallineargroupof n X n matrices,
n •> 5, in
thefieldZp. In {}3wewillchoose
G to beoneof these
groups.
See[7] [10] for
missing proofs.
Kervaire and Steinberg provide the following machinery for constructinggroups
with trivial first and second hornology' Given a perfect group G, there is an extension
Gc of G, knownas the universal
centralextension,
whichis characterized
by the
existence
of a surjection
')':Gc • G whosekernelliesin thecenterof Gc andwhich
factors uniquely through any other central extension of G.
Milnorconstructs
Gc asfollows[7]' Let F be a freegroupwhichmapsontoG
and let R be the kernel of the map. Since G is assumedperfect, [F,F]/[F,R]
maps onto G, and the kernel, (R ChIF,F] )/[R,F]
also
is clearly central. Milnor providesan
elegant
proofthatGc "• [F,F]/[F,R]. Notethatthecenter
(R C3[F,F])/[R,F] ofGc
is, according to Hopf [3], precisely H.•(G;Z).
2.1 LEMMA.For an),perfectgroupG, H2(Gc) '• Hi(Gc) '• O.Furthermore,
if
H2(G)= H1(G) = O,thenGc '• G.
PROOF.By [7, Theorem5.3] Gc is perfectand,for (Gc)c theuniversal
central
extension
of Gc, theexactsequence
1 •H2(G
c) •(Gc)
c½-•Gc • 1
splits.Let s: Gc -->(Gc)c bethesplitting
(i.e.½s= 1).By definition
of (Gc)c,thereisa
unique
homomorphism
(Gc)
c• (Gc)
csuch
that
½-h
=½.Since
½s½
=½,s½
=1.Then
½
isanisomorphism
andsoH2(Gc)= 0.
If H2(G)'" Hi(G) '" 0, thenker(Gc • G) istrivialandsoGc '" G. Thiscompletes
THOROUGHLY
KNOTTED
HOMOLOGY
SPHERES
277
the proof of 2.1.
REMARK. An (unimportant) corollary of this lemma is that a group G is the
fundamental group of a hornology m-sphere. m •
$ if and only if G is a finitely
presented universalcentral extension. Unfortunately, the presentation [ F,F] / [ F,R] of
Gc givenby Milnoris finitelygenerated
onlyif it istrivial.
Steinberg has studied a class of groups for which a presentation for the universal
extension can be calculated in a different manner. To each of these groups G(Z,F),
called Chevalley groups, is associated a field F and a root system Z of a semi-simple
Lie algebra. The most important example to topologists has been the root system of
type An_1. In thiscaseG(An_1,F) isSL(n,F),thegroupof unimodular
n X n matrices
with coefficients in F. For n •
5, its universal central extension St(n,F)-•
SL(n,F),
generated
by theSteinberg
symbols,
haskernelMilnor'sK2(F).
For F a finitefield,K2(F) = 0. Thuswehave
2.2 PROPOSITION. (Steinberg). For F a finite field and n • 5, SL(n,F) is a
finite universal central extension.
õ3. The algebraic construction. In this section we present a push-out diagram
satisfying the hypotheses of 1.l,with H the dodecahedral space. The existence of a
thoroughly knotted imbedding of the double of the dodecahedral space (in fact,
examples in all dimensions) will then follow from 1.3.
Let K= •rl(H)= SL(2,5) and G= SL(5,5). There is a naturalinjectionj:
SL(2,5)ß SL(3,5)-• SL(5,5)inducedby thedecomposition
(Z5 ß ZS)ß (Z5 ß Z5 ß
ZS). SL(2,5) has one non-trivialnormalsubgroup,
the centerC '-' Z2 with one
non-zero
element
(b
1•). Ingeneral,
SL(m,n)
issimple
ifmand
n- 1are
relatively
prime [2]. In particular SL(3,5) and SL(5,5) are simple.Let cr',fi'• SL(3,5) have
non-trivialcommutator[cr',fi'] and let cr= j(1 ß or'),• = j(1 ß •'). For G = SL(5,5),
define•-: G-• G by •-(g)= crgcr
-1. Clearly
•-[j(SL(2,5)ß
1) is theidentity,
sothe
homomorphism
•1: K• Z• G•. Z givenby •l(k,t) = (j(k• 1),t)• G•. Z is
well-defined and injectire.
Let•2: K ß Z • K betheprojection,
andconsider
thepush-out
K ß z ....
.....
•2'N"-,..
K./*2
278
MARTIN
SCHARLEMANN
3.1 PROPOSITION. The abovepush-outsatisfiesproperties(a) - (e) of õ1.
PROOF.(a) Foranyt G Z, SO2(1,t)
= 1,so•lsol(1,t) = 1. Thusker(•l ) contains
the normalsubgroup
generated
by so
l(1 ,t). Then for any(g,0)C G*r Z andt C Z,
el(g,0)
= • i(sol(1,t)-(g,0)-sol(1
,-t))= ½l(rt(g),0).
Inparticular,
forg=/T1and
t = 1,
•,1(18o•-1oe-l,0)=
1.
Since[/•,oe]
4= 1, ker •1 (3 G C G*r Z is a non-trivial
normalsubgroup
of G.
SinceG is simple,ker• 1 contains
G aswellasZ. Hence• 1 is trivial,so•3 istrivial
and, finally, X = 1.
(b) K = SL(2,5)= •r1 (Dodecahedral
space).
(c) solisinjective
by definition.
(d) SinceG'" SL(5,5) is finite,G*r Z is finitelypresented.
SinceG is
simple,Hi(G) = 0 andHi(G* r Z)= Z. To showH2(G* r Z)= 0, let Y be a K(G* r
Z,1) space,The homomorphism
•rl(Y)-+ Z yieldsa covering
K(G,1) spaceY. Then,
sinceH2(G)= 0, H2(Y) = Hi(Y) = 0. HenceH2(G*r Z)= H2(Z)= 0 [6, Corollary
7.3].
(e) Follows from the definitions.
3.2 COROLLARY.
There are thoroughly knotted hornology n-spheresfor all
n•>3.
PROOF. By 1.3 it suffices to produce for each n•> 3 a hornologyn-splaerewith
•rl(H) "• SL(2,5).For n = 3, let H be the dodecahedral
space.Forn > 3, usea(H_ X
Dn-2).
õ4. Algebra 4= Geometry for n = 3. The purpose of this section is to show that
the diagram of 3.1 is a counterexample to 1.4 in the casem = 3. The fundamental tool
is
4.1 THEOREM. SupposeN is a compact orientable 4-manifold with connected
boundary
M suchthat •rl(M)"• SL(2,5)and•rl(N) mapsontoSL(5,5).Thenthe
composition
•rl(M)-+ •rl(N)-+ SL(5,5)cannot be the naturalinclusion
SL(2,5)•
SL(5,5).
First we show how 4.1 produces a counterexample.
Suppose
thereisasmooth
imbedding
of a hornology
3-sphere
H inS5 such
that
the induced push-out diagram is that of 3.1 ß
THOROUGHLY
KNOTTED
SL(2,5)½Z
HOMOLOGY
SPHERES
2'/9
) SL(5,5)OrZ
1
SL(2,5)
>1
Leti: H X D2•--•S5 bea tubular
neighborhood
of H.
Consider
theinfinitecycliccoverX of S5 - (H X D2).X hastwoendsand
boundary H X R. Extend the projection H X R-• R to a proper map f: X-• Rand
homotope rel boundary to make f transverseto 0 in R. By standard surgery arguments
N = ffl(0) maybechosen
sothatN isconnected,
and,since
G isfinitely
generated,
so
thattheinclusion
i•: rr1(N)->Gissurjective.
Thiscontradicts
4.1.
We can eliminate the possibility of such a non-smoothablelocally fiat imbedding
H• S5 in eitherof twoways.Thefirst'applythe transversality
theoryof [9] to
produce a hornology manifold satisfying the hypotheses of N in 4.1 but not the
conclusion, and observe that the proof of 4.1 applied also to N a homology m'anifold.
Thesecond
method'
addtoHanon-smoothable
knotS3• S5withrr1(S5- S3)• Z,
to make H smoothable [ 11 ß
We begin the proof of the theorem.
LEMMA1. The3-S)'lowsubgroups
of SL(2,5)andSL(5,5)are,respectin,ely,
Z3
and Z3 • Z3.
PROOF. A calculation [2, page 491 ] shows that the highest power of 3 dividing
the orders of SL(2,5) and SL(5,5) are 3 and 9 respectively. Thus the 3-Sylow subgroup
of SL(2,5)is Z3 and the inclusionZ3+ Z3C SL(2,5)½SL(2,5)C SL(5,5) is a
3-Sytow subgroupof SL(5,5).
For V a vector space,let V-V denote the symmetric product of V with itself.
LEMMA2.
Thereis a natural
isomorphism
of Z3-rector
spaces
H2(Z3-•
Z3,.Z).H2(Z
34-Z3,.Z)•-,
H4(Z
3+Z3:Z)giren
byso(a
'b)=at3b.
PROOF.
In general
Hq(ZP:Z)
•IZp
qeven
0
It
q odd
[6].
follows from the integral Bockstein sequence that the Bockstein map fi:
280
MARTIN
SCHARLEMANN
H2k-I(z3;z3)->
H2k(z3;z)
isalso
anisomorphism,
andfrom
theuniversal
coefficient
theorem
thatthereduction
homomorphism
r: H2k(z3;z)->
H2k(z3;Z
3) isalso
an
isomorphism.
Letggenerate
H2(Z3'Z).
Sincethe 3-skeleton
of K(Z3,1) is a Lensspace,
hencea manifold.
it followsby
Poincare
duality
inthefieldZ3 thatfi-l(g)
Uggenerates
H3(Z3:Z3),
sorfi(/•-l(g)
rg)= rgUrg+fi-l(g)Ufi(rg)
generates
r(H4(Z3:Z)).
Since
rgisintheimage
of
/•(rg)
= 0. Hence
r(gU g)generates
r(H4(Z3:Z3))
andsogOggenerates
H4(Z3:Z).
Denote g k_J
g by h.
Since
(H*(Z3:Z)*
H*(Z3;Z))5
= 0, theK•inneth
formular
provides
a natural
isomorphism,
(H*(Z3;Z)
•) H*(Z3:Z))4
• H4(Z
3+Z3;Z),
soH4(Z
3+Z3;Z)isgenerated
byhX 1,gXg,I Xg.
LetPi:Z3+ Z3->Z3betheprojections,
i = 1,2,andgi= P•(g)'
Then
H4(Z
3+
Z3;Z)isgenerated
byp•(h)= gl Ogl, gl U g2,andp•(h)= g2U g2'
The Kunneth formula also givesa natural isomorphism
(H*(Z3:Z)
©H*(Z3:Z))2
->H2(Z
3+Z3;Z)
sending
g• 1togland1g>gtog2'Thus
H2(Z
3+Z3;Z).H2(Z
3+Z3iZ)
isgenerated
by gl'gl' gl'g2, andg2'g2.
Since
s•maps
abasis
oftheZ3-vector
space
H2(Z
3+ Z3:Z)-H2(Z
3+ Z3;Z)
toa
basis
oftheZ3-vector
space
H4(Z
3+Z3:Z),
• isanisomorphism.
Q.E.D.
Let
7={_01
_11}
inSL(2.5).
Then
73=1.so7generates
asubgroup
Z3C
SL(2,5)
• SL(5,5).
LEMMA
3. ThemapH3(SL(5,5):Z
3)-->H3(Z3,'Z3),
i•duced
byi•tclttsio•t.
is
surjectire.
PROOF. Let N C SL(5,5) be the norrealizer of the 3-Sylow subgroup generated
by the matrices
a=Q and
b= 7
I
1
.
ThenZ3C N isgenerated
bya. ThemapH3(SL(5,5);Z3)->
H3(N:Z
3) isan
isomorphism
[4].Thus
itsuffices
toshow
H3(N:Z
3)->H3('Z3;Z
3)issurjective.
THOROUGHLY KNOTTED HOMOLOGY SPHERES
281
From the Lyndon spectral sequence we deduce that the 3-primary torsion of
H4(N;Z)
isisomorphic
tothesubmodule
ofH4(Z
3+ Z3;Z)
fixed
bytheaction
(via
conjugation)
of N onZ3 + Z3 [6, page117,3521.Wenowexamine
thisaction.
Write x G N C SL(5,5) as
Since
xGN,xax
-1= anb
TM
forsome
n,m
GZ3.Then
xa=anbmx
implies
X2=•,nx2,
so(I-•,n)x2=0.
Butifn•0inZ3then(I-•,n)=(
I _-l)or(_-I
l).Ineither
case
the
two rowsof X2 areequal.
Similarly
xb=an'bm'x
forsome
n',m'
GZ3,soX1=•,n'x
I and
(I-•,n')x
1=0.
Thenif n' • 0, X 1 hasbothrowsequal.Sincedetx 4=0, oneof n orn' mustbezero.
Furthermore,
since
X4=bmX4
and
X3=bm'X3,
it follows
inasimilar
manner
that
one of m or m' is zero.
n
n
Since
conjugation
byx isanautomorphism
ofZ3 + Z3,thematrix
(m m') must
be non-singular. Hence it is one of the following eight matrices
or
0
+I
---I
0
The universal coefficient theorem gives an isomorphism
H2(Z
3+Z3;Z)
• Ext(Hi(Z
3+Z3;Z),Z).
S/race
Ext is conSravariant
in its firstvariable,
an automorphism
of Hi(Z3 + Z3;Z)
represented
bya matrix
A,induces
anautomorphism
ofH2(Z
3+ Z3;Z)represented
byAT onitsnatural
basis
(gX I,I Xg).Thus
any
xinNoperates
onH2(Z
3+Z3:Z)
by a transposeof one of the above matrices,hence by one of the abovematrices.
ByLemma
2,theaction
ofxonH4(Z
3+Z3;Z),with
respect
tothebasis
given
in
the proof of that lemma, is given by the symmetric product of one of these eight
matrices with itself. There are four possibilities:
---I
,
1
---I
I
282
MARTIN
SCHARLEMANN
Inanycase
x fixes
glU gl+ g2u g2,so
theimage
ofH4(N;Z)-*
H4(Z
3+ Z3;Z)
i*
contains
glUgl+g2u g2'Hence
H4(N;Z)
--*H4(Z3;Z)
isonto.
Since
the3-primary
partofH4(N:Z)
isasubmodule
ofH4(Z
3+ Z3:Z),
each
of
itselements
has
order
atmost
3.Hence
i* issplit
byamap
s:H4(Z3;Z)
-*H4(N;Z).
Consider
the
commutative
diagram
with
rows the
integral
Bockstein
homomorphism
H3(N;Z3)
• , H4(N;Z
) 3 , H4(N;Z
)
i*
s
H3(Z3;Z3
) /• 'H4(Z3;Z)-•'•H4(Z3;Z)
3
Since
H4(N;Z)
-• H4(N;Z)
istrivial
ons(H4(Z3;Z)),
there
isanelement
xCH3(N;Z3)
such
thati*/•(x)generates
H4(Z3;Z).
Butsince
/•i*(x)=
i*/•(x)
and/•:
H3(Z3;Z3)-•
H4(Z
3;Z)isanisomorphism,
i*(x)generates
H3(Z3;Z3).
Q.E.D.
LEMMA4. Thehomomorphism
Hs,(Z3,'Z)
-* H3(SL(5,5),'Z)
isinjectire.
PROOF. Consider the commutative diagram obtained from the
coefficient
universal
theorem:
0
0
Ext(H2(SL(5,5);Z),Z
3) •Ext(H2(•3
;Z);Z
3)
H3(SL(S,S);•3
)
' H3(Z3;Z
1)
, Hom(H(Z3:Z),Z3)
Hom(H3(SL(5,5):Z),Z3)
$
0
35
0
SinceH3(SL(5,5);Z3)
-• H3(Z3:Z3)is onto, Hom(H3(SL(5,5);Z),Z3
Hom(H3(Z3;Z),Z3)isonto.ThenH3(Z3;Z)-• H3(SL(5,5);Z)mustbeinjective.
PROOF
OF4.1. Interms
ofthebordism
groups,
Theorem
4.1claims:
If o•:M3 -•
K(SL(2,5),I)
is a map inducing an isomorphism on fundamental groups, then the
inclusion
•3(SL(2,5)) -> •3(SL(5,5)) doesnotmapo•to zero.
Sincerrl(M) is finite,rr2(M)= 0, andsotheuniversal
coverof M isa homotopy
3-sphere.Thus K(SL(2,5),I) may be obtainedfrom M by attachinga si•gle4-celland
cells of dimension greater than 4. Hence we may take a to be the inclusion, and
THOROUGHLY KNOTTED HOMOLOGY SPHERES
283
furtherdeduce
thatH3(SL(2,5);Z)
isgenerated
byor,[M].
Since the composition
H3(Z3;Z
) ->H3(SL(2,5):Z)
->H3(SL(5,5);Z
)
is non-trivial, the generator cr,[M]
of H3(SL(2,5),Z)must be non-trivialin
H3(SL(5,5):Z).
From the commutative diagram
a G123(SL(2,5))
o•,[M]G H3(SL(2,5):Z)
• 123(SL(5,5))
) H3(SL(5,5);Z)
it thenfollowsthatcrmustbenon-trivial
in I23(SL(5,5);Z).
REFERENCES
1. S. CappellandJ. Shaneson,
Topological
knotsandknotcobordism,
Topology,12(1973),33-40.
2.
D. Gorenstein,Finite groups,Harper and Row, 1968.
3. H. Hopf, Fundamentalgruppe
und zweite BettischeGruppe, Comment.Math. Helv.,
14(1941-42),257-309.
D.L. Johnson,A transfertheorem•br the cohomologyof a finite group,InventionesMath.,
7(1969), 174-182.
5. M. Kervaire,Smoothhornologyspheres
and theirfitndamentalgroups,Trans.Amer.Math. Soc.,
144(1969), 67-72.
6. S. MacLane,Homology,Springer-Verlag,
1963.
7. J. Milnor,Introductionto AlgebraicK-theo•;v,Princeton,1971.
8. M. Scharlemann,Isotopy
and cobordismof hornologyspheres
in sl•heres,to appear.
9. M. Scharlemann,
Tranversality
theories
at dimension
jbur, Inventiones
Math.,33(1976), 1-14.
10. R. Steinberg,
Lectureson Chevalley
groups,(mineographed),
Yale, 1967.
4.
Universityof California,SantaBarbara
Santa Barbara,California93106
ReceivedDecember 16, 1976