INOUE, TSUYOSHI

Osaka City University Repository

Inoue, T.
Osaka J. Math.
27 (1990), 421-430
A NOTE ON THE EQUIVARIANT WHITEHEAD
GROUPS OF DIHEDRAL GROUPS
Dedicated to Professor Shόrό Araki on his 60th birthday
TSUYOSHI INOUE
(Received March 23, 1989)
0. Introduction
This note is intended as "The equivariant Whitehead torsions of equivari-
ant homotopy equivalence between the unit spheres of representations I I " .
Therefore, we shall use the notations in [11], In this note, restriction maps in
Whitehead groups play an importnat role. To illustrate this, we begin with
an example pointed out by M. Masuda. Let C
n
 and D
n
 be the cyclic group and
dihedral group of order n and 2n respectively. As we remarked in [11], a gen-
erator of Wh(C5) appears as the reduced equivariant Whitehead torsion of any
C5-homρtopy equivalence
where V
a
 ( Λ = 1 , 2, 3) denotes the complex C5-module C with g^C5 acting as mul-
tiplication by exp2τr/α/5 and S(V) denotes the unit sphere of C5-module V.
Since the torsion does not depend on the choice of /, wre can assume that / is
the map due to T. Petrie (see §2). By the complex conjugation, C5-modules
V
a
 can be regarded as Z)5-modules. Then the Petrie's map / turns out to be a
Dg-homotopy equivalence. The reduced equivariant Whitehead torsion ψDs(f)=
P*τD5{f) of / as a D5-homotpoy equivalence lies in WhDs(*)^Wh(D5) wherep%:
WhDs(S(V3® F2))-» WhDs(*) is the induced map by the obvious map p: S(F30 V2)
—>*. It is obvious that the restriction map from D5 to C5 sends the torsion to the
generator of WhCs(*)^Wh(C5). Therefore the restriction map induces an iso-
morphism of the Whitehead groups because Wh(D5) is a free abelian group of
rank 1 (see [3], [21], [19], [20] and [17]). Moreover we see that the torsion is a
generator of Wh(D5). Our main result (Theorem A) is a generalization of this
observation.
Theorem A. The restriction map induces an isomorphism
422 T. INOUE
where Wh
τep(G) denotes the subgroups of WhG{*) generated by the reduced torsions
of G-homotpoy equivalences between the unit spheres of G modules.
By the Thorem A, the same conclusion as [11, Theorem C] holds for dihed-
ral groups.
Corollary B. Whtep(Dn) is of finite index in WhDn(*) if and only if n=
8, 9, 12, 16, 18, p or 2p for odd prime integers p.
In §1, we discuss the restriction maps of Whitehead groups from dihedral
groups to cyclic groups. We give a sufficient condition for the restriction map
being an isomorphism. In §2, we investigate the C^-homotopy equivalences
between the unit spheres of C
n
-modules due to T. Petrie. In §3, we state the
main results and prove them. We also exhibit an example concerning genera-
tors of Whitehead groups of dihedral groups in §3.
The author owes to Professors Shόrό Araki and Mikiya Masuda by useful
discussions and advices, and would like to express here his hearty thanks to
them.
1. The restriction maps from dihedral group to cyclic group
In this section, wτe shall investigate the restriction map of Whitehead groups
from a dihedral group to a cyclic group. First, we consider the standard involu-
tion on Whitehead groups. Let G be a finite group. The assignment "gi-*g~ι"
in G induces a conjugation ~: Z[G]—>Z[G]. This conjugation induces the
standard involution ~: Wh(G)->Wh(G). The following lemma is fundamental
in our investigation.
Lemma 1.1. Let G be an abeliangroup. Then, each element of(Z[G])*l±G
is represented by a unit u^(Z\G\)* such that u—U. In particular, if Wh(G) is
torsion free, each element of Wh(G) is represented by a unit u£z(Z\G\)* such that
u=U.
Proof. It is well known that the standard involution on Wh\G)=Wh(G)/
torsion is trivial (see [24], [2] or [16]). According to the proof of [2] for this fact,
for each wG(Z[G])*, there exists g
o
^G such that u-(U)~1=^zg0. Applying the
augmention map Z[G]-*Z to both sides of the identity, we see w(u)~1=g0.
Here, we consider an involution θ: G->G, θ(g)=g
o
g~1. If we put u=*Σ.agg
)y the identity u (U)~1=g0 implies
ag = aθ(g) for each
Therefore, θ must have a fixed point because Σ ag=±l. The fixed point of
θ, say g^G, satisfiesg2=g0. If we put v=g~
ι
 u, v is a required element because
v=u in (Z[G\)*I±G and
EQUIVARIANT WHITEHEAD GROUPS 423
v=g-1u = g-1g
o
π = gπ = ϋ. Q.E.D.
NOTATION 1.2.
D
n
: the dihedral group of order 2n generated by two elements s and t
with relations tn=s2=ί and sts=t~1.
C
n
: the cyclic subgroup of D
n
 generated by t.
In later sections, we shall consider the equivariant Whitehead group of D
n
(called the generalized Whitehead group of D
n
 by Rothenberg). Therefore, we
shall treat the classical Whitehead groups and the equivariant Whitehead groups
at the same time. To do this, we need the following lemma.
Lemma 1.3. WhDm(*)=®ilΛ Whψd)
and the following diagram commutes
WhDn{*)
^BRes z ? ' f
U Wh{Dd) ^ - > c " ®dinWh(Cd).
Proof. For a subset A of D
n
, we denote by ζAy the subgroup generated
by A. Since <rf*, stmy=<Jίk~m, stmy in D
nj any subgroup of Dn has a form <**>
or <**, stm>. On the other hand,
[ <**, ίί> if m is odd,
ζtk, stmy is conjugate to .
(ζr, sy if m is even.
Moreover, if n is odd, <ί*, sty is conjugate to ζtk9 sy. Therefore, C(Dn), the
conjugacy classes of the subgroups of D
n
, is
ίi(<f>)Λ<*>*>)\d\»} if/* is odd,
I { Wy)> K^> sy)> i^y sty I ^  I ^  } if w is even.
Moreover, we have
D
n
, W<?> = N<t<>l<t<> = Dd ,
if d is odd, Γ 1 if rf is odd,
if d is even, ' I C2 if rf is even,
if d is odd,
 ΎΎΎ/A v (1 if J is odd,
C2 if d is even,
where iVfiΓ denotes the normalizer of HdD
n
 in D
n
 and WΉ" denotes NH\H.
Since PFA(C2)=0, we have
WhDM
f  if d is
L 2 if d is
(</,rf> if rf i  ,
 y ^ (
, sty = ; 2 Ϊ F ^ , to> =
I <Γ/2, ίί> if J is even, I
424 T. INOUE
By the definition of RescJJ: WhDJ*)-+WhcJ*)f we have the commutative diagram
Res?;
WhDn{*) 1 WhCn(*)
ί ί
WhD.{*,{<f») —* Whcu{*,{<f»)
This completes the proof. Q.E.D.
Lemma 1.4. Res?- Wh(D
n
)-^Wh(C
n
) and Res?": «%*,(*)->W^CM(*) or*
monomorphisms.
Proof. By Lemma 1.3, it is sufficient to show that Res^: Wh(D
n
)->Wh(C
n
)
is a monomorphism. We note that Wh(D
n
) and Wh(C
n
) are free abelian groups
of the same rank by [21], [19], [20] and [17]. Moreover
Resg; Indg; y = y8 for each y<Ξ Ψ7z(C
w
).
Therefore Ind^: Wh(C
n
)->Wh(D
n
) is a monomorphism and its image is a sub-
group of finite index. So, for each # e Wh(D
n
), there exixt m^Z adn^/e Wh(C
n
)
such that Λ;w=Indc^ J>. Suppose that Res?" x=l, then
1 - (Res?« *) " = Res?: xm - Res?: Indg y = y2.
Since Wh{C
n
) and W%(Z)W) are torsion free, we have y=\ and JC=1. This com-
pletes the proof. Q.E.D.
Now we shall observe the classical restriction homomorphism of the unit
groups. The point of our observation is to consider C2n and Dn parallelly.
Let r be a generator of C2M. Identifying t~r
2
y we can regard Cn as a subgroup
of C2M. Because each element of Z\Dn] can be expressed by a+sb, a,
we can define a homomorphism
(Z[D
n
])* -> (Z[CJ)*
Similarly, we can define a homomorphism
(Z[C2n])* -> (Z[CJ)*
The above two homomorphisms are the classical restriction homomorphisms
in the following sense.
Lemma 1.5. The following diagrams commute.
EQUIVARIANT WHITEHEAD GROUPS 425
(Z[D
n
ψ — (Z[C
n
])*
{Z[C2nψ -^ (Z[CJ)*
Proof. If we regard a+sb<=(Z[D
n
])* as a Z[CB]-isomorphism
Z[D
n
] and take basis 1 and s of Z[D
n
] as a Z[C
n
]-module, then α+s& is ex-
pressed by a matrix
b
Since
\b a)
det [ J = aa—bb ,
\b a)
we have the commutativity of (1) by the definition of Res?". By the same argu-
ment, we have the commutativity of (2). Q.E.D.
Using the above lemma, we have the following.
Proposition 1.6. If Resgf: Wh(C2n)->Wh(Cn) is an epimorphism, Res?^:
Wh{D
n
)-> Wh(C
n
) is an isomorphism.
Proof. By lemma 1.4, it is sufficient to show that Res?" is an epimorphism,
i.e., for each x<= Wh(C
n
), there exists y<= Wh(D
n
) such that Rescϋ y=x. By the
assumption, there exists a yf^Wh(C2n) such that Resc^wy=#. According to
Lemma 1.1, yf is represented by a unit a-\-rb^(Z[C2n])* such that a-\-rb=a-\-rb.
Since the condition a-\-rb=a-\-rb implies a=a and b=br2=bt, it is easy to see
that a-\-sb is a unit of Z[D
n
]. By lemma 1.5, Res?" sends a-\-sb to aa—bb —
a
2
—tb2 at the unit level. On the other hand Res£f sends a+rb to a2—tb2.
Therefore a-\-sb represents the required y. Q.E.D.
EXAMPLE 1.7. Res£f Wh(C2n)->Wh(Cn) is an epimorphism in the following
cases.
(1) n:odd.
(2) n=S or 12.
But if n=2k(k^tA), Res£f is not an epimoephiam.
Corollary 1.8. // n is odd or n=8,12, Res?": Wh(D
n
)->Wh(C
n
) and
Res?": WhDn(*)->WhCn(*) are isomorphisms.
Proof of Example 1.7. In the case (2), since the generator of Wh(C
n
) is
426 T. INOUE
known (see [11]), a direct computation shows that RescΓ is an epimorphism.
By the following Lemma 1.9, it follows from [5, Theorem 3] that Rescf is an
epimorphism if n is odd. The example that Res£f is not an epimorphism is
given by [9, Theorem 1.1]. Q.E.D.
Lemma 1.9. The following are equivalent to each other:
(1) ResgΓ: Wh(C
mn
)^Wh(C
n
) is an epimorphism.
(2) t7: (RcJ*l±C
mn
-+(RcM)*l±Cn is an epimorphism where RCn=Z[Cn]l(Σgecng)
(see [5] and [9] for the definition of tr).
(3) Any free C
n
-action on S2k+1(k^2) extends to a free C
mn
-action.
Proof. [5, Theorem 4] shows that (2) and (3) are equivalent to each other.
To show (1)«=>(2), we note that there exists a split extension
1 - Wh{C
n
) - ( i ?J*/±C B i (Z/nZ)*/±l -+ 1
where A: (i?cB)*/± CH-* {ZlnZ)*l± 1 is induced by the augmentation. Moreover
we have the commutative diagram
1 -> Wh{C
mn
) - {R
c
J*l±C
mn
 - {Zlmn
1 Resgr tr" [ I
\->Wh{C
n
) -
where (Z/mnZ)*l^t\-^(ZjnZ)*j±l is the natural map. A simple diagram
chasing shows that (1) and (2) are equivalent to each other. Q.E.D.
2. The Petrie's maps
In this section, we shall discuss an interesting example of maps between
C
n
-modules due to T. Petrie.
NOTATION 2.1.
V
a
: The complex C
n
-module C with g^C
n
 acting as multiplication by
exp 2πiajn.
Let a and b be integers which are relatively prime and prime to n. Choose
integers^, q such that —ap-\-bq=l. It is well known that the Petrie's map
/: V.®Vt -
(x,y) .-»(«»jH, * » + / )
is a C
β
-homotoρy equivalence. This induces a CB-homotpoy equivalence
(x,y) M. f(x,y)l\\f{x,y)\\
EQUIVARIANT WHITEHEAD GROUPS 427
which will be also called Petrie's map.
Lemma 2.2 Let V and V be complex C
n
-modules such that C
n
 acts freely
on S(V) and S(V). If S(V) and S(V) are C
n
-homotopy equivalent, then one
can choose a C
n
-homotopy equivalence as composition of suitable suspension of
Petrie's maps, inverse of Petrie's maps, and a complex conjugation.
Proof. Let 0 V
a
. be a direct sum decomposition of V to irreducible O
n
-
modules. Since C
n
 acts freely on S(V), each a{ is prime to n. Relacing a{
with a{-\-mn, we can assume a{ (i=ίy •••,./) are mutually distinct prime integers.
Now we have a composition of Petrie's maps
/: S(V) =
k
Similarly for V'= 0 Vb., we have a composition of Petrie's maps
/': S(V) - SiV& ' Θ ^ Θ ^ J .
Since S(V) and S(V) are C
κ
-homotopy equivalent, we have
j = k and a
x
 aj = db^ &y (mod n)
In case al'"aj
:
=b
ι
'"bj (mod ή)y f'~ι° f is a required CΛ-homotρoy equivalence.
In case a^ -aj^—b^ bj (mod ή)yf'~1ocofisa. required one where
is a suspension of a complex conjugation. Q.E.D.
Since D
n
=O
n
Y\O2y Va can be considered as a real 2)w-module on which
ί ^ C 2 acts by complex conjugation. The following lemma was pointed out by
M. Masuda.
Lemma 2.3. The Petrie's map
is a D
n
-homotopy equivalence.
Proof. A direct computation shows that h is a D
n
-map. Therefore it is
sufficient to show that h is homotopy equivalence on the fixed point set of each
subgroup H of D
n
. We shall show that
/ : R2 -> R2
428 T. INOUE
has degree ± 1 . This is sufficient because h is C
n
-homotopy equivalence.
To calculate the degree of/, we consider the image of S1={(cosθ, sin0)ei2 2 |
by/. W e p u t S ί l Λ = { ( c o s β , s i n β ) | ^ β ^ ^ . T h e n S ^ S j ^ U
Slf3ηe/2 U SIK^W We shall distinguish the following four cases.
(1) a: odd, b: even, p: odd and q: even.
(2) a: even, h: odd, >^: even and q: odd.
(3) α: odd, b: odd, >^: even, and q: odd.
(4) «: odd, b: odd, >^: odd, and q: odd.
We note that the other cases do not occur by the choice of ay b, p and q. Since
the arguments for the cases (1), (2), (3) and (4) are similar, we shall only discuss
the case (1). In this case,
/(Sl^) is a loop at (0, 1) in i(
f{Sl/2t«) is a loop at (0, 1) in {(x9
f(Sl.*ώ is a path from (0, 1) to (0, -1) in i(xyy)\x^0} and
f(S\*n.*) is a path from (0, -1) to (0,1) in \(xyy) \x^0}.
Therefore / must have degree + 1 . Q.E.D.
Using the above lemma, we have
Propisition 2.4. Let U and U' be real O
n
-modules such that S(U) and
S(U') are C
n
-homotopy equivalent. Then there exist real D
n
-modules V and V
such that
(1) Res?" V=U and Res?: V = U',
(2) S(V) and S(V) are D
n
-honιotopy equivalent.
Proof. We write
U=®HCCn U(H) and U' = ®SCCH U'(H)
where U(H) and U\H) collects the irreducible submodules of U and V res-
pectively which have kernel H. It is well known that S(U) is homotopy equi-
valent to S(U') if and only if S(U(H)) is homotopy equivalent to S(U'(H)) for
each HdC
n
. Therefore, it is sufficient to show this lemma for each U(H) and
U\H). In case H=C
n
 or the subgroup of index 2, it is obvious. Since C
n
\H
acts freely on S(U(H)), we may assume that C
n
 acts freely on S(U) and S(U').
If we can choose a C
n
-homotopy equivalence S(U)->S(U') as a Petrie's map
(or its suitable suspension), the Petrie's map itself gives a D
n
-homotopy equiva-
lence by Lemma 2.3. Of course, the complex conjugation gives a D
n
-homotopy
equivalence. This together with Lemma 2.2 completes the proof. Q.E.D.
3. Main results
Finally, we state our main results which are easy consequences of previous
EQUIVARIANT WHITEHEAD GROUPS 429
sections.
Theorem A. RescJ: Wh
τep{Dn)-> WhTep(Cn) is an isomorphism.
Proof. Since Wh
τep(Dn) and Whlep(Cn) are subgroups of WhDn{*) and
WhCn(*) respectively, Lemma 1.4 shows the injectivity. On the other hand
Proposition 2.4 shows the surjectivity because the reduced torsion depends only
on G-modules if WhG(*) is 2-torsion free. Q.E.D.
Using [11, Theorem C], we have a corollary to Theorem A.
Corollary B. WhIep(Dn) is of finite index in WhDn(*) if and only if n=
8, 9, 12, 16, 18, p or 2p for odd prime integers p.
We shall conclude this note by referring the generators of Whitehead
group of dihedral groups.
EXAMPLE. The generators of Wh(D5), Wh(D8) or Wh(D12) are given by
the reduced torsions of D
Γ
homotopy equivalences between the unit sphere of
D
r
modules. The units which represent the generators of Wh(D5), Wh(D8)
and Wh(D12) are
(1) l+( ί+r 1 )- ( ί 2 +Γ 2 )+5(-2+( ί 2 +Γ 2 ) ) in case Wh(D
s
),
(2) -\+(t2+t-2)+s(t-?-t*+Γ2) in case W%(Z)8),
(3) 4+2{t+r1)-{f+r2)-(t4+rA)-(f+r5)-f
+s(3+t-t2-t3-t*-?-f-r5-r*-t-3+r2+3Γ1) in case Wh{D
ι2).
Proof. We note that the generators of Wh(C5), Wh(CB) and Wh(Cι2) ap-
pear as the reduced torsions of the Petrie's maps *S f(F20F3)-»5 l(F1φF1),
5(Γ 3 ΘF 5 )->5(F 1 0Γ 7 ) and S{V^®V7)-^S(VX®V^ respectively (see [11]).
Therefore the reduced torsions of the above Petrie's maps (as Z)
w
-homotopy eq-
uivalences) represent each generator of Wh(D5)> Wh(D8) and Wh(D12). Using
the method of Proposition 1.6, we can find the elements of (1), (2) and (3).
Q.E.D.
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Department of Mathematics
Osaka City University
Sugimoto-3, Sumiyoshi-ku
Osaka 558, Japan
and
Intelligent Wave Inc.
C. ItohBldg. 8F 2-5-1,
Kita-Aoyama, Minato-ku,
Tokyo 107, Japan


A note on the equivariant Whitehead groups of dihedral groups (Dedicated to Professor Shόrό Araki on his 60th birthday)

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A note on the equivariant Whitehead groups of dihedral groups (Dedicated to Professor Shόrό Araki on his 60th birthday)

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