NAGASAKI, IKUMITSU

Osaka City University Repository

Nagasaki, I.
Osaka J. Math.
22 (1985), 895-905
HOMOTOPY REPRESENTATIONS AND SPHERES
OF REPRESENTATIONS
IKUMITSU NAGASAKI
(Received November 6, 1984)
0. Introduction
T. torn Dieck and T. Petrie have introduced and studied homotopy re-
presentations in [7] and [8]. Let G be a finite group. A G-CW-complex X
is called a homotopy representation of G if the ίf-fixed point set Xs is homo-
topy equivalent to a (dim .X^-dimensional sphere for each subgroup H of G.
(If XH is empty, then we set dim XH= — \ and 5"1 is empty.) A homotopy
representation of G is called linear if it is G-homotopy equivalent to
a unit sphere of a real representation of G. (See [7].) We denote the set
of G-homotopy classes of homotopy representations [resp. linear homotopy
representations] of G by V+(G, h°°) [resp. V+(G, /)]. These sets are commuta-
tive semi-groups with addition induced by join. Let F(G, λ) be the Grothen-
dieck group associated to V+(G, λ) for \=/ or h°*. We call V(G, h°°) the homo-
topy representation group of G and V(G, /) the linear homotopy representation
group of G. The group F(G, h°°) has been studied by torn Dieck and Petrie
([7], [8]) and the group V(Gy /) has been studied by many authors. (See [1],
[4], [10], [11], [13], [15], [18] and [19].)
Let φ(G) be the set of conjugacy classes of subgroups of G and C(G)
be the ring of all integer valued functions on φ(G). For any homotopy re-
presentation X, DimJ£eC(G) is defined by (ΌimX) (H)= dim X*+l, which
is called the dimension function of X. Since Dim ^Γ*y=Dim J£"+Dim Y
(* means the join), the homomorphism Dim: F(G, λ)->C(G) is induced by
the assignment JΪWDim X. This homomorphism is called the dimension
homomorphism of V(Gy λ). The kernel of Dim is denoted by v(Gy λ). torn
Dieck and Petrie have shown that v(G, h°°) is isomorphic to the Picard group
Pic(,4(G)) of the Burnside ring of G in [7].
We are interested in the difference between F(G, /) and F(G, A00). We
observe the homomorphisms which are induced from the inclusion V+(G, ί)
v(G,h~).
896 I. NAGASAKI
These homomorphisms are injective ([7]). We obtain the following results.
Theorem A. The homomorphίsm IG is isomorphic if and only if G is a
cyclic group or a dihedral 2-group D
n
(n>2).
Here D
n
 = <*, b \ a2"'1 = b2 = 1, b~lab = a'1) .
Theorem B. Suppose that G is nilpotent. Then the homomorphism ίG
is isomorphic if and only if G is a cyclic group or a dihedral 2-group D
n
 (n>2).
T. Petrie has already announced the analogous theorem of Theorem A
for oriented homotopy representations in [16], Our Theorems are the un-
oriented versions.
This paper is organized as follows. In section 1 we shall have the neces-
sary conditions. In section 2 and section 3 we shall compute the order of
v(G, h°°). In section 4 the proofs of the main theorems will be completed.
The author would like to thank Professor K. Kawakubo for many helpful
conversations and suggestions.
1. Necessary conditions
Using the results in [5] and [11], we shall show the following proposition.
Proposition 1.1. Let G be an abelian group. Then the homomorphsίm
iG is an isomorphism if and only if G is a cyclic group or Z2 X 2Γ2.
Proof. It is known that the group v(G, h°°) is isomorphic to
ΠZ|G//Ϊ1*/{±1} ([5])
ff^G
and the group v(Gy /) is isomorphic to
([11]).
G /H : cyclic
Here Z\G/H\* is the group of invertible elements in Z\G/H\. Hence it is easy
to see this proposition since iG is injective.
Let H be a normal subgroup of G and X a homotopy representation of G.
Since Xs is a homotopy representation cf G\Hy the correspondence ' X\-+X
H
induces a homomorphism/: v(G, \)-*v(GIH, λ) for λ=/ or h°°. The following
lemma is elementary.
Lemma 1.2. Under the above situation,
i) The homomorphism f is surjective.
ii) The following diagram is commutative.
HOMOTOPY REPRESENTATIONS AND SPHERES OF REPRESENTATIONS 897
v(G, hr)
if
v(GIH, ί) — — > v(G/Hy h00)
iii) If iG is isomorphic 9 then iG/H is also isomorphic.
Proof, i): Let /: v(GIHy λ)->^(G, λ) be the homomorphism which
is induced by considering G/ίf-spaces as G-spaces via the canonical projection
G->G/£Γ. Then, the composition fol is the identity on v(G> λ) and so / is
surjective.
ii): This follows from the definitions of /and iG.
iii): This follows directly from the injectivity of iG/Hy i) and ii).
Let us apply lemma 1.2 to the commutator subgroup [G, G] of G. Then
we have:
Proposition 1.3. // iG is an isomorphism, then G/[G, G] is a cyclic group
or Z2 X Z2. Furthermore suppose that G is nilpotent, then G is one of the following
groups',
(1) cyclic group
(2) dihedral 2-group D
n
 (n > 2)
D
n
 = <G,b\ a2"'1 = b2=l, b~lab = a~ly
(3) quaternion 2-group Q
n
 (n>3)
Q
n
 = <β, b\a2"-1 =l,b2 = cΓ~\ b"lab = α"1)
(4) semi-dihedral 2-group SD
n
 (n>4)
SD
n
 = <a, b\a2"~l = b2=l, b"lab = α-1+2>l'2>
Proof. The first half is clear from Proposition 1.1. Suppose that G
is nilpotent. We may put G=P1X ~χPr, where Pj is a 2-group and P, is
a/>
r
grouρ (p{: odd prime) for z>l. If G/[G, G] is cyclic, then G is also cyc-
lic. In the case G/[G, G]=Z2xZ2, Pil[Ph P, ] is trivial for ί>l. Hence P,
must be trivial for ί>l, so that G is 2-group. Therefore, if G is abelian, then
G is D2 (=Z2xZ2). If G is not abelian, then G is Dn (/z>3), Qn (n>3) or
SD
n
 (n>4) by [9, Chap. 5, Theorem 4.5].
2. The order of the Picard group of the Burnside ring
The group v(Gy h°°) is isomorphic to the Picard group Pic(^4(G)) of A(G),
where A(G) is the Burnside ring of G. In this section we compute the order
of Pic(-4(G)). We recall the Burnside ring A(G). The set of G-isomorphism
898 I. NAGASAKI
classes of finite G-sets becomes a commutative semi-ring with addition induced
by disjoint union and multiplication induced by cartesian product. The Groth-
endieck ring of the semi-ring is called the Burnside ring. Let S be a finite
G-set. The correspondence Si-*!*?^! induces the ring homomorphism %:
A(G)-+ Π ZH, where ZH~Z. As is well-known, the ring homomorphismCff)eφCΦ
% is injective and if we consider A(G) as a subring of Π ZH via %, then
A(G) = {(dff)<= Π ZH\ congruences (*)# for (if)eφ(G)} ,Cff)
where
("V Σ INH/NHnNK\φ(\KIH\)d
κ
==Q mod \NHIH\ .
CE)
The sum is over the ΛΓ/ί-conjugacy classes (K) such that if is a normal sub-
group in K and K/H is cyclic, φ denotes the Euler function and NH denotes
the normalizer of H in G. We can rewrite (*)ff as the following:
(**)*: Σ*(H,K)d
κ
=0 mod\NH/H\,
C-K)
where each n(H, K) is a certain integer and n(H, H)=l for any (if), (K)^φ(G)
such that ^//f is cyclic. The sum is over the G-conjugacy classes (K) such
that H is normal in K and K/H is cyclic (see [6]).
We put φ(G) ={(#!)> •••, (#„)} and assume that (Ht)<(Hj) implies t
>/, where (/f, )<(/fy) means that if,- conjugates to a certain subgroup of £f; .
We set
Note that Λ^Z and R
n
=A(G).
Lemma 2.1. jR^ w ίfe subring of Πiii ^  ,
Proof. It is trivial that Λ
Λ
 is an additive subgroup of ΠΛi -£#,-• We
note that there exists (dffi)1^i^n^A(G) such that dHi=άrHi, \<i<k, for any
(dH^i<k^Rk Let (dHύι&i&k and (eH^^k be any two elements in Λft. Since
^4(G) is the ring, (d^fΉ^^n is in -4(G). Hence (dHieH^^k satisfies (**)jyy,
l<j<k, since (dΉfΉ^^ satisfies (**)ffy, l<j<n. Therefore
is in Λ
Λ
.
We define a map^>: Rk^-^ZlWffkh WHk=NHk/Hk, by
^~ Σ nHk, Hdff modlήpBj/ffk : cyclic
Lemma 2.2. ^> is the ring homomorphism.
Proof. It is trivial to be an additive homomorphism. For (dH^i^k [resp.
A-i, we choose (^ )ι^ ^» [resp. (^is^Je^G) like the proof of
HOMOTOPY REPRESENTATIONS AND SPHERES OF REPRESENTATIONS 899
(2.1). Then
Λ
S
- Σ y
: cyclic
ts- Σy n(Hk, Hj)dHj mod \WHk\lήpSj/Sk : cyclic
<&4 = - Σy n(Hk, #,.)%, mod | WHk\ .
: cyclic
Let siRfr-^Rk^ and r: Rk-*Z be the ring homomorphisms defined by
We have the following lemma. (See [5].)
Lemma 2.3. TTitf following diagram is the pull-back of ring.
_
Ί I*
Z - > Zlwakl (2<k<n)
Here q is the canonical projection.
Proof. It is easy to show this lemma from the definitions of Rk> s, p and r.
The next proposition is the main result in this section.
Proposition 2.4. Let G be any finite group. Then
I Yιc(A(G)} I = 2- I A(G)* \ Π ψ( I NH/H \ ) ,
Cff)eφ(^)
where n= \ φ(G) \ and A(G)* is the unit group of A(G).
Proof. The pull-back diagram in Lemma 2.3 yields the Mayer- Vietoris
exact sequence of the Picard group [6]. That is, the sequence:
0->Rk*->Rk_*<
-> Pic Rk -* Pic Rk^ 0 Pic Z -*> Pic Z\WHk\
is exact, 2<k<n. Since PicZ= 0 and Pic 2 ,^1=0 by ([2], chap. 2, 5), we
obtain the exact sequence:
-» Pic Rk -» Pic Rk^ -> 0 (2<k<n).
900 I. NAGASAKI
Inductively, Pic Rk has a finite order and then we have
_φ(\WHk\) \Rk*\
2 1 Γ) s{c II -K k—i I
Therefore,
1 I J? * I n
—
 L
 lκ» I TT m(
Since R,=A(G), Rl=Z and ?>( | WHl \ )=1, the desired result holds.
Corollary 2.5. | β(G, A-) | =2'" 1 4(G)* | Π φ( I JVH/ff | ), n= \ φ(G) | .
CZQ
3. Examples
We indeed compute the order of Pic(^4(G)) (i.e. v(G, h°°)) for some groups.
EXAMPLE 3.1. Let G be D
a
 (n>2) in Proposition 1.3. Then \Pic(A(G))\
=2", where JV=(β-2) (n-3)/2.
Proof. Conjugacy classes of subgroups of D
α
 are the following:
(A,)
,* = 0,1, .•.,»-!
and WH of these subgroups in G are the following:
WD
n
=\
2
'> = A+i, * = 0, 1, .. ,n-
= Z2) « = 1, 2, ••-, w-1 .
Hence |φ(G)|=3»-l and Π φ(\ WH\)=2^"-^. We see that \A(G)*\ is
CH)
2«+2 by Theorem 4.1 or Example 4.8 in [14]. By Proposition 2.4,
I Pic(A(G)) I - 2-<3*-1> x 2n<n-»'2 x 2n+2
__ 2^v
One can show the following Examples 3.2 and 3.3 by the same argument.
EXAMPLE 3.2. Let G be Q
n
 (ra>3) in Proposition 1.3. Then |Pic (A(G))\
=2N+\ where N=(n-2) («-3)/2.
EXAMPLE 3.3. Let G be SD
n
 (rc>4) in Proposition 1.3. Then | Pic(A(G)) \
HOMOTOPY REPRESENTATIONS AND SPHERES OF REPRESENTATIONS 901
=2*, where N=(n-2) (n-3)/2.
EXAMPLE 3.4. If v(G, h°°)=Q, then \G\=2" or 2nρ1ρ2 - ρr, where ρ{,
l<i<r, are distinct odd primes of forms 2e+l.
Proof. Put \G\=2npιi ••• pe
r
r
, whereph l<i<ry are distinct odd primes.
Then φ(\G\)=2--\p1-l)p^1 - (pr-\)p^\ If »(G, A")=0, then φ(\G\)
must be 2-ρower by Corollary 2.5. Hence £,— 1 and pi—1 is 2-power for
any t.
EXAMPLE 3.5. Let S
r
 be the symmetric group on r letters, where r—3,
4 or 5. We have the following table.
G
S3
S4
S5
V(G,i)
z3
z5
z7
V(G, h")
Z4
Z8
ZK®Z2
Proof. We note that F(G, λ)=Dim V(G, \)®v(G, λ) and Dim F(G, X)
is a free abelian group, λ=/ or A00. The rank of Dim F(G, /) is equal to the
number of conjugacy classes of cyclic subgroups of G by [6]. The rank of
Dim V(G, h°°) is equal to the number of conjugacy classes (H) of subgroups
H of G such that H/[H, H] is cyclic by [7]. By these facts Dim F(G, λ) is
computable. For the symmetric group on n letters S
n
, v(S
n
, /)=0 by [12]
or [6], Now we compute v(S
r
, h°°) for r=3, 4 or 5. In case r— 3, the diagram
of conjugate subgroups of S3 is the following:
Z3
\
\ /
1
and WS3=l, WZ3=Z3, WZ2=\ and Wί=S3. Hence |φ(53)|=4 and Π φCH)
( I WH\)=2. The order of ^(S3)* is 8 by using the congruences (*)a in sec-
tion 2. Therefore V(S3y h°°)=0. In cases r=4 and r=5, using the diagrams
of conjugate subgroups of S4 and /S^ in [14], one can show 1^(54, h°°)\ =1 and
\v(S5, h°°) I =2 respectively. Hence v(S4, h°°)=0 and v(S^h°°)=Z2.
EXAMPLE 3.6. Let G be the symmetric group on n letters. Then iG is
902 I. NAGASAKI
an isomorphism if n<4 and is not an isomorphism if n>4.
Proof. By [12], v(S
n> /)=0. for any positive integer n. If n>6y then
v(SM,h-)*Q by Example 3.4. Since v(Sryh~)=Q (r<4) and v(S5,h°°)=Z2 by
Example 3.5, the desired result follows.
REMARK. v(G, h°°)=Q implies that the stable G-homotopy class of the
homotopy representation X is decided by its dimension function Dim X> where
homotopy representations X and Y are stable G-homotopy equivalent if there
exists a homotopy representation Z such that X*Z and Y*Z are G-homotopy
equivalent.
4. Proofs
We shall prove Theorem B by compairing orders of v(G, /) and v(Gy h°°).
Let m be the exponent of G and u
m
 a primitive m-th root of unity. Let Γ be
the Galois group Gal(Q(u
m
)/Q). Γ acts on the set Irr(G, R) of real irreducible
characters of G by (γ χ) (g)=γ(X(g)) for %e!rr(G, Λ), <yeΓ and #eG. We
need the following theorem in [4] and [6] in order to compute the order of
Theorem 4.1 (T. torn Dieck). Let G be α p-group. Then v(G, /) is iso-
morphic to 0 Γ/Γ^, where X=Iπ(G> R)fΓ and Γ^=
REMARK. For any group G, ^(G, /) is isomorphic to a quotient group of
(See [4] or [6].)
Lemma 4.2. For Z)M, Qn ^/zJ 5Dn m Proposition 1.3, α;£ A^z β |^(Z)n, /)|
=2", \v(Q
ny J)\=2N and \v(SDny t)\=2N~\ Here N=(n-2) (n-3)/2.
Proof. We need the real irreducible character tables of D
ny Qn and SDn.
By [17, Chap. 13, 2.], we have the next tables.
In the D
n
 case:
*1
02
3^
04
%A
fl*
1
1
(-1)*
(-1)*
u^Λ-u-^
a
kb
1
J
(-i)4
(_l)m
0
Here l<h<mβ, m=
HOMOTOPY REPRESENTATIONS AND SPHERES OF REPRESENTATIONS 903
In the Q
n
 case:
In the SD
n
 case:
01
02
03
04
X*
ψ,
a"
1
1
(-1)*
(-1)*
u^"+u^hk
ΨΛβ*)
α*ό
1
-1
(-1)*
(-!)*«
0
0
01
02
03
04
%*
^5
cf
1
1
(-1)*
(-1)*
u^+u^
Ψ.(«*)
α*ό
1
-1
(-1)*
(_!)*«
0
0
Here ψ
 s
(α*)=«L2s"1)*+M»(2s-1)*
,
 M(m/2-(2s-l))A_j_M-(m/2-(2s-l))A
, l<s<m/8, m=2"-1.
By the irreducible character tables of D
n
, Q
n
 and »SD
Λ
, we have
X = Irr(G, Λ)/Γ
=
} if G = Z),,
-3} if G=Q
β
{{θ,}, At, C|l<^4, l<ί<«-3> if G=SDH
Here A= {%* I h=?^ mod 2'}
904 I. NAGASAKI
Hence,
'
TAt =
 \ 2"-*-' if G = SD
Λ
 or Q
n
 ,
and
|Γ/ΓC| = 2«-4.
Therefore, by Theorem 4.1, we have
|5| Π 141 =2" i f G = Q.
I C I Γ M Λ I =2"-' i(G=SD
n
.
t=l
Proposition 4.3. The homomorphism ίG is an isomorphism if G=Dn (n>2)
and is not an isomorphism if G=Q
n
 (n>3) or SD
n
 (n>4).
Proof. The desired result follows from Examples 3.1, 3.2 and 3.3 and
Lemma 4.2.
Corollary 4.4. In the case G=Q
n
 or SD
ny v(G, /) ύ a subgroup of index 2
ofv(Gyh~}.
Proof of Theorem B. This follows directly from Propositions 1.1, 1.3
and 4.5.
Proof of Theorem A. Theorem A follows from Theorem B and the theo-
rem of torn Dieck and Petrie ([3], [7]), that is, Dim F(G, /)=Dim F(G, h°°)
if and only if G is nilpotent. Indeed, if IG is an isomorphism, then G is nil-
potent since Dim V(G, /)=Dim V(G, h°°). Since iG is also isomorphic, G is
a cyclic group or D
n
 by Theorem B.
Conversely, suppose that G is a cyclic group or Z)
Λ
, then iG is an isomor-
phism and Dim V(G, /)—Dim V(G, h°°). It is sufficient to show that IG is
surjective. Let x be any element of F(G, h°°). Then there exists an element
u^v(G, /) such that Dim #=Dim u. Hence x—IG(u) is in v(G, h°°). Since
iG is an isomorphism, there exists an element v^v(G, /) such that x—IG(u)
=iG(v). Hence x=IG(u-\-v). Therefore IG is surjective.
HOMOTOPY REPRESENTATIONS AND SPHERES OF REPRESENTATIONS 905
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Department of Mathematics
Osaka University
Toyonaka, Osaka 560
Japan



Homotopy representations and spheres of representations

Nagasaki, Ikumitsu

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Homotopy representations and spheres of representations

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