SHIBATA, KATSUYUKI

Shibata, Katsuyuki

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A note on the formal group law of unoriented cobordism theory

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Shibata, K.
Osaka J. Math.
10 (1973), 33-42
A NOTE ON THE FORMAL GROUP LAW OF
UNORIENTED COBORDISM THEORY
KATSUYUKI SHIBATA
(Received May, 9, 1972)
Introduction
This is a continuation of the author's previous work [6] on the cobordism
generators defined by J.M. Boardman in [1]. Previously we have used the
Landweber-Novikov operations to calculate the coefficients %2i and z4i+1 of a
primitive element
n
This time we use the Steenrod-tom Dieck operations in the unoriented
cobordism theory ([2], [8]) to deduce that the coefficient z^ for the "canonical
primitive element' ' P0 is represented by the "iterated Dold manifold' ' (Rl)a(P2b)
for ί=2β(2ft+l), where R1(M)=S1x(MxM)/aχT (Theorem 3.2).
In other words, let L=Z2[eί_1: /Φ2*] be the Lazard ring of characteristic
2 and F(x, y)=g-l(g(x)+g(y)) with g(x)=^
ί
ei_1x
i(e0=\ί ezk_=ϋ) be the univer-
ί'^ l
sal formal group law. Then the canonical ring isomorphism of Quillen [5]
φ: L->9ί* sends the generator e^ to [(R1)a(P2b)] for ί=2β(26+l).
We also study the behaviour of the Dold-tom Dieck homomorphism
Rf. 9&*-»3l2*+y defined by Rj([M])=[S^x(MxM)lax T]. In particular, we
present the following product formula (Lemma 2.2)
*X*y)= Σ (ΣΠ[P2,r^W»ωy>ft + «>o »>o
In the final section, we examine the relation between the algebra structure of
yi*(BO(l))^yi*(Z2) and the coalgebra structure of 5ft*(5O(l)). As an applica-
tion, we obtain the following formulas for the Smith homomorphism Δ ([3]);
, β]) [5", a]+[Sm, a] (Δ[S", fl])+[PJ(Δ[5", «]Δ2[5", a]
, and
34 K. SHIBATA
for 2k>m^0 (Corollary 4.3). The former equation would be an answer to a
question of J.C. Su [7] on the relation between Δ and the multiplication in
•Jϊs^ Z j). The latter formula for &<3 was first proved by Uchida [9] utilizing the
multiplicative structures of S1, *S3 and S7.
In the appendix, we state brief comments on the unrestricted bordism ring
of involution I*(Z2) ([3], IV 28). We define the "switching involution" homo-
morphism S: yi^-^I2^(Z2)9 which is a ring monomorphism with a left inverse.
We see, by definition, that Rj=KjoS with Kj the 3^-homomorphism studied
by Conner-Floyd in [4], and thus give a proof for the well-definedness of the
Dold-tom Dieck homomorphism Rj.
The author would like to acknowledge stimulating conversation with Profes-
sor Tammo torn Dieck. The author also wishes to express his gratitude to Pro-
fessor Larry Siebenmann for cordial and constant encouragement.
1. Formal group law in the unoriented cobordism theory
As in [6], let
ft*
be the comultiplication defined by the /f-space map.
The cobordism first Stief el- Whitney class W
ί
 is mapped by μ* to a
formal power series
(1.1)
The formal power series defined by these coefficients
(1.2) F(x,y)=x+y+ Σ atjxfy*
i,j>ί
is a commutative formal group law [5] it satisfies the following properties
(1.3) (1) F(*,0)=0,
(2) F(F(x, y), g)=F(x, F(y, *)) ,
(3) F(x,y)=F(y,x).
The following lemma explains the relation of primitive elements in ίJί1
(BO(l)) to the formal group law F(x,y) of (1.2) .
Lemma 1.4. An element g(W1)=W1+^ίzί_lW1i of yi\BO(l)) is primitive
i>2
if and only if F(x,y)=g~1(g(x)J
Γ
g(y))y where g~\x) is the inverse of g(x)\ g(g~\x))
=g-W)=χ
Proof, If g(W^ is primitive, then
FORMAL GROUP LAW 35
=g'
Conversely, if F(x, y)=g'\g(x)+g(y)), then
μ*g(Wl)=g(μ*W1)=g(F(W1®l,l®Wl))=g(W1®l)+g(l®W1)
Lemma 1.5. Concerning the coefficients of the formal group law (1.2), we
have the following formulas for every integer & > 1 .
(1) *,. *-,=<>.
(2) Σ fllliy[P«*-yJ=0 .k>j>Q
(3)
 tΣo«1>2,[P*_y]2=[P2J.
k- j: even
In the above formulas , P, denotes the real protective space of dimension ί .
Proof. Putting m=\ in (3.4) of [2] (p. 190), we obtain
where H(l, n) is Milnor's hypersurface in PjXP,,. But [P2t _J=0 and [H(l, n)]
=0 for every n^l ([!]). So Σ «ιfy[P»_y]=0. Letting n= 2k— 1, we have
« - / : even
Σ
 Λ
ι,2y-ι [^>2C*-y)]==0 an<i Part (1) follows by induction on k. Analogously
k>j>ι
letting n=2k^2, part (2) follows. Now, from part (2).
βι.*+ [Pa]= Σ^y^α-p] -
So [P2fe]=«1,2*+ Σ ( Σ*>/>o y>»I>
=fl,,2*+ Σ βι.,y ( Σ
*-2>;>0 ^-y-l^
This yields part (3) since ϋft* is a Z2-vector space.
2. Steenrod-tom Dieck operations
T. torn Dieck has defined in [8] the stable cohomology operations
Sf: W*(X) -> 9l*+i(X) (-oo<ί<oo)
such that
(2.1) (a) For
(b)
(c)
36 K. SHIBATA
(d) For the natural transformation μ=$l*( )-»#*( Z2), it holds
that μoRk=Sq
k
oμy (Sgk: Steenrod operation, 5/=0 for K<0).
On the other hand, torn Dieck has also defined in [8] the following mapping
Rji 9Z*-»5K2*+y for j>0; for a closed differentiable manifold M, let Λ/M) be the
orbit space of the free involution (SJχ(MχM), ax Γ), where α is the antipodal
involution and T is the switching map. It was proved in [8] that, if M is bor-
dant to M', then Rj(M) is bordant to -Ry(M') and that this construction yields
consequently a mapping of the bordism set Rji %l*-*3l2*+j
The mappings 7?y are expressed by the operations on 3l*(pt)
and vice versa as in the following lemma. (Recall that we are always identifying
ϊίt with 9Ϊ~*' via the Atiyah-Poincare duality.)
Lemma 2.2.
(1) For xt=yi
m
, Rf(x)= Σ [P/_/]#~w~y(#), βwd consequently,
(2) Λχ*+y)=JR
(3) R,(Xy)= Σ
summation runs through all the sequences of non-negative integers
(n0, n19 ••-, wt , •••) ίw^A ίAβί 2 2i+1ni=j—(k+m) .
i>0
Proof. Part (1) follows easily from (14.1) of [8]. Since the R* are stable
cohomology operations, they are additive and so part (2) follows from (1). For
and eSR,,,
(2.3) Λ/*y)= Σ [ΛJ( Σ Λ-Λ^ 11
y>2, >o ft + ff=«y-2,
by part (1) and 2.1 (c). On the other hand,
Σ [P*]( Σ Λ^Λ^y))
y>2ι>0 k + q=j-2i
=Σ [^ .] [Λ]( Σ [PJ( Σ Λ— '(*)/z-"-'Cy))) by (i)
«,* y>2ι>o « + /=cy-β-*)-2»
=Σ [ΛJ [P«]^_,c
e+B(*y) by (2.3)
Σ
Substituting repeatedly the latter part of the right hand side, we obtain part (3).
REMARK 2.4. In 2.4 below, we give a complete description of the mapping
R1 with respect to the "canonical ring generators" of 9i*. This would be a
partial answer to a question of torn Dieck ([8]) on the behaviour of the mappings
FORMAL GROUP LAW 37
Corollary 2.5. Let 1 = [ρi\ e 9i* fo ίAe Mm'ί element. Then, for —°°<j
<00,
Ri
w
=\fi (f10 ( 7 :
Proof. Fory>0, the assertion is clear by 2.1 (a) and (b) . So let j =—i
(i<0). Then /?,.(!)= [P, ] by definition. On the other hand, by 2.2 (1),
So jR~''(l)= 2 [P,_ y]jR~'(l) and the assertion follows by induction on i. (Of
»>y>o
course, this result can also be obtained directly from the definition of the opera-
tion RΛ)
Corollary 2.6. Let P= H^+ΣX -iWY be a primitive element in W
ι>2
Then, for every integer j( — °°<y<l), RJ(P) is also primitive.
Proof. μ*RJ(P)=&(μ*(P))=R'(Pχ 1 + 1 xP)
= Σ (R'WxR'-'M+R'MxRJ-'p)) by 2.1 (c)
, =a-βO
=R'(P) x 1+ 1 X Λ>(P) by 2.5.
Lemma 2.7. Lei -X^ΣX-i WV *e an arbitrary element of W(BO(\)).
»">!
Then
=Σ( Σ
β Λ l f 2 y Λrβ ^  coefficients in (1.1).
Proof.
=Σ Σ Λ^K -O^WiO by 2.1 (c)
ι>l _/=-«>
=Σ Σ R-^-JRWfi by 2.1 (b)
ί>ι »>y>i -i
} by 2.1 (a)
It was observed in [2] (p. 141) that
y j&t
by 1.5(1)), and /2~'(*i_i)=^i(*<-i) by 2.2 (1). Therefore the lemma follows.
38 K. SHIBATA
3. Determination of Boardman's generators
Let P0=ίF1+Σ*ί-ι^ι''e5K
1CBO(l)) be the (unique) primitive element such
ι>2
that z2k_1=0 (&>1) (see [1], [6] Introduction).
Then we have
Lemma 3.1 R\P0)=P0 .
Proof. By 2.6, R\P0) is primitive, and by 2.7, R\P0) is of the form W,+
Σ#
ί
 -ιW/ιί'> with #2*-ι= ^ ι(Λ?2*-1-ι)= 0 (Λ>1). So, by the uniqueness of such a
ι>2
primitive element ([!]), the lemma follows.
Theorem 3.2 Γλe coefficient zi_l of the canonical primitive element
P0=ίΓ1+Σ>ί_1ίΓ1
f
e5R1(5O(l)) wίίA ^2*-ι=0 (&>1) is the cobordίsm class of thej'^2
"iterated Bold manifold" (/?1)β(PlS)=JR1( . (Λ1(PlΛ)) ) /or ί=2"(2i+l) (α>0, i
>1).
Proof. We prove by induction on fl>0, using 3.1 and 2.7.
(1) In case α=0. By 3.1 and 2.7, we have
Z2b= Σ (*2*)X,2y2ft + >=»*
So #2— (00)2 Λ1|2=[P2] by 1.5 (2), and inductively on b we can deduce, by 1.5 (3),
that*
a
>= Σ'[P2*]2fl1>2> = [P2J.2jfe + y=6
(2) If we suppose that the theorem holds for a— 1>0, then for i=2α(2i+l),
3.1 and 2.7 imply that #i_i=/?i(*y_i) withy=2α~1(2ό+ 1). So, by induction hypo-
thesis, ar
ί
.1=/?1([(/Z1)β 1(Λ5)])=[(Λ1)β(P2»)] as desired.
Corollary 3.4.
(1) ΓA^ cobordism class [(R1)a(P2b)] can be taken as a ring generator of 31*
in dimension 2β(2A+l)— 1.
(2) Denoting [(Rλa(P2b}] by X(a, ft), an additive basis for 5ft* is given by {
(α, ft)2λcα,6)+βcα,6).
 χ
^ ^ >Q, 1 > £ (β, ft) > 0, λ(β, ft) = f(β, ft) =
number of pairs (a, ft)}.
(3) ϊϊΎίA respect to this basis, the additive homomorphίsm
is determined by the following formula
RI( Π X(a, i)««*
«>0,6>1
A)( Π
FORMAL'GROUP^LAW 39
Proof. Part (1) and (2) are the consequence of the fact that the coefficients
#,-_! of the primitive element P0 are indecomposable in Sft* ([!]). Part (3) follows
from 2.2 (3) and the definition of the X(ay b).
4. Bordism algebra of free involutions
In this section we consider the relation between the algebra 9 (^50(1)) with
the multiplication
9?*
and the coalgebra 9έ*(.BO(l)) with the comultiplication
μ*:
3?*
via the cap product ([2] p. 186)
Π : W(X)®MJ(X)
Let η
n
: P
n
-*BO(l) be a classifying map of the canonical line bundle over
PM, and denote by {n} the singular bordism class [Pny ηn]^3ln(BO(l)). It is
well-known that 5R*(fiO(l)) is a free ^-module with basis {{0}, {!},•-, {n}y •••}.
Let α
Λ
(w, n)e5Jl
w+Λ_A. be the element such that
It is equivalent to define [5m, α] [5", α]=Σ«*(^ *)[$"> a} in ^*(Z2) ([9]).
Theorem 4.1.
(1) ak+1(m, ri)= X] at tjak(m~ i, n—j), where the aitj are the coefficients in
ι'f j>0
(1.1).
(2) Σ # -ι#*+ί(w> w)=2'8r/-i(a*(m~' z> W)+<^A(W> w~ 0)> wA^re ίA^ ^^j «rβ
» >o ί>o
ίA^ coefficients of a primitive element P (Uchida [9])).
(3) 2 a2i+1(m, n)[P2i] = [H(m, n)}, where H(m, n) denotes Milnor's hyper sur-
Proof. The proof of [2] XIII (3.3) shows that
{n})=W^ (Σ ak+1(m, n
= Σ Λί.yMί^—O® {»—>})= Σ <iij(ΊloLk(m-ί
ί,y>o 'y i,y>o A
=Σ( Σ a^aάm-ί, n-j)){k} .
k i,j>0
Comparing the coefficient of {k}, part (1) follows. Analogously,
40 K. SHIBAT A
Pn M*( W® M)=Σ(Σ>, -ι«*+, K »)){*}
Part (3) follows from the proof of [2], XII (3.3).
Corollary 4.2. In 3l^(Z2), the following multiplicative relations hold.
(1) [S\ a] [S2n, β]=Σβι>2, [ Sr2""2ί+1, a], where the α l>2/ are determined by the
formula 1.5 (2).
(Uchida [9])
(2) [S\ a] [S2», flHΣ^-DS'"-*"1, a]
i>0
+Σ(«o(2, 2
ί
 -
ί>0
-
1
, α]=Σ{α.(2, 2i-2)+ε(
n
-i)(altiγ+a^1}[S"-2i+\ a]
where 8(n—i)=0 (n—ί: even), =1 (n—ί: odd), and
«0(2, 20=
«.(2, 2f-2;) toiίft α0(2, 0)=0 .
Proof. Letting m=l in 4.1 (1), we have
This yields, by induction on A, the former part of (1) and [S\ a][S2n+\ a]=Q.
Together with 5.1 (2), this in turn gives [S\ fl](Σ[Λy] [S2""2y, β])=Σ( Σ [P*j]
=
^\ a]=[S^\ a]. So [S™+\ a] [S2»+\ a] = [S\ a] [S2Λ+1, a]
[S2m-2^, α])=0.
Analogously, letting m=2 in 4.1 (2), part (2) follows.
Corollary 4.3. Concerning the Smith homomorphism Δ> we have the follow-
ing formulas.
(1) A([Sm, a] [S", «])= Σ βί.yΔ'[5", α] Δy[5", β]
ί.y^o
=(Δ[5", β])[5", α]+[5m, a](A[S», a])
+ [P,](Δ[Sm, a]Δ2[5", α]
(2) Δ2*([5ra, β].*)=[5", α] Δ2» for
(3) Δ2ft([S2*, β] ^ )=[52ft, β] Δ2»
FORMAL GROUP LAW 41
Proof. Part (1) is a paraphrase of 4.1 (1).
Substituting repeatedly the second factor in the right side of 4.1 (1), we obtain
a
s+2k(m, n)=Σ Π ^ iqjq)oίs(m—^iq, n—^jg)
1<Ϊ<2* * q q
= g,K,)rta.O»-2*i, n-2*j)
=α.(», n-2*)+ Σ («, .y)2?*αί(m-2*z) π-2*/) .f>ι»y>o
This yields part (2) and (3).
Appendix. Unrestricted bordism algebra of involutions
In this appendix, we consider the unrestricted bordism module of all involu-
tions (admitting fixed point sets). The basic notations are found in Conner-Floyd
[3], IV 28.
The unrestricted bordism group of involution I*(Z2) has an S^-algebra
structure via the cartesian product. The direct sum *Σβl#(BO(m)) also admits a
multiplicative structure by the formula [M, ξ] [N, ή\ = [MxN, p
Lemma 1. There is the well-defined ring homomorphism
defined by τ[M] = [M, TM], where TM denotes the tangent bundle of M.
Proof. Let W be a manifold giving the bordant relation of M to N\ dW=
M(JN. Then d(W, τ
w
)=(M, τM01) \j (N, τ^φl). So we have
(**.H+ι)*[M, τM]=(intn+1)*[N, rN]
where /
Λ > Λ + 1: BO(n)-*BO(n-\-l) is the canonical map (up to homotopy). But
(i«.«+ι)*: 'WJiBO(n))-+ϊfl
n
(BO(n+l)) is a monomorphism ([3], 26.3). So [M, TM]
— [N, TN] The assertion that r is a ring homomorphism is clear from the
definitions.
Corollary 2. There is the ring homomorphism
defined by S([M])=[MχM, 71], where T(x, y)=(y, x).
Proof. Consider the ring monomorphism i* : I*(Z2) -+ *Σ$l*(BO(m)) of [3]
m
(28.1). By the definition of ί* and the proof of [3] (24.3), i*([M X M, T])=[M,
TM] and i*([NxN, T^])=[^> TΛT] Therefore by the preceding lemma, [MxM,
T]=[NxN, T\ if [M]=[N]. Next we show that S is additive. 5([M]+[JV])
= [(MUΛOx(MUΛO, T]=S([M])+S([N])+[(MxN)\J(NxM), T]. Since
42 K. SHIBATA
any free involution bords in /*(Z
a
), [(MxN) U (NxM), T]=0 and S is additive.
The multiplicativity of S is clear by difinition.
Corollary 3. R—K^Sy i.e. the following diagram commutes
where Rj is the Dold-tom Dieck homomorphism of (2.2) and Kj is the %l*-homo-
morphism defined by Kj([M, μ])=[SJ'xM/aXμ] (Conner-Floyd [4]).
The proof is obvious from the definitions.
Corollary 4. As a ring, I*(Z2) contains the polynomial subalgebra Z2[*S'(^_1) :
/— 1Φ2*-— 1] as a direct summand.
Proof. Let 6: *Σpfl*(BO(m))-*yi* be the augmentation homomorphism in-
duced by the constant map. Then £ oi^oS= id: 9ϊ*~^ ϊϊ*
 an
d £, ί% and S are all
ring homomorphisms. So the corollary follows.
OSAKA UNIVERSITY AND UNIVERSITE DE PARIS (URSAY)
References
[1] J.M. Boardman: Stable Homotopy Theory, Ch. IV, (mimeographed) Warwick
Univ., 1967.
[2] T. Brδcker-T. torn Dieck: Kobordismentheorie, Springer-Verlag, 1970.
[3] P.E. Conner-E.E. Floyd: Differentiable Periodic Maps, Springer-Verlag, 1964.
[4] : Fibering within a cobordism class, Michigan Math. J. 12 (1965), 33
-47.
[5] D. Quillen: On the formal group laws of unoήented and complex cobordism theory,
Bull. Amer. Math. Soc. 75 (1969), 1293-1298.
[6] K. Shibata: On Boardman's generating sets of the unoriented bordism theory, Osaka
J. Math. 8 (1971), 219-232.
[7] J.C. Su: A note on the bordism algebra of involutions, Michigan Math. J. 12(1965),
25-31.
[8] T. torn Dieck: Steenrod Operationen in Kobordismen-Theorie, Math. Z, 107 (1968),
380-401.
[9] F. Uchida: Bordism algebra of involutions, Proc. Japan Acad. 46 (1970), 615-619.
