SHIBATA, KATSUYUKI

Shibata, Katsuyuki

Osaka University Knowledge Archive

On Boardman's generating sets of the unoriented bordism ring

Osaka City University Repository

Shibata, K.
Osaka J. Math.
8 (1971), 219-232
O N BOARDMAN'S GENERATING SETS OF
THE UNORIENTED BORDISM RING
KATSUYUKI SHIBATA
(Received September 24, 1970)
Introduction
For a pointed finite CWFpair (X, A), define as usual the ^ -dimensional un-
oriented cobordism group Sίlk(X, A) of (X, A) by
3ί*(-Y, A) = lim [S"-"(XIA), M0(n)],
n
and denote Σ 1flh(X, A) by <3t*(X, A).
We identify the coeficient ring 31* with the unoriented bordism ring Sft* by
the Atiyah-Poincare duality [2]
D : yik -> 91-* .
Let PM be the w-dimensional real projective space and ??„ be the canonical
line bundle over P
n
. Define
= lim
n
where W1= lim WJji^) is the cobordism first Stiefel-Whitney class [4]. On
n
account of the Kunneth formula, the homomorphism
induced by a continuous map μ
m
 „ satisfying μ%,
n
v
m
+
n
 = π*V
m
®π*VH gives
rise to the comultiplication
μ* :
Let
P = ^ 1 + ^ ί
be a primitive element in ?ί*(βθ(l)) with respect to this comultiplication. Such
220 K. SHIBATA
elements exist ([3]). Fix once and for all a primitive element P of such kind.
Following Novikov [8, appendix II], we define in section 1 a cobordism stable
operation ΦP which is a multiplicative projection characterised by the formula
ΦP(W1) = P.
The restriction of the natural transformation
μ I Image ΦP : Image ΦP -> H*(X, A Z2)
is a natural ring isomorphism in the category of finite CW pairs. And this induces
a natural Si*-algebra isomorphism
5ft*(X, A) » 91* ® H*(X, A;Z2).
Conversely, any such natural isomorphism, commuting with suspensions, is
induced by ΦP for some choice of a primitive element P.
In section 2, we study the relation between the operations S
ω
 and S
ω
 defined
in [8]. The result is applied in section 3 to prove that the coefficient z^ of a
primitive element P is the bordism class [P2k] °f the real projective space for
each k^O.
And the coefficient z4k+1 is shown to be the class [P(l, 2k)] of Dold manifold
[5] in section 4.
The coefficients z£ of dimensions i other than 2k and 4&+1 are expressed as
very complicated polynomials in the generators of Dold [5] or of Milnor [7].
The present paper is motivated by the following classification theorem stated
in the proof of Theorem 8.1 in [3].
Theorem. P. (Boardman [3])
For an arbitrary family of decomposable elements {jv'-i ί JV-i^^2f'-i> z'^1},
there exists one and the only one primitive element
in ^*(£O(1)), satisfying
The coefficients zk_x with k not a power of 2 are a set of polynomial generators
for 5ft*.
Moreover, if z2t_1=z/2i_1for 1 fgz ^n for primitive elements
P = W1+z2W1
3+ztW*+z5W1
e+z6WS+z7W°+
and
P' =
BOARDMAN'S GENERATING SETS 221
then zk-i
=zk-i for k not a multiple of 2n+1.
The author wishes to thank Professors M. Nakaoka and F. Uchida for their
advices and encouragement.
1. Operation ΦP
Let e^*(0)= 2 c^ WO) denote the ring of stable operations in the un-
oriented cobordism theory. There is an isomorphism of ϋft*-modules ([6], [8])
Ψ : oϊ*(0) - 5R* ® Z2[[Wl9 W2y ..., Wk9 •••]],
where 9Ϊ* is identified with Jί* by the duality and (g) denotes the complete tensor
product.
For a partition ω=(i19 i29 •••, ίr)> denote PFω the symmetrized monomial of
the Wk and the operation SωEΞ^?*(0) is defined by 5 β=Ψ" 1(PFω).
For a primitive element
P = W1+z2W1
3+ziW1
5+z5W*+zW+z7W*+-
in 9l*(.BO(l)) and for a partition ω = ( ί l f ί2, •••, ir), we denote the product
Z i 1 '
Z i 2 ' "
Z i
r
 a S Z
ω
Following the line of Novikov [8 appendix II], we define an operation
Φpeoί°(0) by
where the summation runs through all the partitions.
Lemma 1.1.
(1) ΦP(χ.y) = ΦP(x
(2) φp(z0) = z0 for zo^3lo and
ΦP(y) = 0 for y^% ( i>0) .
(3) (ΦP) 2 = Φ P .
Proof.
(1). By the definition of ΦP and from the Cartan formula for Sω ([6], [8]),
part (1) is easily derived.
(2). It is obvious by definition that φP(z0)=zQ for zo^yio.
It is known that S
ω
(ίF1)= W,k+1 if ω=(k) for some k^O and that S β (ϊF 1 )=0
otherwise ([6], [8]). Thus Φ ^ H ^ ^ P . By the naturality of Φ P , {ΦP)\Wλ)
=ΦP(P) is also a primitive element with the leading term Wλ. So it follows
from Theorem P in the introduction together with the fact that ^ ^ ^ ^ { O }
that
222 K. SHIBATA
for some decomposable elementsy^j-
On the other hand,
Comparing both formulas, we see that φp{zk_^)=Q for Λ^7. So Φp(#8_i)
= 0 since ar7 is decomposable. So 3>7=0 and it follows Theorem P that j 1 6 y + 7 = 0
for ally ^ 0. Repeting this procedure, we can inductively deduce that ΦP(zk_1)=0
for all k^3. At the same time we have proved that (ΦP)2(W1)=ΦP(W1).
Now (Φp)2 is also a multiplicative operation. As in the weakly complex case
([8]), a multiplicative operation of the unoriented cobordism theory is easily seen
to be uniquely determined by its value on W\. Therefore ( Φ P ) 2 = Φ p . This
completes the proof of Lemma 1.1.
Notation. For a partition ω=(ily i2y •• ,ir)> ^ e t llωll—zΊ+^H \-irbeits
degree and | ω | =r its length. And we call ω non-dyadic if none of the component
ik of ω is of the form 2
m
 — 1.
Theorem 1.2. On the category of finite pointed CW pairs and continuous
maps, there is a natural direct sum splitting as a graded Z2-vector space
W*(X, A)= e * Γ Φ P ( S J Ϊ * ( X , A)),
ω; non-dyadic
where (1) the restriction
μ I Image ΦP : ΦP(^(X, A)) -> H*(X, A Z2)
is a natural Z2-algebra isomorphism, and (2) the scalar multiplication
* Γ ^ : ΦP($ft*(X, A)) -> ^ p ) Φ P (? i*(X, A))
is a graded Z2-module isomorphism of degree — | |ω| | if ω is non-dyadic.
Therefore we obtain a natural equivalence of graded Ήl*-algebras
3l*(X, A) - ^ f t * ® H*(X, A Z2)
eϋAzVA commutes with suspension. (Suspension S and a bordism element x act on the
π^/tf iy *Sf(jy®β)=3;®'S'(α) and x(y®a)=x y®a, respectively.)
Moreover, the converse holds such an equivalence is induced by © #ί,P)Φp
ω; non-dyadic
for some choice of a primitive element P.
For the proof of the above theorem, we need the following operations which
BOARDMAN'S GENERATING SETS 223
are just the unoriented analogue of those defined in [8].
Lemma 1.3.
For an indecomposable element y ^ ^ , define an operation Ay.= Σ ,?«*""
Σ
((/)*= (ί, i9 ..., i) the k copies of ΐ)
Then
Ayi(a*b) = Ayi(a).b+a.Ayi(b)+yrAyi(a).Ayi(b)
and, in particular,
The proof of the lemma is straightforward from the definition of Δ^. and
the fact that SQO(yt)=\<=Z2.
Proof of Theorem 1.2.
First we prove property (1). By (2) of Lemma 1.1, property (1) holds for
(X, A)=(S°, P). Since Φ P commutes with suspensions, (1) also holds for (X, A)
=(Sny P) for n^l. Since ΦP is a projection, ΦF(5Jl*(,)) is also a cohomology
theory. So the general cases are proved by induction on the number of cells in
X—A, using the five lemma.
Next we prove property (2). The multiplication
z^v. φp($l*(Xy A)) -> z™ΦpCSl*(X9 A))
is obviously a graded Z2-module epimorphism of degree — | |ω| |.
Suppose z(J) a=0 for aGΦp(ϊί*(Z, ^4)) and for a non-dyadic ω. Order the
components of ω=(/1, ί2y •••, ir) a s h^h= %"^kK a n < i define the operation A^ by
Δ . Γ = Δ.,oΔ, |o...oΔ,, .
Ί *2 r
Then a=A2
ζJ\zlFκa)=AzlP\0)=0 by Lemma 1.3. This proves property
(2)
Totally order the set of all non-dyadic partitions by ω'<ω if (a) | |a/| |<||a>||
or(b) | |ω' | | = | |ω|| and i
r
=j
s9 ~,ir-m+i=js-m+i> K-m>U-m ^ r some m^O, where
<»'=(ii, i2y ~ >ir) and ω=(jlyj2, —,;,) with iλ<,i2^ — <,ir and y ^ j ^ . - ^ / , .
We show that
Φ P Δ Z £ > ( * Γ Φ P ( y)) = 0
for any homogeneous element 3; if ω'<ω. In case | |ω ' | |< | |ω | | , Lemma 1.3
implies that
for some elements u^W* and j y ^ Φ p ^ * ^ , ^4)) with dimtt f.^| |ω| | — | |ω' | |>0.
Thus, by Lemma 1.1 (1), (2),
224 K. SHIBATA
<M Σ M. J . ) = Σ ΦΛ«i)
In case | |ω' | | = | |ω|| and i
r
=j
s
, —, i
r
-
m
>j
s
_
m
,
- 0 .
The last equality follows from the preceding case.
Let Σ ztPΦP(pi*(X, A)) be the graded vector space spanned by all
z^Φp^^X, A)) with ω'<ω.
It follows from the above fact that
Σ z$>ΦP(yt*X, A))A^P)ΦP(9fϊ*(X, A)) = 0
for each ω, so that there is a direct sum splitting
ω ; non-dyadic ω; non-dyadic
Since it can be proved similarly as above that Image (ΦPoΔ^P ))
= Image Φ P for each non-dyadic ω, we have proved that there is a natural linear
endomorphism of degree zero
ω; non-dyadic ω; non-dyadic
It is clearly an automorphism for (X7 A)=(S°, P) and therefore an automor-
phism for every finite CWpair by the effect of suspensions and of the five lemma.
Thus
0 #1P)ΦP($TC*(X, A)) = 9ί*(-Y, A).
*«>; non-dyadic
Since z^ΦP(y)'Z^ΦP(yr) = z^Φp(ymy'), we have obtained a natural
equivalence of graded 31*-algebras
, A) β 9ί* ® //*(X, ^ Z2)
which commutes with suspension.
Conversely, each such equivalence Θ induces a natural monomorphism of
a graded Z2-algebra
λ = Θ"
11 H*(X, A;Z2): H*(X, A;Z2)-> 9l*(X, A).
Then the composition \oμ is a stable miltiplicative operation in cJ?*(0) and
\oμ{W1)=\(w1)=P is a primitive element in 9i*(fiO(l)). And the element P
has the leading term W1 since
BOARDMAN'S GENERATING SETS 225
is an ϊi*-algebra isomorphism. Therefore
θ = θ {1 ® (μ I Image ΦP)} :
ω; non-dyadic / \
yt*(X, A)= φ z^ΦP{Vl*{X, A)) -> φ K P ) ® H*{X, A Z2)}
ω; non-dyadic ω; non-dyadic
This completes the proof of Theorem 1.2.
2. Operations S
ω
Let W
ω
 denote the symmetrized monomial of the cobordism normal charac-
teristic classes Wk. (Wω(ξ)=Wω(—ξ) for every stable vector bundle ξ.) The
operation S
ω
 is defined in [8] by S
ω
=Ψ~
1(W
ω
)y where Ψ is the additive isomor-
phism mentioned in section 1.
Notation 2.1. (Landweber [6])
For a partition ω=(ι1, β ,i r) l e t rJJ) denote the occurrences of the integer i
in ω. And define
. v ίO if ^ < | ω | = r
lω/~ ^
ω
— j if
The modulo 2 reduction of ( n ) is denoted by ( n ) .
\ ω / \ ω /2
Similarly to the weakly complex case [8], we can easily determine the value
Lemma 2.2.
(1) S
ω
[Pk] =
(2) S
u
[Pk] = (2P~k~l) [p ] forp such that 2*>&+l.
v ω /2
Proof. By the geometric interpretation of the action of ^?*(0) on 31* given
in [6], [8], S
ω
[Pk]=SWω(τP)=s(k+ί) ί F 1 l l ω | l = f*+1\ [Pk_UωU] Part (2)
\ CO /2 \ CO ' 2 .
is proved similarly. Now we give some relations between S
ω
 and S
ω
.
Lemma 2.3.
(1) If the occurrence r
ω
(i)<^l in ω for all i, then S
ω
=S
ω
.
226 K. SHIBATA
(2) S
ζi* = Σ S,.. and dually
l l ω l l = *
= Σ
llω !l
where ί*ω w meant a partition (i j 1 9 i'j2> •••> i'jr)for ω=(ji>J2y "*>jr)'
After Landweber [6] we denote the partition .(/)* by AΔX and the totality of
linear combinations of the S
ω
 by ^4*(0). ^4*(0) is proved a Hopf algebra over
Theorem 2.4. (Landweber [6])
The set {S2*Δ1> S ^ ^^0} provides a minimal set of generators of A*(0).
Corollary 2.5.
The set {S2*Δ1, S2*Δ2 £ ^ 0 } provides a minimal set of generators of A*(0).
Proof of Lemma 2.3.
By the Whitney product formula, it follows that Σ W W
ω
 = 0 ifωΦ (0).
ω = s ω i ω 2
Therefore H C^fo=Wi:ί) f° r all ί ^ l and we see by induction on the lengths of
partitions that W
ω
=W
ω
 if r
ω
(i)<^l for all /. Part (1) follows from this and
from the definition of S
ω
 and S
ω
.
Put
for a sufficiently large s.
Then part (2) of the lemma is proved by induction on k as follows
w^= Σ wiOiW^->= Σ ( Σ w^yw^-,
= Σ { Σ (Σ(«ί)y («i)M}{Σ(«i) («ί-ι)}
= Σ (Σ(«ί)'ι-(«i)'-)( Σ ( *. ) )
ιι
ω
ιι=* V 0
Part (2) follows from this.
Proof of Corollary 2.5.
It follows from Lemma 2.3 and Theorem 2.4 that
S2*Δj = S2*Δl+£2*-iΔ2+decomposables in -4*(0), and
S2*Δ2 = 52*Δ2+decomposables in ^4*(0).
Thus the corollary follows from Theorem 2.4.
BOARDMAN'S GENERATING SETS 227
3. Even dimensional coefficients
Following suit of Novikov [8, appendix I], we obtain the following. We omit
the proof.
Lemma 3.1.
For a partition ω and for a positive integer k=2p(2q-\~l) (/>^0, #=^1), the
following formula holds if 11 ω 11 ^ 2P
2
2
') = 0 '
z{ denote the coefficients of a fixed primitive element P as in the introduc-
tion.
Now we prove the following theorem.
Theorem 3.2.
The coefficient z2k of a primitive element
4
P = wi+z2W1'+ziW1
5+z5W1
G+z6W1
7+z7W1
8+-'
in 9i*(5O(l)) is equal to the bordism class [P2k]for all k^l.
Proof. For k—1, the theorem is clear since z2 is indecomposable from
Theorem P in the introduction.
Assume that the theorem holds up to dimension 2(&—-1)^2.
In order to show that S
ω
{z2k^r[P2k])=Wω{z2k+[P2k])=Q for all ω with | |ω||
=2ky it saffices from Theorem 2.4 to prove
S s (z 1.4-1 P J} — 0 (i — 1 2^
To prove this, we see from Lemma 3.1 and the induction assumption that it
is sufficient to show
Σ S
m
^i[P2k_ni](2k+ί-ni) = 0 ( ί = 1,2).
This is obvious in case 2si>2k or s=0 since
by Lemmas 2.2 (1) and 2.3 (2).
For the remaining cases, it suffices to prove the following lemma.
Lemma 3.3.
(1) Σ ( Σ (k-n))(k~n) = 0 (mod 2) f
m
 + «=>Λ||ω||=,w\ ω / / \ ^ /
or
for
228 K. SHIBATA
(2) Σ ( Σ (k-2n))(k-2n)
 Ξ
 0 {mod 2) fo
Proof.
(1) Put
A(ky s) = w Σ Γ Σ ^~"nW*"""n) (k^O, s^O), and
5(ft, s) =
m
Σi
a
(
u
Ίl
u
(k^
Then it holds in general that
/A_
w
\ /b — tj 1\ /fe_
w
_1N
//v ' ^ \ / / v #* J. I i / /v ' * A
(
 n
 ) = (
 M ) + ( „ - ! ,
ι|Σm(*~M) = o ί Σj A ~^~ 1 ) .
So we obtain that ^
(*) ^(A, j )= Σ A(k-1, s')+ Σ -4(^-2, s") and
/ψ
ψ
\ Tί^ ίί Λ V* TMb 1 c^_J_ SΠ
Part (1) clearly holds when k=s=2.
Assume, by induction, that (1) holds for such (k9 s) that ko>k*t2 and
Thus, for (k
oy s0) with
o) = Σ A(k
o
-h s')+ Σ A(ko-2, s") = 0 (mod 2)
by the induction hypothesis and by the fact that A(k, s) = \ for k^s and ί = 0 , 1.
And for (k0, ko)> the iterated application of (*) shows that
K) = A{ko-h k
o
)+A(k
o
-2, ft
o
-l)
= A(l, k
o
)+ Σ -4(0, s") = 0 (mod 2).
Part (2) of the lemma is proved similarly, using the formula (**) repeatedly.
This completes the proof of Lemma 3.3 and Theorem 3.2.
REMARK 3.4. Theorem 3.2 has been proved independently by F. Uchida
[9] by a geometric method.
4. The coefficients of dimensions
A. Dold has defined in [5] manifolds P(m, n) which are the identification
BOARDMAN'S GENERATING SETS 229
spaces of SmxCP
n
 with (#,*)=(—#, *). He proved that, for 2p(2q+l) — l
(pi^l, #2^1), the bordism class [P(2p—l, 2pq)] provides a polynomial generator
of Ji* in the corresponding dimension.
Theorem 4.1.
The coefficient z4k+1 of a primitive element
P
 =
 W1+z2W*+zJV*+z5W*+zJV1
1+z1W1
s+--
in 9t*(BO(l)) is equal to the bordism class [P(l, 2k)] for all k^l.
For the proof of this theorem, we need the following notations.
Notation 4.2.
(1) Let cp(m) denote the coefficient of 2P in the dyadic expansion of the in-
teger m
m = r
o
(w)+^(m).2+c2(m).22+..., Ci(tn) = 0, 1 .
(2) For a partition ω, we denote by ω(cp) the partition determined by
r
ωC , (i)=cp(rω(i)) for all i ^ l . Thus ω= Π (ω(cp))2p. For brevity,
Tl(ω(cp))2p~2 and ω ^ ^ ω^o) are denoted as w and 5, respectively;
ω=(ώ)4ώ.
Lemma 4.3.
(n\
 =ΐί
(Cβ{n)\
 Ύhu/n\ /n-φ)-2-co(n)\ fφ) 2+φ)\ ^
V ω /2 o$Aa)(^)/2 V a) /2 ( 4 IV ω A
\ ω ' z
Proof. By definition,
In- Σ r
ω
(i)\
. ( — )
Then, by Lucus' theorem [1],
(n Σ r.(0\ / /^(« Σ r
β
(0)\ \
π
 is<st
-
1
 = π π
 iSi£ί
-
1
'
<n)
- ^ - η
This completes the proof.
Now we calculate all the normal Stiefel-Whitney numbers of P(l, 2k). It is
easily seen that the cobordism Stiefel-Whitney numbers of manifolds agree with
230 K. SHIBATA
the cohomological ones ([6], [8]). So, by abuse of a notation, we denote both
Stiefel-Whitney numbers by W
ω
 (and the normal ones by W
ω
).
W
a
[P(ί, 2k)] =
Lemma 4.4.
0 if \ώ\^3 and ώφ3Δ
x
 or ^ = (1),
) if ω = 3Δi or
2 ^ | ώ | ^ l and δ φ ( l ) ,
where p is any integer with 2 / >>Λ+l.
Proof.
According to Dold [5].
H*(P(\, 2k) Z2) « J / * ( P l X C P r t Z,)
as a ring. Let ί: and rf denote the 1- and 2-dimensional generators of
#*(P(1, 2A) Z2). The total Whitney class is given in [5] by
and thus
l, 2A) =
where/) is any integer with 2 / >>&+l and t2=t1 t2=d and ί i + t ^ c .
By formula (26) in [5],
f ϊ '+f i^O and t\t+1+tlt+1 = cd*.
The lemma follows from these facts and the preceding lemma.
Proof of Theorem 4.1.
Theorem P in the introduction asserts that # 4 + 1 =[P(l, 2)]. Assume, by
induction, that #4ife/+1=[P(l, 2k')] for k'^k—1.
By Lemma 3.1 and Theorem 3.2, together with Lemma 2.2 (2), 4.3 and 4.4,
( ω 2 ) 4 ω 2
for ω such that | |ω | |—4^+1. (The terms with | |ω 2 | |=2 vanish by Lemma 4.3.)
Therefore, by the induction hypothesis and by Lemma 4.3, together with the
fact that I &t \ + | ώ2\ = | ώ \ +4/ (/^0), it can be shown that
S
ω
( W > = Σ 0 + Σ 0 = 0 if \&\ϊ>5.
BOARDMAN'S GENERATING SETS 231
In case 5=(2t, 2», 4/, 4(k-\\ω\\-i-j)+l),
= Σ {(^-l-(*-IWI-O)(*-IWI
ω»ω1ω2l\ δJj / 2 \ ω2
by the induction hypothesis and by Lemma 4.3.
Suppose (2pf— 1 — (k— | |ω 2 | | — i)\ ίk— | |ω 2 | | — A = 1 for some separation
V ω1 / 2 \ ω2 /2
ω1ω2 of ω.
Since cp(2p/— \ — {k— \\w2\\ — t))^cp(k— \\ω2\\ — i) for eachp9 there is at most
one i£>l such that cp{rJJ)) Φ 0. Let r be the number of such odd integers
2z'+l ^>1 that satisfy r
s
(2z+l)>0. Then, by Lemma 4.3, the numbers of such
separations ω1ω2=ω and ω1'ω2'=ω that satisfy
/2>'-l-(k-\\m2\\-i)\/k-\\m2\\-i\ = 1 a n d
V 6)i / 2 \ Wo / 2
M
" "
κ
 = 1 ,
respectively, are both 2r.
The situation is the same if we suppose
s2P'-l-(\\π\\+j+k-\\ω2\\)\n\ω\\+j+k-\\m2\\\ = ί
V ω1 s?\ ω2 ' 2
for some separation ω1 ω2=δ>.
Therefore 5
ω
(^ 4 A f + 1 )=0+0=0 or = 1 + 1=0 if ώ=(2j,2j,4k,
4(s-\\ω\\-j-k)+l).
We can prove analogously in other cases when ώ=(l) or | ώ | ^ 3 and ώΦ 3Δj
that S
ω
(z4k+1)=0.
When I ώ| =2, from dimensional reasons, ώ=(2j, 4(ί— ||ω|Q — 2/+1) for
some y ^  1. In this case
ώ2φ(0)
+ Σ
= Σ
δ=»ώ co
_(2p-\-k\
~\ m A
232 K. SHIBATA
as required.
When 5=3AX or ώ=Ain+1 (w^l), analogous arguments show that Sω(z4k+1)
\ ω I
Comparing these facts with Lemma 4.4, we deduce that S
ω
(z4k+1)
= W
ω
(z4k+1)=Wω(P(l, 2k)) for allω with | |ω | |=4Λ+l. This completes the
proof of Theorem 4.1.
OSAKA UNIVERSITY
References
[1] J. Adem: The relations on Steenrod powers of cohomology classes, Algebraic Geometry
and Topology, Princeton University Press, 1957, 191-238.
[2] M.F. Atiyah: Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200-
208.
[3] J.M. Boardman: Stable Homotopy Theory, Ch. IV, (mimeographed) Warwick
Univ., 1967.
[4] P.E. Conner - E.E. Floyd: The Relation of Cobordism to K-theories, Springer-
Verlag, 1966.
[5] A. Dold: Erzeugende der Thomschen Algebra % Math. Z. 65 (1956), 25-35.
[6] P.S. Landweber: Cobordism operations and Hopf algebras, Trans. Amer. Math. Soc.
129(1967), 94-110.
[7] J. Milnor: On the Stiefel-Whitney numbers of complex manifolds and of spin mani-
folds, Topology 3 (1965), 223-230.
[8] S.P. Novikov: The method of algebraic topology from the view point of cobordism
(Russian), Izv. Akad. Nauk SSSR 31 (1967), 855-951.
[9] F. Uchida: Bordism algebra of involutions, Proc. Japan Acad. 46 (1970), 615-619.


ON BOARDMAN'S GENERATING SETS OF THE UNORIENTED BORDISM RING

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ON BOARDMAN'S GENERATING SETS OF THE UNORIENTED BORDISM RING

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