奈良教育大学学術リポジトリNEAR
ON THE ACTION OF REDUCED POWERS IN AN ASSOCIATIVE H‑SPACES
著者 OCHIAI Shoji
journal or
publication title
奈良教育大学紀要. 自然科学
volume 24
number 2
page range 1‑4
year 1975‑11‑15
URL http://hdl.handle.net/10105/2575
ON THE ACTION OF REDUCED POWERS IN AN ASSOCIATIVE
//- SPACES
Dedicated to Professor Kiiti Morita on his 60th Birthday
SHOJI 0CHIAI
(Department of Mathematics, Nara University of Education, Nara, Japan)
(Received April 30, 1975)
I. The purpose of this paper is to investigate the action of the redui^d powers in an associative //-space. E.Thomas gives the complete answer to the action of the Steenrod squares in the mod 2 cohomology algebra of an //-space [5], [6].
On the other hand, in the mod p (p; odd prime) case, although, A.Borel and J.
P.Serre determined the behavior of Steenrod reduced powers in the mod p cohomo- logy of classical groups [1], few results are known [2], [3].The following Theorem 2givesa partial result on the action of the Steenrod reduced powers in the mod p cohomology of an associative //-space.
First we explain some notations and state Theorems. Let B be a polynomial algeber on generators X, (?'à¬/) and Dá"B be the two sided ideal generated by the elements U.cz,XvU)
, Sv(i)Stn. The element OGB, a$DZB is called the inde- ' ie/ _
composable element where B means the positive dimensional part of B. We call an quotient algebra B/D B a truncated polynomial algebra of height m. We donote this algebra by A.
Theorem. 1. Let A be a truncated polynomial algebra over the mod p Steenrod algebra and height of A is n, where raiSp+l. We assume further that A is an algebra on even dimensional generators of dim^/c. If ws is a generator with dimen- sion h such thath>(k+2(p-l))/2, h=2(pi+j), l^i, \<j<p-l, then we get
for some generators ui and v a^^iiij =Plv modD2A ^t^Zp
Corollary . Assume further, all generators have distinct dimensions. Then, if the dimension of a generator x satisfies the above conditions in Theorem 1, then there exists a generator y such that
x=Ply mod DZA.
Theorem 2. Let X be a finite polyhedron which is an associative //-space and has no p-torsion. The generators of H (X; Zp)have distinct dimensions and dim^k.
If v is the generator such that dim v> (k-1+2 (p-1 ))/2, dim v-2(pi+j)-l, ISj, lsSjS-p-1, then we obtain for some generator v
* We admit m=°o
Shoji Ochiai
Corollary is obtained immediately from Theorem 1. Theorem 2 follows from Theor-
rem1 bythe usual way. From the assumption of Theorem 2, we see that H (X;Zp)
=A (»j,•E•E•E«,) dim vi odd. Therefore we see (vv•E•E•Evt) is the simple system of generators for H* (X: Zp) and v. are universally transgressive. From these facts,
we obtain by the generalized Borel's theorem, that H (BX; Z ) is a polynomial
algebra generated by the transgressions of the generators in the simple system for
H (X; Z ) (4], (5], [6). By considering the fact that the transgression operator
commutes with the reduced powers, Theorem 2 follows from Theorem 1.
2. In this section we shall prove two Propositions which are used to prove The-
orem 1. The method is similar to Thomas's one. Let aà¬=A be a indecomposable
element and / be a monomorphism of graded module /: M-åºA such that o£/ (M)
where M is M=A/DZA.
Suppose r(M) be a free associative commutative tensor algebra generated by
M. Then there exists an algebra epimorphism J"such that follwing commutativity
holds {.
M » r(M)
/ /
where £ is the inclusion map. As the kernel of/ is DnA (ra2;p+l),then we see A
~T (M)/DnA. Therefore if we identify A with T (M)/D"A by f, then we get the
direct sum decomposition of A hy M. k
A=Z ©M© [M(g)M] / {M,M) 0-v©Jtf(8)-"(g)M/ {M,M] +higher terms, where
P ft i
[M,M] denotes the subspace of_M(X)---(g)M spanned by the elements m^-^ffl.®1"
)?ȣ m^M. Assume that V is the subspace
>;. m. )rk.
of A satisfying following conditions. A={a]+[a ]-\ \~[ap ]+V, DPACV
where Cal. Ca],'"ia 3 is the subspace ofA spanned by the elements a, a,•E•E•E
a respectively.
Proposition 1. Let A be a truncated polynomial algebra over Zp. Then any
indecomposable element a satisfies a" *$V-A where V has the above mentioned meaning.
Proof. From the assumption on V, V can be written as follows.
V=Wl+W2+---+Wp_1+ (M<g)""ngfM) / [M, M) + higher terms^ where
W means next vector space. Let V. be the vector space such that M(5<)•E•E•E(X)M
=a®^'<g) a+Vr Then W. is given by Wi=Vl/ [M,M]. pAsM(X)---(g)M/
[M,Af| has the direct sum decomposition MCg)---(g)M/ [M,.Af] =a(g)---(g)a+Wp, then
_p-
we can get V-AC\M(g)---(g)M/ [M,M]=(Wt•E>!+t^.A+-typ-ri4+phigher terms) f)
~"~(g)M) / [M,M] =W , From this we see op^V•EADM<g)---(g)M/ [M,M] , and
so apt$V-A
Proposition 2. Let A be a truncated polynomial algebra over the mod p (p; odd
prime) Steenrod algebra and a^A be an indecomposable element of dimension 2( pi
+j), l^i, l<j<p~l. If all elements a, aZ,---av~l are not contained in P1A+DPA,
then Ppia$D2A holds.
ProofSince a, a å •Eå ap ^.P A+DPA, we can choose the vector space V
+ [ap~1] +y, PlA+DpA(ZV. Suppose PPia^DZA,
C.A such that A=[a]+[a2]+
then we see ap=Ppi+j (a)=X(P1)>Ppi(a) S (P1)
This contradicts Proposition 1. This concludes the proof.
3. In this section we shall prove Theorem 1.
Proof of Theorem 1. In the first place, we shall show Ppius^DlA. In order
to show this, it is sufficient to show that the inequality h-\-2pi (p-1) > k å holds.
But this inequality is easily checked by considering the assumption on h, i, and j.
#) for some m (ISmSp-l),
{D2A) aP1A- A CV-A
«:=
Then we get from Proposition 2 the equation
P 1 gS*,-,,-«:«,.j+*X+(|*,,.«,, +d2) +dp- (*) where «,,«
',; II, S are
generators of A, d is an element of D A and d2 is a decomposable element which
dose not contain monomials u'su{. (i=l---f, j=zl---g), urg. If m=\, there is nothing
to prove. Suppose mi£2. Comparing the dimension of both sides, we get (m-r)h=
2 (p-1). But this means h=2 (p-1) / (m-r) <2p. This contradicts the fact A=2
(pj+i), l^i, 1^7=p-1. Therefore we see kr=0. Next we consider the coeffi-
cient k g. We get easily dimu =mh-2 (p-l)S2/i-2 (p-1)>A:. But this contra-
dicts the assmuption that k is the highest dimension of generators of A. Then we
obtain k =0. Consider the monomial k. uu.. such that m-j2s2. By the analogo-
us way as above, we obtain the contradiction. So we obtain
yy/ g it.
i"i=.1 'å å > ',j
=2*.g 7=1
uIB-1«_
Therefore the equation
jr.
* m-1
m-llj
reduced to
;+d2) +dr
rvi\
-i-m-l' J m-1'j
From the fact that the coefficient of ug in P (d2) is zero and mfip-1, we can
get the conclusion i.e., P (A j, wm_i> å ) is the indecoposable element containing
ug. for some generator «m_j, •E
Acknowledgment. The author is indebted to Professor R. Nakagawa for his helpful
advices.
References
1. A.Borel and J.P.Serre, Groupes de Lie et puissances reduites de Steenrod,
4 Shoji Ochiai
Amer.J.Math., 75, 409-448 (1953).
2. P.J.Kumpel, Jr, Lie groups and product of spheres, Proc. Amer. Math. Soc. 16
1350-1356 (1965).
3. M.Mimura and H.Toda, Cohomology operations and the homotopy of compact Lie
groups, I, Topology, 9 317-336 (1970).
4. J.C.Moore, Algebre homologique el des espaces classifiants, Seminar Cartan et
Moore (1959-1960) Expose 7.
5. E.Thomas, Steenrod squares and H-spaces, Ann. of Math.,77, 306-317(1963).
6. E.Thomas, Steenrod squares and H-spaces, II, Ann. of Math., 81, 473-495
(1965).
