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~qx&gy Vol. 12. pp. X33-195. Pergamon Press, 1973. Printed in Great Britain. A SPECTRUM WHOSE COHOMOLOGY OVER THE STEENROD IS A CERTAIN ALGEBRA CYCLIC MODULE EDGAR H. BROWN, JR. and SAMUEL GITIXRI. (Received 4 June 197 I ) $1. INTRODUCIION LET A be the mod 2 Steenrod adopt the notation Y"(X), for a space X, denotes Let H be the spectrum algebra and x : A -+ A the canonical by Frank Adams, its q-homology and cohomology whose qth term is an Eilenberg antiautomorphism. We suggested namely, if Y is a spectrum, groups with respect to Y. MacLane space of type (Z, , q). Y,(X) and Let M(k), k 2 0, be the module over A defined by M(k) = A%A{x(Sq') 1 i > k}. The main result of this paper is: THEOREM I. 1. There is a spectrum B(k) satisfying: (i) H*(B(k)) z M(k) as A modules. (ii) I f u : B(k) -+ H is the map representing 1 E M(k), a* : W),(X) -+ H,(X) is an epimorphism for any C W complex X and q < 2k + 2. COROLLARY if v is the norm& bundle of M embedded in R"+k (k large) and T(v) is its Thorn space, then there is u Thorn class (I E B({<n%2>})'( T( v)) which when restricted to ajbre of B({<n%2>})k(Sk) z 2, . I .2. If M is a compact n-manifold. M is B( {<n%2>}) orientable in the sense that Sk giues the generator Proof. Recall (l.l(ii)), M + is an n + k S-dual of T(v). Hence, by Alexander duality and @{<nl2l)'(T(v)) --t H'(T(b*)) is an epimorphism. (1.2) now follows by pulling back the Thorn class in Hk(T(v)). Remark. Suppose V E H"( T( v)) is the Thorn class. {a E A 1 aV = 0 for all n-manifolds is the ideal A{X(Sq')lZi > n}. Hence, in some sense, B{<n%2>} is the smallest spectrum respect to which all n-manifolds are orientable. M} with A straight-forward calculation shows: 7 This work was carried out while the first named author was a visitor to Centro de Investigation Estudios Avanzados Mexico, and subsequently, while both authors were supported by NSF GP 21510. y de 283 |
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284 E . H. BR OWN, J R . and s. GIT L E R T HE OR E M 1.3. {x(Sq') 1 S,' is admissible, I = (i,, . for M(k). M(k), = 0 for q > 2k - r(k), where r(k) is the number of ones in the dyadic expan- sion of k. , i,) and i, 5 k) is an additire basis From (1.3) we see that we may choose B(k)2, to be a finite CW complex and B(k), = Sq-2kf?(k)zk for q > 2k. Let Zk be a t S-dual of B(k)?, for some large t and let T(n) be the spectrum defined by T(n), = SrZ{<.,Z>} where r = q + n - t + 2{<n,'2>}. T HE OR E M 1.4. T(n) satisfies. (i) T(n) is a(n) - 1 connected. (ii) H,(T(n)) = Z, , q = n = 0, q > n. (iii) If X is a C W complex and c E H,(X), I, such thatf, : H,(T(n)) there exists u map f: T(n)i + S'X, for some -+ H,(X) maps the generator into c. Remark. (1.4) is a strong form of the well known theorem that if a E A,_i and i < a(n), then u : H'(X) -+ H"(X) is zero. Proof of (1.4). We may assume X is a finite C W complex. Take f to be an S-dual of a map representing the Alexander dual of c in Hteq(X*). Let X* be a t S-dual of X. which maps into an element of B({<nj2>})'-q(X*) (i) follows from (1.3) and (ii) is obvious. Probably the above results also hold for odd primes but we have not checked through the details. In constructing in slightly obscure ways. The obscurity Since the important our arguments make perfectly good sense. the spectrum B(k) we sometimes comes about from having infinitely part of B(k) is in dimensions consider spectra and modules over A many generators. 0 to 2k we could truncate everything so that The organization of this paper is as follows: The sections are ordered Unfortunately, struct B(k) by building rather indirect extends x : A + A (viewing A as maps of H to itself). Applying a sort of dual tower and it is this which we first construct analogue "-" on A-modules so as to successively is not very illuminating a generalized way. In 96 we describe a duality introduce the steps in the proof of (1.1). with respect to motivation. tower for it but we build this tower in a functor x from spectra to spectra which x to a Postnikov for B(k). The operation which is described in 92. Its relation this ordering We con- Postnikov tower yields x has an to x is: H*(x(Y)) = H*(Y) for some spectra Y. In paragraph 2 we construct " - " of such a resolution. dual resolution. In $4 we use the homology we construct the dual tower and in 97 we construct a free acyclic resolution In $3 we construct operations of M(k) by first constructing MacLane spectra realizing this Qi to prove a curious lemma. In $5 B(k). Eilenberg |
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SPECTRUM WHOSE COHOMOLOGYIS CERTAIN CYCLIC MODULE OVER STEENROD ALGEBRA 285 92.ALCEBFtAIC PRELIIMINARJES Let x be the free associative module the relations: algebra over Z2 with unit generated by %i, i = - 1. 0, 1, 3 _... If 2k < I, (2.1) Let A be the quotient {<I>}. In {<I>} it is shown that: of K by the left ideal generated by I._,. A is the algebra described in LEMMA 2.2. {i.i, i.i1 . . . l.il % 0 s i, , 2i, 2 ii+ 1} is an additive basis for A. If Z = (i1, i,, il), let J1 = Ji, JiJ ... E.i,. We say that ll.1 is admissible if it is as in (2.2). We grade A by dim i.i = -ii. Let A @ A be the tensor product as algebras. We denote a @ p by up. Let 6 : A 0 A -+ A 0 A be the A-linear map defined by ( 6%l = 1 sq'+'%l, i s- 1 11 ) where p E A. LEMMA 2.3. 66 = 0. Proof. It is sufficient to show that 1 Sqif'Sqic' this as a sum li"_j = 0. Using (2.1) one may express 2zjaij 0 E.iAj. One easily calculates that the aij's are the Adem relations. If Z= (il, . . , il) and I' = (ii, . . . , i;) let I > I' if ij = ij for 1 =< t < j and i, > il. Let l(i,, i,, . . . , il) = 1. Let 1 = 1, where Z = ( ). LEMMA 2.4. For any I which is not admissible, I., = c llIj where E$, is admissible and Ij > I. Proof. (2.4) follows by induction of f(Z) and (2.1). Let J, be the left ideal of A generated by I., , A,, . . . j.k_l. (2.2) and (2.4) yield LEMMA 2.5. A @ J, is a subcomplex sible, I > 0 and i, < k} is an additive basis for J, . of {A @ A, S} and {I_,1 I = (iI, . . . , ir) with Z admis- Let " - " denote the functor from the category of modules over A to itself defined by R = Hom,(M, A) where A acts on M by (au)(m) = +)x(a). |
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286 E. H. BROWN,JR. and S. GITLER (mq is the homomorphisms {ml}, we call {hi) c R a dual basis of R if which raise dimension by q.) If M is a free module with basis Zi(rnj) = bij. Note dim fit = -dim mi . The following is trivial to check: LEMMA 2.6. If M is free with basis {mi}, R is free with basis { fii}. If M, and M, are free with bases {mi} and {nj} andf: M, -+ M, is given by fCmi) = C aijnj then f(Ej) = 1 x(aij)iEi. Let A4 be the subspace of A generated by J., where l(I) = q. Note 6 : A @ A4 + A @I AQ+' and A @J,, is a subcomplex with respect to this grading. Let C, = C,(k) = (A @ hq%A @ Jlrq). We may view C, as contained Note 6: C,+ C,_,. in (A @A'). Let {I,} be the dual basis of {II 1 I admissible}. THEOREM 2.7. C, is a free A module with basis {A, 1 I = (il . . . , il), i, 2 k and I admissible}. F~, = C X,(~i L,)X(Sq'+ ')X, where the sum is over J such that 1, is a basis element of C,-,. Furthermore, a -+C,--,C,_,+~~+C,~M(k) where E(i) = 1, is exact. Proof Except for exactness, (2.7) is immediate. Exactness at C,: 5ii = x(sqi+ ')i and {X,{< i 2 k} is a basis for C,. Exactness at C,, q > 0: Since 66 = 0, 88 = 0. If u = 2 b,&,, rj 2 I~, . Suppose u E ker 8. Note by the above formula let Z(U) = Ijo where for 62, and (2.4), if I = (i, J) 61, = x(Sq'+ ')A, + c b,, 2,. .l'>J Z(u) = (i, J) where I(J) 2 0 and U = 1 UjX(j,J) + C Cr*X,* jti where 1' = (i', J'), J' > J. Hence 0 = 6~ = 1 ajX(S$+')J, iri in the usual sense, Sq('* L, is admissible +,,T,~J*I.J*. Note if SqL is admissible the first entry of L, is larger than i. Express each aj as a sum of x(Sq') where L is admissible. or Sq"* ') = c SqL' where |
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SPECTRUM WHOSE COHOMOLOGY IS CERTAIN CYCLIC MODULE OVER STEENROD ALGEBRA 287 Suppose Say (i + 1, f.,) is not. L, = (j + 1, K) where 2i i j. Ui = 1 X(SqL')X(Sqi+') = X(Sq (i+l* ',)). All of the (I' + 1, L,) cannot be admissible. 6(&j, i. I)> = X(sqj+')i(i. ,) +J>;,)CJiJ. Adding Continuing increases with T. (dim Ii = -dim F(X(SqK)I,j, i, ,J to I( replaces in this way, we express u = 6 W + terms in 1, with Tarbitrarily ,Ii = i). Hence u = 3 W. x(SqLo)xci. ,) by terms involving Jr, high. But dim Jr T > (i, J). If M is a module over A, let T,(M) be the quotient {Sq'm 1 i > dim m + I}. Note if f :M -+ N is a module r,(f): ?;(W+ T,-,(W. of ,Cf by the submodule map of degree q, it induces generated by a map LEMMA 2.8. T,(a) : T,(A 0 (A%Jd) + T- ,(A @ (A!Jd) iszeroifl=<2k+ 1 andifI=2k-+-2 T,(6)1, = 0 or T,(6)& = sq'+'1., I.1 wherej+ 1 =I-- 1 +dimlj,I,. Proof. We first prove by induction on I(I) that Jr I.1 c J, where s = {<(2f - dim 4)%z>}. Suppose I(1) = 1. Suppose j < 1. If 2j 2 i l-j,'.i E J i c J {<(i+*l)% Z>}. If% < L AjLi E J ~(i+x),21 b Y ( 2.1). Suppose I = (i, J), I(J) > 0 and our formula is true for J. JI lI = (J1 Ai)AJ c J {<i+Z1%2>} AJ c J, We now prove (2.8) 61, = 1 Sq'+'E.jl.r. AjE J,+l. Hence ,Ij ,I1 E J, where r = {<(2(j + 1) sum ranges over j such that r 2 k + 1, i.e. j+122k+l the desired result now follows. - dim Z)%Z>}. Hence in A @ (A%JL). the above + dim E., i., $3. PRELIMINARIES ABOUT SPECTRA Unless otherwise specified all spectra will be R-spectra. will denote homotopy classes of maps from B to C of degree q (u : B, -+ C,,,). A, = {<H, HIP and Hq(B) = {<B, If>},. If u E {<B, Cl,, If B and C are spectra, {<B, C>}, Note C:E%+B |
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