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Spectra of $\text {BP}$-linear relations, $v_n$-series, and $\text {BP}$ cohomology of Eilenberg-Mac Lane spaces

Hirotaka Tamanoi

1999
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Transactions of the American Mathematical Society
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On Brown-Peterson cohomology groups of a space, we introduce a natural inherent topology, BP topology, which is always complete Hausdorff for any space. We then construct a spectra map which calculates infinite BP-linear sums convergent with respect to the BP topology, and a spectrum which describes infinite sum BP-linear relations in BP cohomology. The mod p cohomology of this spectrum is a cyclic module over the Steenrod algebra with relations generated by products of exactly two Milnor
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... ly two Milnor primitives. We show a close relationship between BP-linear relations in BP cohomology and the action of the Milnor primitives on mod p cohomology. We prove main relations in the BP cohomology of Eilenberg-Mac Lane spaces. These are infinite sum BP-linear relations convergent with respect to the BP topology. Using BP fundamental classes, we define vn-series which are vn-analogues of the p-series. Finally, we show that the above main relations come from the vn-series. Part (I) says that each element in L * (X) can be thought of as an infinite sum BP-linear relation in BP cohomology. Thus we call L the spectrum of BP-linear relations. Part (II) shows that in L-theory, there exist Milnor operations q i , namely θ • q i for i ≥ 0. But any product among them is zero, since q j • θ = 0 for any j in the cofibre sequence (1-6). Part (III) is a reflection of this fact, and it shows that there are no other relations in the mod p cohomology of L. We can also consider finite BP-linear sums in BP n * (X) of the form The spectra map which calculates this summation is the following composition of BP-module maps: (1-10) 5142 HIROTAKA TAMANOI Let L n be the cofibre of κ n . Then L n is a BP-module spectrum with properties corresponding to a finite version of Theorem 1-4 [Theorem 4-1]. These BP-module spectra L n fit into the following tower [Proposition 4-5]: This tower can be used to construct infinite sum BP-linear relations in BP * (X) from finite sum BP-linear relations in BP n * (X) . The BP cohomology theory and mod p cohomology theory are closely related by the Thom map ρ * : BP * (X) → HZ * p (X). Through ρ * , BP-linear relations in BP * (X) translate into a certain property of the action of Milnor primitives on the mod p cohomology of X. Proposition 1-5 [Proposition 5-1]. Let X be a space and let k be a positive integer. Suppose we have pb 0 + v 1 b 1 + · · · + v n b n + · · · = 0 in BP k+1 (X) for some elements b n ∈ BP k+2p n −1 (X) for n ≥ 0. Then there exists an element x ∈ HZ k p (X) such that in mod p cohomology we have ρ * (b n ) = Q n (x) for all n ≥ 0. (1-12)

doi:10.1090/s0002-9947-99-02484-8
fatcat:qs5u5xehjfbkzeryaw2vjzxkpe