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On Brown-Peterson cohomology of \(QX\). (English) Zbl 0985.55006

If \(X\) is a spectrum, let \(\Omega^\infty X\) denote its zeroth infinite loop space. If \(Y\) is a space, then the suspension spectrum \(X = \Sigma^\infty Y\), determined by \(Y\), has spaces \(X_i\) given by \[ X_i \cong Q\Sigma^i Y \cong \text{colim}_s \Sigma^{i+s} \Omega^s Y. \] For \(h_*(-)\) a generalized homology theory, one wants to relate \(h_*(\Omega^\infty X)\) to \(h_*(X)\), that is, in the case \(X = \Sigma^\infty Y\), to relate \(h_*(QY)\) to \(h_*(Y)\). When \(h_* = H(\mathbb{Z}/p\mathbb{Z})_*\), ordinary mod \(p\) homology, this relation is given in terms of the Dyer-Lashof-Araki-Kudo operations. Generalizations to \(h_* = K_*(\;;\mathbb{Z}/p\mathbb{Z})\), mod \(p\) K-theory, for certain spaces \(Y\) are due to Hodgkin, Miller, Snaith, and McClure. In this paper, the author determines BP(\(\Omega^\infty X\)) in terms of BP(\(X\)) for certain spectra \(X\). The generalization of the case \(h_* = H(\mathbb{Z}/2\mathbb{Z})_*\) uses work of J. Lannes and S. Zarati [Math. Z. 194, 25-59 (1987; Zbl 0627.55014)] in which they express the primitives, \(PH^*(QX; \mathbb{Z}/2 \mathbb{Z})\) in terms of the left derived functors of destabilization (left adjoint to the forgetful functor from unstable algebras over the Steenrod algebra to modules over the Steenrod algebra). The analogue of destabilization for BP is constructed by the author for the categories \({\mathcal M}_{\text{BP}}\) of stable BP-cohomology modules and \({\mathcal K}_{\text{BP}}\) of unstable BP-cohomology algebras as defined by Boardman and his coworkers. Let \({\mathcal K}_{0\text{BP}}'\) denote the category of augmented unstable BP-cohomology algebras that are well-presented, that is, \(A\) is in \({\mathcal K}_{0\text{BP}}'\) if \(A\) is isomorphic to the cokernel of a morphism of stable BP-cohomology modules \(F^1\to F^0\) which are both free (in the image of the left adjoint to the forgetful functor to graded sets). The augmentation ideal functor from \({\mathcal K}'_{0\text{BP}}\) to \({\mathcal M}'_{\text{BP}}\) (well-presented stable BP-cohomology modules) has a left adjoint which the author calls destablization, \(\mathcal D\). The main theorem deals with spectra that are well-generated, that is, the mapping \(\text{BP}^*(X)\widehat{\otimes}_{\text{BP}^*} (\mathbb{Z}/p\mathbb{Z}) \hookrightarrow H^*(X; \mathbb{Z}/p\mathbb{Z})\) is an injection. Being well-generated implies being well-presented. The author proves that if \(X\) is a connected space such that \(\text{BP}^*(X)\) is Landweber-flat and well-generated, then the natural mapping \({\mathcal D}\widetilde{\text{BP}}^*(X) \to \text{BP}^*(QX)\) is an isomorphism in \({\mathcal K}_{0\text{BP}}'\). The conditions on the space \(X\) are shown to be equivalent to a plethora of stronger or equivalent formulations in terms of other generalized homology theories making the result very concrete.

MSC:

55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55P47 Infinite loop spaces
55S12 Dyer-Lashof operations

Citations:

Zbl 0627.55014
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