Homology decompositions for \(p\)-compact groups.

*(English)*Zbl 1156.55014By a homology decomposition of a space \(X\) with respect to a generalized homology theory \(h_*\) the authors mean an \(h_*\)-equivalence \(f:Z \to X\) with \(Z\) a space that is constructed as a homotopy colimit from a diagram of spaces which themselves are homotopy-theoretically easier to understand than the given space \(X\). A typical context for homology decompositions is the theory of classifying spaces where homology decompositions for the classifying space of a topological group are assembled from suitable classifying spaces which evolve from certain families of subgroups of the group.

For a given prime \(p\) there are two (by now) classical \(mod\;p\)-homology decompositions of the classifying space of a compact Lie group \(G\): the centralizer decomposition, established in [S. Jackowski and J. McClure, Topology 31, No. 1, 113–132 (1992; Zbl 0754.55014)], and the subgroup decomposition, established in [S. Jackowski, J. McClure and B. Oliver, Ann. Math. (2) 135, No. 1, 183–226 (1992; Zbl 0758.55004)]. In the centralizer decomposition one realizes the \(mod\;p\)-homology type of \(BG\) by gluing together the classifying spaces of the centralizers of the \(p\)-elementary abelian subgroups, while in the subgroup decomposition one glues together the classifying spaces of the so-called \(p\)-stubborn subgroups. In [B. Dwyer and C. Wilkerson, Contemp. Math. 181, 119–157 (1995; Zbl 0828.55009)] the centralizer decomposition was generalized from compact Lie groups to so-called \(p\)-compact groups. \(p\)-compact groups are certain homotopy-theoretic analogues of compact Lie groups, which were introduced in [ibid., Ann. Math. (2) 139, No. 2, 395–442 (1994; Zbl 0801.55007)], and which have extensively been studied ever since. For a first introduction to \(p\)-compact groups see e.g. [W. Dywer, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 433–442 (1998; Zbl 0912.55005)]. One important feature of \(p\)-compacts groups is of course that many concepts from the theory of compact Lie groups, like e.g. the existence of maximal tori, carry over. In particular the notion of a \(p\)-compact toral subgroup makes sense.

In the paper under review the authors establish a variant of the subgroup decomposition above, which applies to \(p\)-compact groups. They proceed in two steps. First the authors show that the existence of the Dwyer and Wilkerson centralizer decomposition for a \(p\)-compact group \(X\) is equivalent to the existence of a subgroup decomposition for \(X\) where the indexing family of subgroups for the homotopy colimit is the family of centric \(p\)-compact toral subgroups. In a second step they show that one still obtains a \(\bmod p\) homology decomposition of \(X\) if one uses the smaller family of radical centric \(p\)-compact toral subgroups. It is not obvious to see that the authors’ new homology decomposition actually generalizes the subgroup decomposition of Jackowski, McClure and Oliver, however, the authors clarify this issue in an appendix to the main text. In a second appendix the authors give a short exposition of the basic theory of \(p\)-compact groups and provide some relevant background material. This appendix may also serve very well as a quick introduction to the theory of \(p\)-compact groups for a novice with a serious background in algebraic topology.

For a given prime \(p\) there are two (by now) classical \(mod\;p\)-homology decompositions of the classifying space of a compact Lie group \(G\): the centralizer decomposition, established in [S. Jackowski and J. McClure, Topology 31, No. 1, 113–132 (1992; Zbl 0754.55014)], and the subgroup decomposition, established in [S. Jackowski, J. McClure and B. Oliver, Ann. Math. (2) 135, No. 1, 183–226 (1992; Zbl 0758.55004)]. In the centralizer decomposition one realizes the \(mod\;p\)-homology type of \(BG\) by gluing together the classifying spaces of the centralizers of the \(p\)-elementary abelian subgroups, while in the subgroup decomposition one glues together the classifying spaces of the so-called \(p\)-stubborn subgroups. In [B. Dwyer and C. Wilkerson, Contemp. Math. 181, 119–157 (1995; Zbl 0828.55009)] the centralizer decomposition was generalized from compact Lie groups to so-called \(p\)-compact groups. \(p\)-compact groups are certain homotopy-theoretic analogues of compact Lie groups, which were introduced in [ibid., Ann. Math. (2) 139, No. 2, 395–442 (1994; Zbl 0801.55007)], and which have extensively been studied ever since. For a first introduction to \(p\)-compact groups see e.g. [W. Dywer, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 433–442 (1998; Zbl 0912.55005)]. One important feature of \(p\)-compacts groups is of course that many concepts from the theory of compact Lie groups, like e.g. the existence of maximal tori, carry over. In particular the notion of a \(p\)-compact toral subgroup makes sense.

In the paper under review the authors establish a variant of the subgroup decomposition above, which applies to \(p\)-compact groups. They proceed in two steps. First the authors show that the existence of the Dwyer and Wilkerson centralizer decomposition for a \(p\)-compact group \(X\) is equivalent to the existence of a subgroup decomposition for \(X\) where the indexing family of subgroups for the homotopy colimit is the family of centric \(p\)-compact toral subgroups. In a second step they show that one still obtains a \(\bmod p\) homology decomposition of \(X\) if one uses the smaller family of radical centric \(p\)-compact toral subgroups. It is not obvious to see that the authors’ new homology decomposition actually generalizes the subgroup decomposition of Jackowski, McClure and Oliver, however, the authors clarify this issue in an appendix to the main text. In a second appendix the authors give a short exposition of the basic theory of \(p\)-compact groups and provide some relevant background material. This appendix may also serve very well as a quick introduction to the theory of \(p\)-compact groups for a novice with a serious background in algebraic topology.

Reviewer: Michael Joachim (Münster)

##### MSC:

55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |

55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

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\textit{N. Castellana} et al., Adv. Math. 216, No. 2, 491--534 (2007; Zbl 1156.55014)

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