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Quillen Grassmannians as non-modular homotopy fixed points. (English) Zbl 1065.55009
In [Ann. Math. 139, 395–442 (1994; Zbl 0801.55007)], W.G. Dwyer and C.W. Wilkerson introduced the concept of \(p\)-compact groups, and \(p\)-compact groups for an odd prime \(p\) recently have been classified: the isomorphism classes are in 1:1-correspondence with the isomorphism classes of finite pseudo-reflection groups over the \(p\)-adic numbers. These pseudo reflection groups have been classified by D. Notbohm in [Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guixols, 1994), Prog. Math. 136, 337–352 (1996; Zbl 0855.20003)], based on the according classification of the finite pseudo reflection groups over \({\mathbb Q}\sphat_p\) in [A. Clark and J. Ewing, Pac. J. Math. 50, 425–434 (1974; Zbl 0333.55002)]. Generalized Grassmannians are \(p\)-compact groups which correspond to the groups \(G(q,r,n)\) in the Clark-Ewing list of pseudo reflection groups 2a in [A. Clark and J. Ewing, loc. cit.]. Here \(q,r,n\) are integers which satisfy \(r| q| p-1\); the corresponding \(p\)-compact groups are denoted \(X(q,r,n)\).
In [Fundam. Math 155, 1–31 (1998; Zbl 0896.55013)], D. Notbohm showed that unstable Adams and some other related operations can be defined on the classifying spaces \(BX(q,r,n)\). In the paper under review the author investigates the homotopy fix point sets of these actions. In particular she shows that \(BX(q,1,[n/q])\) arises as the homotopy fix point set of the Adams operation of order \(q\) acting on the \(p\)-completion of \(BU(n)\). Similarly she shows how \(BX(q,1,n)\) can arise in various ways as the homotopy fixed point set of unstable Adams type operations acting on other generalized Grassmannians. The latter then is used to describe the set of homotopy representations of elementary abelian \(p\)-groups into the \(p\)-compact group \(X(q,1,n)\).

MSC:
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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