On The homotopy theory of p-completed classifying spaces
نویسندگان
چکیده
Let G be a discrete group and let BG denote its classifying space. Recall that a group is said to be perfect if it is equal to it’s own commutator subgroup. If G is an arbitrary group, then write BG for the Quillen “plus” construction applied to BG with respect to the unique maximal normal perfect subgroup ΠG of G [32]. The space BG can be obtained by attaching 2 and 3 cells to BG and has the defining properties: 1. There is a natural map BG BG, which induces a homology isomorphism (with any simple coefficients), and 2. π1(BG ) ∼= G/ΠG. Since Quillen’s first defined the higher algebraic K-groups of a ring R, using the “plus” construction, and computed the K-theory of finite fields, the homotopy type of BG, for G finite and perfect has been a subject of interest as the transition between the homotopy theory of BG andBG is dramatic. Two important classical examples illustrate this transition. 1. G = Σ∞ is the colimit of the n-th symmetric group and BG is the space Q0S . [2, 11, 31] 2. G = SL(q) is The colimit of the special linear groups SL(n, q) over the field of q elements Fq and BG gives, after localization at a suitable prime, the space “image of J” [32]. In addition, Kan and Thurston showed that any path-connected CW -complex has the homotopy type of BG for a suitable group G (which is an extremely “large” infinite group in their construction) [15]. However, if G is assumed to be finite, then there are strong restrictions placed on the homotopy type of BG. One feature is that the homotopy groups are entirely torsion and are non-trivial in arbitrarily large degrees [22]. In addition, in the cases for which G is finite, it will be seen below that BG behaves through the eyes of homotopy groups as if it were a finite complex. Furthermore, the homotopy groups of BG for G finite frequently have direct summands given by the homotopy groups of various classical finite complexes such as spheres and mod-p Moore spaces [8, 22, 23, 18]. In contrast, for finite groups G the space BG does not admit an essential map to any simply-connected finite
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