On the $I(G)$-adic Topology of the Burnside Ring of Compact Lie Groups

A Segal conjecture for p-completed classifying spaces

“…The result is similar in spirit to Minami's description of A(G) ∧ I for a compact Lie group G in [12]. Indeed, when K is the trivial group one recovers Minami's result for finite G. This result is Theorem 3.4 in the text.…”

“…Taking K = 1 one recovers a special case of Minami's description of the I (G)-adic completion of the Burnside ring in [12] (he allows G to be compact). It should be noted that even though one can regard A(G, K) as a submodule of A(K × G) when K is finite, the results in this paper do not follow from Minami's results, as his result involves the I (K × G)-adic completion whereas we are interested in the I (G)-adic completion, and the two actions are hard to reconcile.…”

Hecke algebras and cohomotopical Mackey functors

“…We now offer a topological proof of this particular case, which is a simplified version of our original proof in [31]. Although the proof requires the Segal conjecture, we hope this would give the reader better ideas about the underlying relationship between algebra and topology.…”

“…[31]) and the topology on A(H p ) ∧ p+I(Hp) is given by the p-adic topology since H p is a p-group ( [18]). …”

“…p , which can be canonically identified with A(G) ∧ p+I(G) from our knowledge of the I(G)-adic topology and p-adic topology of A(G) (see [18,31]). …”