Segal’s Burnside Ring Conjecture for Compact Lie Groups

Equivariant Stable Homotopy Theory

“…The situation for general compact Lie groups is still only partially understood; Lee and Minami have given a good survey [43]. One direction of application has been the calculation of stable maps between classifying spaces.…”

The Relative Burnside Module and the Stable Maps Between Classifying Spaces of Compact Lie Groups

“…The author would like to express his gratitude to Ralph Cohen for his suggestion to write the survey paper [LM2], which motivated the author to write this up. He also thanks Bill Richter for his generous TeX-nical assistance.…”

The Geometric Equivariant Segal Conjecture for Toral Groups

“…No-one has given a fully satisfactory conjecture for compact Lie groups of positive dimension which could be true in general (but see [12] for a survey). Much study has centred on the map A{G)i -> n°(BG + ) from the completion of torn Dieck's Burnside ring [3] to the zeroth stable cohomotopy of BG + .…”

“…The form we find most illuminating is the geometric realization of the algebraic statement: when G is finite the map EG + -* S° induces an equivalence DEG + m (SX (0-0) of G-spectra [13] where Z>() denotes functional duality and (5°) 7 A denotes the geometric completion of S° at / [7,8] (see also [6]). No-one has given a fully satisfactory conjecture for compact Lie groups of positive dimension which could be true in general (but see [12] for a survey). Much study has centred on the map A{G)i -> n°(BG + ) from the completion of torn Dieck's Burnside ring [3] to the zeroth stable cohomotopy of BG + .…”

“…However one may conjecture that it has a dense image, and Feshbach has proved this provided the action of the Weyl group on the maximal torus is not 'quaternionic', but it is generally false otherwise [4,1]. More recently C. N. Lee has made substantial studies of the spectra DBG + , at least after p-Sidic completion: see also [9,10,11,12,17] for further discussion of these nonequivariant geometric statements.We seek a fully equivariant and arithmetically global statement analogous to (0.0), which provides an elegant packaging of known results, and suggests more general ones. Ideally, known results should be easily reconstituted from the equivariant statement by routine manipulations, and peculiarities arising from the…”