| Segal Conjecture |
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| CATEGORIES ABOUT SEGAL CONJECTURE | |
| representation theory of finite groups | |
| homotopy theory | |
| conjectures | |
| mathematical theorems | |
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STATEMENT OF THE THEOREM The Segal conjecture has several different formulations, not all of which are equivalent. Here is a weak form: there exists, for every finite group ''G'', an isomorphism
THE BURNSIDE RING The Burnside ring of a finite group ''G'' is constructed from the category of finite ''G''-sets as a Grothendieck Group . More precisely, let ''M(G)'' be the commutative Monoid of isomorphism classes of finite ''G''-sets, with addition the disjoint union of ''G''-sets and identity element the empty set (which is a ''G''-set in a unique way). Then ''A(G)'', the Grothendieck group of ''M(G)'', is an abelian group. It is in fact a Free abelian group with basis elements represented by the ''G''-sets ''G''/''H'', where ''H'' varies over the subgroups of ''G''. (Note that ''H'' is not assumed here to be a normal subgroup of ''G'', for while ''G''/''H'' is not a group in this case, it is still a ''G''-set.) The Ring structure on ''A(G)'' is induced by the direct product of ''G''-sets; the multiplicative identity is the (isomorphism class of any) one-point set, which becomes a ''G''-set in a unique way. The Burnside ring is the analogue of the Representation Ring in the category of finite sets, as opposed to the category of finite-dimensional Vector Space s over a Field (see Motivation below). It has proven to be an important tool in the Representation Theory of finite groups. THE CLASSIFYING SPACE See Also: Classifying space For any Topological Group ''G'' admitting the structure of a CW-complex , one may consider the category of Principal ''G''-bundles . One can define a Functor from the category of CW-complexes to the category of sets by assigning to each CW-complex ''X'' the set of principal ''G''-bundles on ''X''. This functor descends to a functor on the homotopy category of CW-complexes, and it is natural to ask whether the functor so obtained is Representable . The answer is affirmative, and the representing object is called the classifying space of the group ''G'' and typically denoted ''BG''. If we restrict our attention to the homotopy category of CW-complexes, then ''BG'' is unique. Any CW-complex that is homotopy equivalent to ''BG'' is called a ''model'' for ''BG''. For example, if ''G'' is the group of order 2, then a model for ''BG'' is infinite-dimensional real projective space. It can be shown that if ''G'' is finite, then any CW-complex modelling ''BG'' has cells of arbitrarily large dimension. On the other hand, if ''G'' = Z, the integers, then the classifying space ''BG'' is homotopy equivalent to the circle ''S''1. MOTIVATION AND INTERPRETATION
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