|
|
| Line 1: |
Line 1: |
| '''Segal's Burnside ring conjecture''', or, more briefly, the '''Segal conjecture''', is a [[theorem]] in [[homotopy theory]], a branch of [[mathematics]]. The theorem relates the [[Burnside ring]] of a finite [[Group (mathematics)|group]] ''G'' to the [[stable cohomotopy]] of the [[classifying space]] ''BG''. The conjecture was made by [[Graeme Segal]] and proved by [[Gunnar Carlsson]]. {{As of|2006}}, this statement is still commonly referred to as the Segal conjecture, even though it now has the status of a theorem.
| | Irwin Butts is what my spouse loves to call me although I don't truly like being called like that. Supervising is my profession. Doing ceramics is what adore performing. My family life in Minnesota and my family enjoys it.<br><br>Here is my web-site :: [http://www.gaysphere.net/blog/262619 at home std testing] |
| | |
| ==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
| |
| :<math>\varprojlim \pi_S^0(BG^{(k)}_+) \to \hat{A}(G).</math>
| |
| Here, lim denotes the [[inverse limit]], π<sub>S</sub>* denotes the stable cohomotopy ring, ''B'' denotes the classifying space, the superscript ''k'' denotes the ''k''-[[CW-complex|skeleton]], and the subscript + denotes the addition of a disjoint basepoint. On the right-hand side, the hat denotes the [[topological ring#Completion|completion]] of the Burnside ring with respect to its [[augmentation ideal]].
| |
| | |
| ==The Burnside ring==
| |
| {{Main|Burnside ring}}
| |
| | |
| The Burnside ring of a finite group ''G'' is constructed from the category of finite [[group action|''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|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_(mathematics)|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 (mathematics)|field]] (see [[#Motivation and interpretation|motivation]] below). It has proven to be an important tool in the [[group representation|representation theory]] of finite groups.
| |
| | |
| ==The classifying space==
| |
| {{Main|Classifying space}}
| |
| For any [[topological group]] ''G'' admitting the structure of a [[CW-complex]], one may consider the category of [[principal bundle|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 functor|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''<sup>1</sup>.
| |
| | |
| ==Motivation and interpretation==
| |
| The content of the theorem becomes somewhat clearer if it is placed in its historical context. In the theory of representations of finite groups, one can form an object ''R[G]'' called the representation ring in a way entirely analogous to the construction of the Burnside ring outlined above. The stable cohomotopy is in a sense the natural analog to complex [[K-theory]], which is denoted ''KU''*. Segal was inspired to make his conjecture after [[Michael Atiyah]] proved the existence of an isomorphism
| |
| :<math>KU^0(BG) \to \hat{R}[G]</math> | |
| which is a special case of the [[Atiyah-Segal completion theorem]].
| |
| | |
| ==References==
| |
| *{{cite conference
| |
| | author=[[Frank Adams|J.F. Adams]]
| |
| | title= Graeme Segal's Burnside ring conjecture
| |
| | booktitle= Proc. Topology Symp. Siegen
| |
| | year= 1979}}
| |
| *{{cite journal
| |
| | author=G. Carlsson
| |
| | title=Equivariant stable homotopy and Segal's Burnside ring conjecture
| |
| | journal=Annals of Mathematics
| |
| | year=1984
| |
| | volume=120
| |
| | issue=2
| |
| | pages=189–224
| |
| | doi=10.2307/2006940
| |
| | publisher=Annals of Mathematics
| |
| | jstor=2006940}}
| |
| | |
| [[Category:Representation theory of finite groups]]
| |
| [[Category:Homotopy theory]]
| |
| [[Category:Conjectures]]
| |
| [[Category:Theorems in algebra]]
| |
Irwin Butts is what my spouse loves to call me although I don't truly like being called like that. Supervising is my profession. Doing ceramics is what adore performing. My family life in Minnesota and my family enjoys it.
Here is my web-site :: at home std testing