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| In [[mathematics]], '''Sullivan conjecture''' can refer to any of several results and conjectures prompted by [[homotopy theory]] work of [[Dennis Sullivan]]. A basic theme and motivation concerns the [[Fixed point (mathematics)|fixed point]] set in [[group action]]s of a [[finite group]] <math>G</math>. The most elementary formulation, however, is in terms of the [[classifying space]] <math>BG</math> of such a group. Roughly speaking, it is difficult to map such a space <math>BG</math> continuously into a finite [[CW complex]] <math>X</math> in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by [[Haynes Miller]].<ref>[http://www.jstor.org/discover/10.2307/2007071?uid=2129&uid=2&uid=70&uid=4&sid=21100787615331 Haynes Miller, The Sullivan Conjecture on Maps from Classifying Spaces, The Annals of Mathematics, second series, Vol. 120 No. 1, 1984, pp. 39-87]. JSTOR: The Annals of Mathematics. Accessed May 9, 2012.</ref> Specifically, in 1984, Miller proved that the [[function space]], carrying the [[compact-open topology]], of [[base point]]-preserving mappings from <math>BG</math> to <math>X</math> is [[weakly contractible]].
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| This is equivalent to the statement that the map <math>X</math> → <math>F(BG, X)</math> from X to the function space of maps <math>BG</math> → <math>X</math>, not necessarily preserving the base point, given by sending a point <math>x</math> of <math>X</math> to the constant map whose image is <math>x</math> is a [[weak equivalence (homotopy theory)|weak equivalence]]. The mapping space <math>F(BG, X)</math> is an example of a homotopy fixed point set. Specifically, <math>F(BG, X)</math> is the homotopy fixed point set of the group <math>G</math> acting by the trivial action on <math>X</math>. In general, for a group <math>G</math> acting on a space <math>X</math>, the homotopy fixed points are the fixed points <math>F(EG, X)^G</math> of the mapping space <math>F(EG, X)</math> of maps from the [[Covering space|universal cover]] <math>EG</math> of <math>BG</math> to <math>X</math> under the <math>G</math>-action on <math>F(EG, X)</math> given by <math>g</math> in <math>G</math> acts on a map <math>f</math> in <math>F(EG, X)</math> by sending it to <math>gfg^{-1}</math>. The <math>G</math>-equivariant map from <math>EG</math> to a single point <math>*</math> induces a natural map η: <math>X^G = F(*,X)^G</math>→<math>F(EG, X)^G</math> from the fixed points to the homotopy fixed points of <math>G</math> acting on <math>X</math>. Miller's theorem is that η is a weak equivalence for trivial <math>G</math>-actions on finite dimensional CW complexes. An important ingredient and motivation (see [1]) for his proof is a result of [[Gunnar Carlsson]] on the [[Homology (mathematics)|homology]] of <math>BZ/2</math> as an unstable module over the [[Steenrod algebra]].<ref>{{cite journal|last=Carlsson|first=Gunnar|title=G.B. Segal's Burnside Ring Conjecture for (Z/2)^k|journal=Topology|year=1983|volume=22|issue=1|pages=83–103}}</ref>
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| Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on <math>X</math> is allowed to be non-trivial. In,<ref>{{cite book|last=Sullivan|first=Denis|title=Geometric topology. Part I.|year=1971|publisher=Massachusetts Institue of Technology Press|location=Cambridge, MA|pages=432}}</ref> Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and [[Daniel Kan|D. Kan]] for the group <math>G=Z/2</math>. This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer,<ref>{{cite journal|last=Dwyer|first=William|coauthors=Haynes Miller, Joseph Neisendorfer|title=Fibrewise Completion and Unstable Adams Spectral Sequences|journal=Isreal Journal of Mathematics|year=1989|volume=66|issue= 1-3}}</ref> Carlsson,<ref>{{cite journal|last=Carlsson|first=Gunnar|title=Equivariant stable homotopy and Sullivan's conjecture|journal=Invent. math.|year=1991|volume=103|pages=497–525}}</ref> and Jean Lannes,<ref>{{cite journal|last=Lannes|first=Jean|title=Sur les espaces fonctionnels dont la source est le classifiant d'un p-groupe abélien élémentaire|journal=Publications Mathématiques de l'I.H.E.S.|year=1992|volume=75|pages=135–244}}</ref> showing that the natural map <math>(X^G)_p</math> → <math>F(EG, (X)_p)^G</math> is a weak equivalence when the order of <math>G</math> is a power of a prime p, and where <math>(X)_p</math> denotes the Bousfield-Kan p-completion of <math>X</math>. Miller's proof involves an unstable [[Adams spectral sequence]], Carlsson's proof uses his affirmative solution of the [[Segal conjecture]] and also provides information about the homotopy fixed points <math>F(EG,X)^G</math> before completion, and Lannes's proof involves his T-functor.<ref>{{cite book|last=Schwartz|first=Lionel|title=Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture|year=1994|publisher=The University of Chicago Press|location=Chicago and London|isbn=0-226-74203-2}}</ref>
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| ==References==
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| <references/>
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| ==External links==
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| *{{Springer|title=Sullivan conjecture|id=s/s120300|first=Daniel H.|last= Gottlieb}}
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| *[http://books.google.com/books?id=rjMXVqEiA7MC&pg=PA67&lpg=PA67&dq=%22sullivan+conjecture%22&source=web&ots=nI3OJJQN2J&sig=DFx0MSsGdiyvyZlbHT825picIbU#PPA68,M1 Book extract]
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| [[Category:Conjectures]]
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Homotopy theory]]
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