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- Sullivan_conjecture abstract "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 set in group actions of a finite group . The most elementary formulation, however, is in terms of the classifying space of such a group. Roughly speaking, it is difficult to map such a space continuously into a finite CW complex in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by Haynes Miller. Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from to is weakly contractible.This is equivalent to the statement that the map → from X to the function space of maps → , not necessarily preserving the base point, given by sending a point of to the constant map whose image is is a weak equivalence. The mapping space is an example of a homotopy fixed point set. Specifically, is the homotopy fixed point set of the group acting by the trivial action on . In general, for a group acting on a space , the homotopy fixed points are the fixed points of the mapping space of maps from the universal cover of to under the -action on given by in acts on a map in by sending it to . The -equivariant map from to a single point induces a natural map η: →from the fixed points to the homotopy fixed points of acting on . Miller's theorem is that η is a weak equivalence for trivial -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 of as an unstable module over the Steenrod algebra.Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on is allowed to be non-trivial. In, Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group . This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer, Carlsson, and Jean Lannes, showing that the natural map → is a weak equivalence when the order of is a power of a prime p, and where denotes the Bousfield-Kan p-completion of . 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 before completion, and Lannes's proof involves his T-functor.".
- Sullivan_conjecture wikiPageExternalLink PPA68,M1.
- Sullivan_conjecture wikiPageID "13400209".
- Sullivan_conjecture wikiPageRevisionID "603667706".
- Sullivan_conjecture first "Daniel H.".
- Sullivan_conjecture hasPhotoCollection Sullivan_conjecture.
- Sullivan_conjecture id "s/s120300".
- Sullivan_conjecture last "Gottlieb".
- Sullivan_conjecture title "Sullivan conjecture".
- Sullivan_conjecture subject Category:Conjectures.
- Sullivan_conjecture subject Category:Fixed_points_(mathematics).
- Sullivan_conjecture subject Category:Homotopy_theory.
- Sullivan_conjecture type Abstraction100002137.
- Sullivan_conjecture type Cognition100023271.
- Sullivan_conjecture type Concept105835747.
- Sullivan_conjecture type Conjectures.
- Sullivan_conjecture type Content105809192.
- Sullivan_conjecture type Hypothesis105888929.
- Sullivan_conjecture type Idea105833840.
- Sullivan_conjecture type PsychologicalFeature100023100.
- Sullivan_conjecture type Speculation105891783.
- Sullivan_conjecture comment "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 set in group actions of a finite group . The most elementary formulation, however, is in terms of the classifying space of such a group. Roughly speaking, it is difficult to map such a space continuously into a finite CW complex in a non-trivial manner.".
- Sullivan_conjecture label "Sullivan conjecture".
- Sullivan_conjecture sameAs m.03c41r5.
- Sullivan_conjecture sameAs Q17089125.
- Sullivan_conjecture sameAs Q17089125.
- Sullivan_conjecture sameAs Sullivan_conjecture.
- Sullivan_conjecture wasDerivedFrom Sullivan_conjecture?oldid=603667706.
- Sullivan_conjecture isPrimaryTopicOf Sullivan_conjecture.