Sullivan conjecture

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 G. The most elementary formulation, however, is in terms of the classifying space BG of such a group. Roughly speaking, it is difficult to map such a space BG continuously into a finite CW complex X in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by Haynes Miller.[1] Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from BG to X is weakly contractible.

This is equivalent to the statement that the map X → F(BG, X) from X to the function space of maps BG → X, not necessarily preserving the base point, given by sending a point x of X to the constant map whose image is x is a weak equivalence. The mapping space F(BG, X) is an example of a homotopy fixed point set. Specifically, F(BG, X) is the homotopy fixed point set of the group G acting by the trivial action on X. In general, for a group G acting on a space X, the homotopy fixed points are the fixed points F(EG, X)^G of the mapping space F(EG, X) of maps from the universal cover EG of BG to X under the G-action on F(EG, X) given by g in G acts on a map f in F(EG, X) by sending it to gfg^{-1}. The G-equivariant map from EG to a single point * induces a natural map η: X^G = F(*,X)^G→F(EG, X)^G from the fixed points to the homotopy fixed points of G acting on X. Miller's theorem is that η is a weak equivalence for trivial G-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 BZ/2 as an unstable module over the Steenrod algebra.[2]

Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on X is allowed to be non-trivial. In,[3] Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group G=Z/2. This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer,[4] Carlsson,[5] and Jean Lannes,[6] showing that the natural map (X^G)_p → F(EG, (X)_p)^G is a weak equivalence when the order of G is a power of a prime p, and where (X)_p denotes the Bousfield-Kan p-completion of X. 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 F(EG,X)^G before completion, and Lannes's proof involves his T-functor.[7]

References

  1. ↑ 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.
  2. ↑ Carlsson, Gunnar (1983). "G.B. Segal's Burnside Ring Conjecture for (Z/2)^k". Topology 22 (1): 83–103. doi:10.1016/0040-9383(83)90046-0.
  3. ↑ Sullivan, Denis (1971). Geometric topology. Part I. Cambridge, MA: Massachusetts Institute of Technology Press. p. 432.
  4. ↑ Dwyer, William; Haynes Miller; Joseph Neisendorfer (1989). "Fibrewise Completion and Unstable Adams Spectral Sequences". Israel Journal of Mathematics 66 (1-3). doi:10.1007/bf02765891.
  5. ↑ Carlsson, Gunnar (1991). "Equivariant stable homotopy and Sullivan's conjecture". Invent. math. 103: 497–525. doi:10.1007/bf01239524.
  6. ↑ Lannes, Jean (1992). "Sur les espaces fonctionnels dont la source est le classifiant d'un p-groupe abélien élémentaire". Publications Mathématiques de l'I.H.E.S. 75: 135–244. doi:10.1007/bf02699494.
  7. ↑ Schwartz, Lionel (1994). Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture. Chicago and London: The University of Chicago Press. ISBN 0-226-74203-2.

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