Sullivan conjecture explained
. 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.
[1] 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 for his proof is a result of
Gunnar Carlsson on the
homology of
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
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
. 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
→
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.
[7] References
- Haynes . Miller . Haynes Miller . The Sullivan Conjecture on Maps from Classifying Spaces . . 120 . 1 . 1984 . 39–87 . 10.2307/2007071. 2007071 .
- Carlsson. Gunnar. G.B. Segal's Burnside Ring Conjecture for (Z/2)^k. Topology. 1983. 22. 1. 83–103. 10.1016/0040-9383(83)90046-0. free.
- Book: Sullivan, Denis. Geometric topology. Part I.. 1971. Massachusetts Institute of Technology Press. Cambridge, MA. 432.
- Dwyer. William. Haynes Miller . Joseph Neisendorfer . Fibrewise Completion and Unstable Adams Spectral Sequences. Israel Journal of Mathematics. 1989. 66. 1–3. 160–178. 10.1007/bf02765891. free.
- Carlsson. Gunnar. Equivariant stable homotopy and Sullivan's conjecture. Inventiones Mathematicae. 1991. 103. 497–525. 10.1007/bf01239524. free. 1991InMat.103..497C .
- Lannes. Jean. Sur les espaces fonctionnels dont la source est le classifiant d'un p-groupe abélien élémentaire. Publications Mathématiques de l'IHÉS. 1992. 75. 135–244. 10.1007/bf02699494.
- Book: Schwartz, Lionel. Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture. 1994. The University of Chicago Press. Chicago and London. 978-0-226-74203-8.
External links
