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
B
G
of such a group. Roughly speaking, it is difficult to map such a space
B
G
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. Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from
B
G
to
X
is weakly contractible.
This is equivalent to the statement that the map
X
→
F
(
B
G
,
X
)
from X to the function space of maps
B
G
→
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
(
B
G
,
X
)
is an example of a homotopy fixed point set. Specifically,
F
(
B
G
,
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
(
E
G
,
X
)
G
of the mapping space
F
(
E
G
,
X
)
of maps from the universal cover
E
G
of
B
G
to
X
under the
G
-action on
F
(
E
G
,
X
)
given by
g
in
G
acts on a map
f
in
F
(
E
G
,
X
)
by sending it to
g
f
g
−
1
. The
G
-equivariant map from
E
G
to a single point
∗
induces a natural map η:
X
G
=
F
(
∗
,
X
)
G
→
F
(
E
G
,
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
B
Z
/
2
as an unstable module over the Steenrod algebra.
Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on
X
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
G
=
Z
/
2
. 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
(
X
G
)
p
→
F
(
E
G
,
(
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
(
E
G
,
X
)
G
before completion, and Lannes's proof involves his T-functor.
