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\centerline{\bf The Sullivan Conjecture}
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The Sullivan Conjecture in Homotopy Theory usually refers to a theorem
about the contractibility or homotopy equivalence of certain types of
mapping spaces. These results are vast generalizations of two different but
related conjectures made by Dennis Sullivan in 1972.

Haynes Miller achieved the first major breakthrough and is given credit for
solving the Sullivan conjecture. This was published in 1984 and one version
reads: The space of pointed maps $Map_*(B_G, X)$ from the classifying space of 
of a finite group $G$ to a finite CW-complex $X$ is weakly contractible. The 
mapping space has the compact-open toplogy.

An equivalent statement: The space of unpointed maps $Map(B_G, X)$is weakly 
homotopy equivalent to $X$ (under the same hypotheses on $G$ and $X$.

These theorems are still true when $B_G$ is replaced by a CW-complex which
has only finitely many non-zero homotopy groups, each of which is locally
finite and where $X$ can be any finite dimensional CW-complex. This 
improvement is due to Alexander Zabrodsky.  

Equivariant versions of the Sullivan Conjecture come about by considering
the question: How close does the natural map $X^G \rightarrow X^{hG}$ come to
being a homotopy equivalence? Here $X^G$ is the fixed point set of a group
action $G$ on the space $X$ and the homotopy fixed point set is 
$X^{hG} = Map_G(E_G, X)$, the space of equivariant maps from the contractible
space $E_G$ on which $G$ acts freely to the $G$-space $X$. For $G$ acting
trivially on $X$, we see that Miller's version of the Sullivan Conjecture
gives a positive answer to this question. 

Another version of this question is that the fixed point set of a $G$-space
localized at a prime $p$ is weakly homotopy equivalent to the homotopy fixed
point set of $G$ acting on the $p$-localization of $X$. One proof of this
result is via the Segal Conjecture by Gunnar Carlsson. Haynes Miller also
independently proved this result, and J. Lannes has a subsequent proof
using his $T$ functor. 

These theorems have found many beautiful applications at the hands of the 
above mentioned workers as well as W. G. Dwyer, C. McGibbon, J. A. Neisendorfer
and C. Wilkerson, S. Jackowsky to mention only a few.

References:

Miller, Haynes The Sullivan conjecture and homotopical representation theory. Proceedings of
the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 580--589, Amer. Math. Soc.,
Providence, RI, 1987.

Carlsson, Gunnar Segal's Burnside ring conjecture and related problems in topology.
Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 574--579, Amer.
Math. Soc., Providence, RI, 1987.
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 \noindent Daniel Henry Gottlieb, Purdue University.

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