In mathematics, Sullivan conjecture or Sullivan's conjecture on maps from classifying spaces 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
      
    
    {\displaystyle G}
  
. The most elementary formulation, however, is in terms of the classifying space 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
 of such a group. Roughly speaking, it is difficult to map such a space 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
 continuously into a finite CW complex 
  
    
      
        X
      
    
    {\displaystyle 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
      
    
    {\displaystyle BG}
  
 to 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is weakly contractible.
 This is equivalent to the statement that the map 
  
    
      
        X
      
    
    {\displaystyle X}
  
 → 
  
    
      
        F
        (
        B
        G
        ,
        X
        )
      
    
    {\displaystyle F(BG,X)}
  
 from X to the function space of maps 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
 → 
  
    
      
        X
      
    
    {\displaystyle X}
  
, not necessarily preserving the base point, given by sending a point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 to the constant map whose image is 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is a weak equivalence. The mapping space 
  
    
      
        F
        (
        B
        G
        ,
        X
        )
      
    
    {\displaystyle F(BG,X)}
  
 is an example of a homotopy fixed point set. Specifically, 
  
    
      
        F
        (
        B
        G
        ,
        X
        )
      
    
    {\displaystyle F(BG,X)}
  
 is the homotopy fixed point set of the group 
  
    
      
        G
      
    
    {\displaystyle G}
  
 acting by the trivial action on 
  
    
      
        X
      
    
    {\displaystyle X}
  
. In general, for a group 
  
    
      
        G
      
    
    {\displaystyle G}
  
 acting on a space 
  
    
      
        X
      
    
    {\displaystyle X}
  
, the homotopy fixed points are the fixed points 
  
    
      
        F
        (
        E
        G
        ,
        X
        
          )
          
            G
          
        
      
    
    {\displaystyle F(EG,X)^{G}}
  
 of the mapping space 
  
    
      
        F
        (
        E
        G
        ,
        X
        )
      
    
    {\displaystyle F(EG,X)}
  
 of maps from the universal cover 
  
    
      
        E
        G
      
    
    {\displaystyle EG}
  
 of 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
 to 
  
    
      
        X
      
    
    {\displaystyle X}
  
 under the 
  
    
      
        G
      
    
    {\displaystyle G}
  
-action on 
  
    
      
        F
        (
        E
        G
        ,
        X
        )
      
    
    {\displaystyle F(EG,X)}
  
 given by 
  
    
      
        g
      
    
    {\displaystyle g}
  
 in 
  
    
      
        G
      
    
    {\displaystyle G}
  
 acts on a map 
  
    
      
        f
      
    
    {\displaystyle f}
  
 in 
  
    
      
        F
        (
        E
        G
        ,
        X
        )
      
    
    {\displaystyle F(EG,X)}
  
 by sending it to 
  
    
      
        g
        f
        
          g
          
            −
            1
          
        
      
    
    {\displaystyle gfg^{-1}}
  
. The 
  
    
      
        G
      
    
    {\displaystyle G}
  
-equivariant map from 
  
    
      
        E
        G
      
    
    {\displaystyle EG}
  
 to a single point 
  
    
      
        ∗
      
    
    {\displaystyle *}
  
 induces a natural map η: 
  
    
      
        
          X
          
            G
          
        
        =
        F
        (
        ∗
        ,
        X
        
          )
          
            G
          
        
      
    
    {\displaystyle X^{G}=F(*,X)^{G}}
  
→
  
    
      
        F
        (
        E
        G
        ,
        X
        
          )
          
            G
          
        
      
    
    {\displaystyle F(EG,X)^{G}}
  
 from the fixed points to the homotopy fixed points of 
  
    
      
        G
      
    
    {\displaystyle G}
  
 acting on 
  
    
      
        X
      
    
    {\displaystyle X}
  
. Miller's theorem is that η is a weak equivalence for trivial 
  
    
      
        G
      
    
    {\displaystyle G}
  
-actions on finite-dimensional CW complexes. An important ingredient and motivation for his proof is a result of Gunnar Carlsson on the homology of 
  
    
      
        B
        Z
        
          /
        
        2
      
    
    {\displaystyle BZ/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
      
    
    {\displaystyle 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
      
    
    {\displaystyle 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
          
        
      
    
    {\displaystyle (X^{G})_{p}}
  
 → 
  
    
      
        F
        (
        E
        G
        ,
        (
        X
        
          )
          
            p
          
        
        
          )
          
            G
          
        
      
    
    {\displaystyle F(EG,(X)_{p})^{G}}
  
 is a weak equivalence when the order of 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is a power of a prime p, and where 
  
    
      
        (
        X
        
          )
          
            p
          
        
      
    
    {\displaystyle (X)_{p}}
  
 denotes the Bousfield-Kan p-completion of 
  
    
      
        X
      
    
    {\displaystyle 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
          
        
      
    
    {\displaystyle F(EG,X)^{G}}
  
 before completion, and Lannes's proof involves his T-functor.