[<< wikipedia] Homeotopy
In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.


== Definition ==
The homotopy group functors 
  
    
      
        
          π
          
            k
          
        
      
    
    {\displaystyle \pi _{k}}
   assign to each path-connected topological space 
  
    
      
        X
      
    
    {\displaystyle X}
   the group 
  
    
      
        
          π
          
            k
          
        
        (
        X
        )
      
    
    {\displaystyle \pi _{k}(X)}
   of homotopy classes of continuous maps 
  
    
      
        
          S
          
            k
          
        
        →
        X
        .
      
    
    {\displaystyle S^{k}\to X.}
  
Another construction on a space 
  
    
      
        X
      
    
    {\displaystyle X}
   is the group of all self-homeomorphisms 
  
    
      
        X
        →
        X
      
    
    {\displaystyle X\to X}
  , denoted 
  
    
      
        
          
            H
            o
            m
            e
            o
          
        
        (
        X
        )
        .
      
    
    {\displaystyle {\rm {Homeo}}(X).}
   If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that 
  
    
      
        
          
            H
            o
            m
            e
            o
          
        
        (
        X
        )
      
    
    {\displaystyle {\rm {Homeo}}(X)}
   will in fact be a topological group under the compact-open topology.
Under the above assumptions, the homeotopy groups for 
  
    
      
        X
      
    
    {\displaystyle X}
   are defined to be:

  
    
      
        H
        M
        
          E
          
            k
          
        
        (
        X
        )
        =
        
          π
          
            k
          
        
        (
        
          
            H
            o
            m
            e
            o
          
        
        (
        X
        )
        )
        .
      
    
    {\displaystyle HME_{k}(X)=\pi _{k}({\rm {Homeo}}(X)).}
  Thus 
  
    
      
        H
        M
        
          E
          
            0
          
        
        (
        X
        )
        =
        
          π
          
            0
          
        
        (
        
          
            H
            o
            m
            e
            o
          
        
        (
        X
        )
        )
        =
        M
        C
        
          G
          
            ∗
          
        
        (
        X
        )
      
    
    {\displaystyle HME_{0}(X)=\pi _{0}({\rm {Homeo}}(X))=MCG^{*}(X)}
   is the  mapping class group for 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
   In other words, the  mapping class group is the set of connected components of 
  
    
      
        
          
            H
            o
            m
            e
            o
          
        
        (
        X
        )
      
    
    {\displaystyle {\rm {Homeo}}(X)}
   as specified by the functor 
  
    
      
        
          π
          
            0
          
        
        .
      
    
    {\displaystyle \pi _{0}.}
  


== Example ==
According to the Dehn-Nielsen theorem, if 
  
    
      
        X
      
    
    {\displaystyle X}
   is a closed surface then 
  
    
      
        H
        M
        
          E
          
            0
          
        
        (
        X
        )
        =
        
          
            O
            u
            t
          
        
        (
        
          π
          
            1
          
        
        (
        X
        )
        )
        ,
      
    
    {\displaystyle HME_{0}(X)={\rm {Out}}(\pi _{1}(X)),}
   the outer automorphism group of its fundamental group.


== References ==
G.S. McCarty. Homeotopy groups. Trans. A.M.S. 106(1963)293-304.
R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610.