chap1.txt

  
  1 Resolutions in Hap
  
  This  document  is  only concerned with the representation of resolutions in
  Hap.  Note that it is not a part of Hap. The framework provided here is just
  an extension of Hap data types used in HAPcryst and HAPprime.
  
  From  now on, let G be a group and dots -> M_n-> M_n-1->dots-> M_1-> M_0-> Z
  be a resolution with free ZG modules M_i.
  
  The  elements  of the modules M_i can be represented in different ways. This
  is what makes different representations for resolutions desirable. First, we
  will look at the standard representation (HapResolutionRep) as it is defined
  in  Hap.  After  that,  we  will present another representation for infinite
  groups.    Note    that    all    non-standard   representations   must   be
  sub-representations  of  the standard representation to ensure compatibility
  with Hap.
  
  
  1.1 The Standard Representation HapResolutionRep
  
  For  every  M_i  we  fix a basis and number its elements. Furthermore, it is
  assumed  that  we have a (partial) enumeration of the group of a resolution.
  In practice this is done by generating a lookup table on the fly.
  
  In  standard representation, the elements of the modules M_k are represented
  by  lists  -"words"-  of  pairs  of  integers. A letter [i,g] of such a word
  consists  of  the number of a basis element i or -i for its additive inverse
  and a number g representing a group element.
  
  A  HapResolution  in  HapResolutionRep  representation is a component object
  with the components
  
  --    group, a group of arbitrary type.
  
  --    elts,  a  (partial)  list  of (possibly duplicate) elements in G. This
        list  provides  the  "enumeration"  of  the group. Note that there are
        functions  in Hap which assume that elts[1] is the identity element of
        G.
  
  --    appendToElts(g)  a function that appends the group element g to .elts.
        This  is  not  documented  in  Hap  1.8.6 but seems to be required for
        infinite  groups.  This requirement might vanish in some later version
        of Hap [G. Ellis, private communication].
  
  --    dimension(k), a function which returns the ZG-rank of the Module M_k
  
  --    boundary(k,j),  a function which returns the image in M_k-1 of the jth
        free  generator  of  M_k.  Note  that negative j are valid as input as
        well.  In  this  case  the additive inverse of the boundary of the jth
        generator is returned
  
  --    homotopy(k,[i,g]) a function which returns the image in M_k+1, under a
        contracting  homotopy M_k -> M_k+1, of the element [[i,g]] in M_k. The
        value  of  this might be fail. However, currently (version 1.8.4) some
        Hap functions assume that homotopy is a function without testing.
  
  --    properties,  a  list  of pairs ["name","value"] "name" is a string and
        value  is  anything  (boolean, number, string...). Every HapResolution
        (regardless  of  representation)  has  to  have ["type","resolution"],
        ["length",length]  where  length  is  the length of the resolution and
        ["characteristic",char].  Currently  (Hap  1.8.6),  length must not be
        infinity.  The  values of these properties can be tested using the Hap
        function EvaluateProperty(resolution,propertyname).
  
  Note  that  making  HapResolutions  immutable  will make the .elts component
  immutable.  As this lookup table might change during calculations, we do not
  recommend using immutable resolutions (in any representation).
  
  
  1.2 The HapLargeGroupResolutionRep Representation
  
  In  this  sub-representation  of  the  standard  representation,  the module
  elements  in  this resolution are lists of groupring elements. So the lookup
  table  .elts is not used as long as no conversion to standard representation
  takes  place. In addition to the components of a HapResolution, a resolution
  in large group representation has the following components:
  
  --    boundary2(resolution,term,gen),  a  function that returns the boundary
        of the genth generator of the termth module.
  
  --    groupring the group ring of the resolution resolution.
  
  --    dimension2(resolution,term)  a  function that returns the dimension of
        the termth module of the resolution resolution.
  
  The  effort of having two versions of boundary and dimension is necessary to
  keep the structure compatible with the usual Hap resolution.