chap3.txt

  
  3 Contracting Homotopies
  
  
  3.1 The PartialContractingHomotopy Data Type
  
  A  partial  contracting homotopy is a component object that knows the values
  of  a  contracting  homotopy  on  some  subspace of a resolution. It has two
  mandatory components:
  
  --    .resolution a HapResolution on which the contraction is defined.
  
  --    .knownPartOfHomotopy  a  list  of  Records  with components .space and
        .map.
  
  Let  h  be a contracting homotopy. The lookup table .knownPartOfHomotopy has
  one  entry  for  each term of the resolution h.resolution (that is, one more
  than Length(h.resolution)).
  
  The  i  th element of .knownPartOfHomotopy contains a record with components
  .space  and  .map  where  .space  is  a FreeZGWord of the i-1 st term of the
  resolution.     The     component    .map    is    a    list    of    length
  Dimension(h.resolution)(i-1).  The  entries  of  this  list are pairs [g,im]
  where  g  represents  a  group  element  and  im represents the image of the
  contraction. So the entry [g,im] in the kth component of the list .map means
  that  the kth free generator of the corresponding module multiplied with the
  group element represented by g is mapped to im under the partial contracting
  homotopy.  Note  that  the data type of g or im are not fixed at this level.
  They  must  be  specified  by  the  sub  representations.  Also, im need not
  represent  the  actual image under a contracting homotopy. It is possible to
  just  store  a  bit  of information that is then used to generate the actual
  image.
  
  As this is a very general data type, it has very few methods.
  
  3.1-1 ResolutionOfContractingHomotopy
  
  > ResolutionOfContractingHomotopy( homotopy ) ________________________method
  Returns:  A HapResolution
  
  This  returns  the  resolution  of  the  homotopy  homotopy  (the  component
  homotopy!.resolution).
  
  3.1-2 PartialContractingHomotopyLookup
  
  > PartialContractingHomotopyLookup( homotopy, term, generator, groupel ) method
  > PartialContractingHomotopyLookupNC( homotopy, term, generator, groupel ) method
  Returns:  The entry im of the corresponding lookup table
  
  Looks up the known part of the contracting homotopy homotopy and returns the
  corresponding image. More precisely, it returns the image of the generatorth
  generator  times the group element represented by groupel in term term under
  the   partial  homotopy.  The  data  type  of  this  image  depends  on  the
  representation of homotopy.
  
  term has to be an integer and generator a positive integer. groupel only has
  to be an Object.
  
  The NC version does not do any checks on the input. The other version checks
  if term and generator are sensible. It does not check groupel.