`PartialContractingHomotopy`

Data TypeA 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`Record`

s 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 `k`

th component of the list `.map`

means that the `k`

th 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.

`‣ ResolutionOfContractingHomotopy` ( homotopy ) | ( method ) |

Returns: A `HapResolution`

This returns the resolution of the homotopy `homotopy` (the component `homotopy!.resolution`).

`‣ 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 `generator`th 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`.

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