GAP (HAP commands) - Chapter 20: Words in free ZG-modules

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20 Words in free ZG-modules
20.1
20.1-1 AddFreeWords
20.1-2 AddFreeWordsModP
20.1-3 AlgebraicReduction
20.1-4 MultiplyWord
20.1-5 Negate
20.1-6 NegateWord
20.1-7 PrintZGword
20.1-8 TietzeReduction
20.1-9 ResolutionBoundaryOfWord

### 20 Words in free ZG-modules

#### 20.1

##### 20.1-1 AddFreeWords

`‣ AddFreeWords`( v, w ) | ( function )

Inputs two words v,w in a free ZG-module and returns their sum v+w. If the characteristic of Z is greater than 0 then the next function might be more efficient.

Examples:

##### 20.1-2 AddFreeWordsModP

`‣ AddFreeWordsModP`( v, w, p ) | ( function )

Inputs two words v,w in a free ZG-module and the characteristic p of Z. It returns the sum v+w. If p=0 the previous function might be fractionally quicker.

Examples:

##### 20.1-3 AlgebraicReduction

`‣ AlgebraicReduction`( w ) | ( function )

`‣ AlgebraicReduction`( w, p ) | ( function )

Inputs a word w in a free ZG-module and returns a reduced version of the word in which all pairs of mutually inverse letters have been cancelled. The reduction is performed in a free abelian group unless the characteristic p of Z is entered.

Examples:

##### 20.1-4 MultiplyWord

`‣ MultiplyWord`( n, w ) | ( function )

Inputs a word w and integer n. It returns the scalar multiple n⋅ w.

Examples:

##### 20.1-5 Negate

`‣ Negate`( [i, j] ) | ( function )

Inputs a pair [i,j] of integers and returns [-i,j].

Examples:

##### 20.1-6 NegateWord

`‣ NegateWord`( w ) | ( function )

Inputs a word w in a free ZG-module and returns the negated word -w.

Examples:

##### 20.1-7 PrintZGword

`‣ PrintZGword`( w, elts ) | ( function )

Inputs a word w in a free ZG-module and a (possibly partial but sufficient) listing elts of the elements of G. The function prints the word w to the screen in the form

r_1E_1 + ... + r_nE_n

where r_i are elements in the group ring ZG, and E_i denotes the i-th free generator of the module.

Examples: 1

##### 20.1-8 TietzeReduction

`‣ TietzeReduction`( S, w ) | ( function )

Inputs a set S of words in a free ZG-module, and a word w in the module. The function returns a word w' such that {S,w'} generates the same abelian group as {S,w}. The word w' is possibly shorter (and certainly no longer) than w. This function needs to be improved!

Examples:

##### 20.1-9 ResolutionBoundaryOfWord

`‣ ResolutionBoundaryOfWord`( R, n, w ) | ( function )

Inputs a resolution R, a positive integer n and a list w representing a word in the free module R_n. It returns the image of w under the n-th boundary homomorphism.

Examples: 1

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