kenzo/doc/org/01_Chain_Complexes.org

• Chain Complexes
  • Introduction
  • Generators, terms, and combinations
    • Representation of a combination
    • Ordering the generators
      • Examples

Chain Complexes

Introduction

In the Kenzo program, a chain complex¹$(C_p, d_p)$ is a collection of free \(\mathbb{Z}\)-modules $(C_p)$, one for each \(p\in\mathbb{Z}\), together with a homomorphism $d_p : C_p \rightarrow C_{p-1}$, such that, for all $p$, $d_{p-1} \circ d_p = 0$.

In the Kenzo program, a morphism $f=(f_p)$, of degree $k$, from a chain complex $(C_p, d_p)$ to another $(C'_p, d'_p)$ is a collection of homomorphisms

$$f_p : C_p \rightarrow C'_{p+k}.$$

This is expressed by the following diagram, generally not assumed commutative.

Three types of morphisms are most generally considered.

1. $k=0$. If the commutativity relation $d'_p\circ f_p=f_{p-1}\circ d_p$ holds for every $p$, then the morphism $f$ is an ordinary chain complex morphism or chain map.
2. $k=-1$. If $(C_p,d_p)=(C'_p,d'_p)$ and $f_p=d_p$, then $f$ is the differential of the chain complex $(C_p,d_p)$ and, in fact, this differential is implemented in the Kenzo program as a morphism of degree $-1$. Also, if $d_p \neq f_p$ and $(d_{p-1}+f_{p-1})(d_p+f_p) = 0$ for every $p$, then $f_\ast$ is a perturbation of the differential $d_\ast$.
3. $k=+1$. In this case, $f$ is usually a homotopy operator, that is, some relation $$d'_{p+1}\circ f_p+f_{p-1}\circ d_p=g_p-g'_p$$ is satisfied for two (ordinary) chain complex morphisms $g_\ast$ and $g'_\ast$, and $f_\ast$ is a homotopy between $g_\ast$ and $g'_\ast$.

For technical reasons, these three types of morphisms have been implemented in the Kenzo program in a unique type.

Generators, terms, and combinations

To become familiar with the Lisp functions implementing the chain complexes, the best is to begin with an example. Simplicial complexes are good candidates for this purpose, and we shall take as a typical example the following simplicial complex.

/git/mmarco/kenzo/src/commit/4ffed1628434f27ec6f65880d6a10d524657a904/doc/org/diabolo.pdf

In this simplicial complex, called here diabolo, there are 3 non-null chain groups.

• $C_0$, the free \(\mathbb{Z}\)-module on the set of vertices $\{s_0,s_1,s_2,s_3,s_4,s_5\}$
• $C_1$, the free \(\mathbb{Z}\)-module on the set of edges $\{s_{01},s_{02},s_{12},s_{23},s_{34},s_{35}, s_{45}\}$
• $C_2$, the free \(\mathbb{Z}\)-module on the set of triangles (here a singleton) $\{s_{345}\}$

The elements of either of those groups $C_p$ are linear integer combinations of the corresponding basis $\beta_p$, i.e. elements of the form: $$ \sum_{b \in \beta_p} \lambda_b b. $$ For example $2 s_{02} - 3 s_{34} \in C_1$. An element $b$ of the basis $\beta_p$ of $C_p$ is called a generator of $C_p$ and in our specific case this generator will be represented by a Lisp symbol. For instance, $s_{45}$ will be translated into s45. However, the user should be aware that in more realistic usage of the software, generators may be of any type. A product such as $\lambda_i\sigma_i$ is called a term and a sum of terms, a combination.

Representation of a combination

A combination is represented internally in the system by a structure having the following general form:

$$\#\texttt{S(CMBN}\quad\texttt{DEGREE P}\quad\texttt{LIST (}\texttt{(} \lambda_1\cdot\sigma_1\texttt{)}\dots\texttt{(}\lambda_k\cdot\sigma_k \texttt{)))}$$

and containing

1. The degree of the combination corresponding to the index \(p\in\mathbb{Z}\) of the group $C_p$ to which this combination belongs.
2. The list of the internal representation of the terms, namely the list of pairs $(\lambda_i\cdot\sigma_i)$.

This choice of representation implies that only homogeneous combinations will be considered. The type CMBN and a printing method are provided with this internal representation. The external form of a combination is shown in the examples.

Ordering the generators

In order to speed up the execution of algorithms involving combinations, the list of pairs $(\lambda_i.\sigma_i)$ is ordered by an adequate ordering function (e.g., the lexicographical ordering on the symbols). For programming convenience, an enumerated type CMPR has been defined:

(deftype cmpr() '(member :less :equal :greater))

A number of macros, functions and methods have been defined on usual sets (symbols, numbers, lists, etc.) taking their value in the set [:less, :equal, :greater]. Of course, the user may define its own function for a particular case. There exists functions to compare various couples of usual items:

f-cmpr n1 n2 | [Function]

• Return :less, :equal, :greater, according to the result of the canonical comparison of both integers n1 and n2.

s-cmpr symbol1 symbol2 | [Function]

• Return :less, :equal, :greater, according to the result of the Lisp comparison functions on the strings (symbol-name symbol1 ) and (symbol-name symbol2 )

l-cmpr list1 list2 | [Function]

• Return :less, :equal, :greater, according to the lexicographical ordering of the two lists list1 and list2, making sense only if list1 and list2 are lists made of integers, symbols and/or lists; in the last case, the qualifier list is to be understood recursively.

Examples

(f-cmpr 123 789)

:LESS

(s-cmpr 'circulation 'circular)

:GREATER

(s-cmpr 'qwerty 'qwerty)

:EQUAL

(l-cmpr '(1 a b) '(1 a))

:GREATER

---

¹

P.J. Giblin in Graph, Surface and Homology, Chapman and Hall Math. series, 1981.