From 3746758860bc43f73400aed0958eb67489fc16bc Mon Sep 17 00:00:00 2001
From: Gerd Heber <gheber@hdfgroup.org>
Date: Thu, 28 Dec 2017 10:46:14 -0600
Subject: [PATCH] First experiments w/ Org version of the Handbook.

---
 doc/org/00_Overview.org | 1926 +++++++++++++++++++++++++++++++++++++++
 doc/org/Foreword.org    |   48 +
 doc/org/Handbook.org    |   11 +
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 create mode 100644 doc/org/00_Overview.org
 create mode 100644 doc/org/Foreword.org
 create mode 100644 doc/org/Handbook.org

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+
+* Overview
+
+Let us begin by the space $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}, 3)$ described
+as a simplicial set having only three non-degenerate simplices, namely in
+dimension $0$, $3$ and $4$. In the  representation created by the software,
+the 0-simplex (base point), the 3-simplex and the 4-simplex are respectively
+labelled =*=, =M3= and =N4=. The faces 0 and 2 of the 4-simplex =N4= are
+identified with (pasted to) the 3-simplex =M3=, the others being contracted on
+the base point. To create the simplicial set one types simply:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar m23)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf m23 (moore 2 3))
+#+END_SRC
+
+#+RESULTS:
+: [K1 Simplicial-Set]
+
+The system answers that a Kenzo object has been created, with number 1 and
+type =SIMPLICIAL SET=. This object may be referenced by the symbol =m23=.
+
+#+BEGIN_SRC lisp :session cat
+m23
+#+END_SRC
+
+#+RESULTS:
+: [K1 Simplicial-Set]
+
+We may compute the homology groups of this space, using the underlying chain
+complex induced by the simplicial set description. Here we compute the $H_i$
+from 0 to 4 included. When in the answer the component part is void, it means
+that the corresponding homology group is null.
+
+#+BEGIN_SRC lisp :session cat :results output
+(homology m23 0 5)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 0.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+Computing the boundary of the generator 1/1 (dimension 0) :
+,* 
+End of computing.
+
+
+Computing boundary-matrix in dimension 1.
+Rank of the source-module : 0.
+
+
+
+
+Homology in dimension 0 :
+
+
+Component Z
+
+
+---done---
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+
+
+Computing boundary-matrix in dimension 1.
+Rank of the source-module : 0.
+
+
+Computing boundary-matrix in dimension 2.
+Rank of the source-module : 0.
+
+
+
+
+Homology in dimension 1 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+
+
+Computing boundary-matrix in dimension 2.
+Rank of the source-module : 0.
+
+
+Computing boundary-matrix in dimension 3.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+Computing the boundary of the generator 1/1 (dimension 3) :
+M3 
+End of computing.
+
+
+
+
+Homology in dimension 2 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+
+
+Computing boundary-matrix in dimension 3.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+Computing the boundary of the generator 1/1 (dimension 3) :
+M3 
+End of computing.
+
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+Computing the boundary of the generator 1/1 (dimension 4) :
+N4 
+End of computing.
+
+
+
+
+Homology in dimension 3 :
+
+
+Component Z/2Z
+
+
+---done---
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+Computing the boundary of the generator 1/1 (dimension 4) :
+N4 
+End of computing.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 0.
+
+
+
+
+Homology in dimension 4 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-28, 0h 13m 32s.
+
+#+end_example
+
+The homology of =m23= is  $(\mathbb{Z},0,0,\mathbb{Z}/2,0,\ldots)$; the reduced
+homology is only  $\mathbb{Z}/2$  in dimension 3, defining the Moore space up
+to homotopy.
+
+As =m23= is a simplicial set, it is possible to create the cartesian product
+=m23xm23= by the function =crts-prdc= (= CaRTeSian PRoDuCt). This is a new
+simplicial set.
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar m23xm23)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf m23xm23 (crts-prdc m23 m23))
+#+END_SRC
+
+#+RESULTS:
+: [K10 Simplicial-Set]
+
+Being a simplicial set, =m23xm23= is also a chain complex object and we may for
+instance ask for the basis in dimension 6.
+
+#+BEGIN_SRC lisp :session cat :exports both :results verbatim
+(basis m23xm23 6)
+#+END_SRC
+
+#+RESULTS:
+: (<CrPr 1-0 N4 3-2 N4> <CrPr 1-0 N4 4-2 N4> <CrPr 1-0 N4 4-3 N4> <CrPr 1-0 N4 4-3-2 M3> <CrPr 1-0 N4 5-2 N4> <CrPr 1-0 N4 5-3 N4> <CrPr 1-0 N4 5-3-2 M3> <CrPr 1-0 N4 5-4 N4> <CrPr 1-0 N4 5-4-2 M3> <CrPr 1-0 N4 5-4-3 M3> ...)
+
+#+BEGIN_SRC lisp :session cat :exports both
+(length (basis m23xm23 6))
+#+END_SRC
+
+#+RESULTS:
+: 230
+
+As shown by the last command, the number of elements of the basis is quite
+large (230). The user will note that the basis elements are in this particular
+case formed by cartesian products of /degenerated simplices/. In the list, an
+element like =<CrPr 1-0 N4 5-3-2 M3>= means
+$\eta_1\eta_0\mathrm{N4}\times\eta_5\eta_3\eta_2\mathrm{M3}$.
+
+We may construct also the tensor product $\mathrm{m23}\otimes\mathrm{m23}$ from
+the underlying chain complex of the simplicial set =m23=. The tensor product of
+two simplicial sets do not make sense, and Kenzo understands to work on the
+underlying chain complex =m23=. This tensor product is a new chain complex and
+we see that the basis in dimension 6 has only one element:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar t2m23)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf t2m23 (tnsr-prdc m23 m23))
+#+END_SRC
+
+#+RESULTS:
+: [K3 Chain-Complex]
+
+#+BEGIN_SRC lisp :session cat :exports both :results verbatim
+(basis t2m23 6)
+#+END_SRC
+
+#+RESULTS:
+: (<TnPr M3 M3>)
+
+The Eilenberg-Zilber theorem is used to compute the homology groups of the
+cartesian product space: as chain complexes, =m23xm23= and =t2m23= have the
+same homology groups, but the computations using the tensor product are
+considerably faster. The user can check Künneth's theorem.
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(homology m23xm23 0 8)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 0.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/1 (dimension 0) :
+<TnPr * *> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 1.
+Rank of the source-module : 0.
+
+
+
+
+Homology in dimension 0 :
+
+
+Component Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+
+
+Computing boundary-matrix in dimension 1.
+Rank of the source-module : 0.
+
+
+Computing boundary-matrix in dimension 2.
+Rank of the source-module : 0.
+
+
+
+
+Homology in dimension 1 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+
+
+Computing boundary-matrix in dimension 2.
+Rank of the source-module : 0.
+
+
+Computing boundary-matrix in dimension 3.
+Rank of the source-module : 2.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/2 (dimension 3) :
+<TnPr * M3> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 2/2 (dimension 3) :
+<TnPr M3 *> 
+End of computing.
+
+
+
+
+Homology in dimension 2 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+
+
+Computing boundary-matrix in dimension 3.
+Rank of the source-module : 2.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/2 (dimension 3) :
+<TnPr * M3> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 2/2 (dimension 3) :
+<TnPr M3 *> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 2.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/2 (dimension 4) :
+<TnPr * N4> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 2/2 (dimension 4) :
+<TnPr N4 *> 
+End of computing.
+
+
+
+
+Homology in dimension 3 :
+
+
+Component Z/2Z
+
+Component Z/2Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 2.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/2 (dimension 4) :
+<TnPr * N4> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 2/2 (dimension 4) :
+<TnPr N4 *> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 0.
+
+
+
+
+Homology in dimension 4 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 0.
+
+
+Computing boundary-matrix in dimension 6.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/1 (dimension 6) :
+<TnPr M3 M3> 
+End of computing.
+
+
+
+
+Homology in dimension 5 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+
+
+Computing boundary-matrix in dimension 6.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/1 (dimension 6) :
+<TnPr M3 M3> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 7.
+Rank of the source-module : 2.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/2 (dimension 7) :
+<TnPr M3 N4> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 2/2 (dimension 7) :
+<TnPr N4 M3> 
+End of computing.
+
+
+
+
+Homology in dimension 6 :
+
+
+Component Z/2Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+
+
+Computing boundary-matrix in dimension 7.
+Rank of the source-module : 2.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/2 (dimension 7) :
+<TnPr M3 N4> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 2/2 (dimension 7) :
+<TnPr N4 M3> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 8.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+Computing the boundary of the generator 1/1 (dimension 8) :
+<TnPr N4 N4> 
+End of computing.
+
+
+
+
+Homology in dimension 7 :
+
+
+Component Z/2Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 20m 39s.
+
+#+end_example
+
+Let us consider now the space $\mathrm{K}(\mathbb{Z}, 1)$. This is an Abelian
+simplicial group created  in `Kenzo` by the function =k-z=. In this simplicial
+group, a simplex in dimension $n$ is mathematically represented by a sequence
+of integers, known as a /bar/ object:
+$$ [a_1 \mid a_2 \mid \ldots \mid a_n].$$
+In =Kenzo=, a non-degenerate simplex of $\mathrm{K}(\mathbb{Z}, 1)$ in
+dimension $n$ will be simply a list of $n$ non-null integers, for instance:
+=(2 3 4 5)=. In dimension 0, the only simplex is =NIL= (the base point).
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar kz1)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf kz1 (k-z 1))
+#+END_SRC
+
+#+RESULTS:
+: [K38 Abelian-Simplicial-Group]
+
+But this object is also a /coalgebra/ and an /algebra/, and we may see the
+effect of the respective induced /coproduct/ and /product/: 
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (cprd kz1 4 '(2 3 4 5)))
+#+END_SRC
+
+#+RESULTS:
+: 
+: ----------------------------------------------------------------------{CMBN 4}
+: <1 * <TnPr NIL (2 3 4 5)>>
+: <1 * <TnPr (2) (3 4 5)>>
+: <1 * <TnPr (2 3) (4 5)>>
+: <1 * <TnPr (2 3 4) (5)>>
+: <1 * <TnPr (2 3 4 5) NIL>>
+: ------------------------------------------------------------------------------
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (aprd kz1 6 (tnpr 2 '(1 2) 4 '(3 4 5 6))))
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+----------------------------------------------------------------------{CMBN 6}
+<1 * (1 2 3 4 5 6)>
+<-1 * (1 3 2 4 5 6)>
+<1 * (1 3 4 2 5 6)>
+<-1 * (1 3 4 5 2 6)>
+<1 * (1 3 4 5 6 2)>
+<1 * (3 1 2 4 5 6)>
+<-1 * (3 1 4 2 5 6)>
+<1 * (3 1 4 5 2 6)>
+<-1 * (3 1 4 5 6 2)>
+<1 * (3 4 1 2 5 6)>
+... ...
+------------------------------------------------------------------------------
+#+end_example
+
+The printed results are the printed representation of /combinations/, i.e.,
+integer linear combinations of generators resulting from the application of the
+morphisms. The degree of the combination is indicated by the information:
+=CMBN n=. 
+
+In the same way, we may create the Abelian simplicial groups
+$\mathrm{K}(\mathbb{Z}/2\mathbb{Z}, n)$:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar k-z2-2)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf k-z2-2 (k-z2 2))
+#+END_SRC
+
+#+RESULTS:
+: [K64 Abelian-Simplicial-Group]
+
+#+BEGIN_SRC lisp :exports both :results output
+(homology k-z2-2 4)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 2.
+
+
+;; Clock -> 2017-12-27, 14h 34m 52s.
+Computing the boundary of the generator 1/2 (dimension 4) :
+<<Abar[4 3]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 34m 52s.
+Computing the boundary of the generator 2/2 (dimension 4) :
+<<Abar[2 1][2 1]>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 3.
+
+
+;; Clock -> 2017-12-27, 14h 34m 52s.
+Computing the boundary of the generator 1/3 (dimension 5) :
+<<Abar[5 4]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 34m 52s.
+Computing the boundary of the generator 2/3 (dimension 5) :
+<<Abar[2 1][3 2]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 34m 52s.
+Computing the boundary of the generator 3/3 (dimension 5) :
+<<Abar[3 2][2 1]>> 
+End of computing.
+
+
+
+
+Homology in dimension 4 :
+
+
+Component Z/4Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 34m 52s.
+
+#+end_example
+
+Let us play now with the sphere $S^3$ and its loop spaces. $S^3$ and
+$\Omega^2 S^3$ are created by respective calls to the functions =sphere= and
+=loop-space=. Then we compute the $H_4$ and $H_5$ of $\Omega^2 S^3$:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar s3)
+#+END_SRC
+
+#+BEGIN_SRC lisp :ession cat :exports both
+(setf s3 (sphere 3))
+#+END_SRC
+
+#+RESULTS:
+: [K174 Simplicial-Set]
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar o2s3)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf o2s3 (loop-space s3 2))
+#+END_SRC
+
+#+RESULTS:
+: [K191 Simplicial-Group]
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(homology o2s3 4 6)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 3.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 1/3 (dimension 4) :
+<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 2/3 (dimension 4) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 3/3 (dimension 4) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 5.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 1/5 (dimension 5) :
+<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 2/5 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 3/5 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 4/5 (dimension 5) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 5/5 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+
+
+Homology in dimension 4 :
+
+
+Component Z/3Z
+
+Component Z/2Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 5.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 1/5 (dimension 5) :
+<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 2/5 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 3/5 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 4/5 (dimension 5) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 5/5 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 6.
+Rank of the source-module : 8.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 1/8 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][5 <<AlLp[2 S3][2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 2/8 (dimension 6) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 3/8 (dimension 6) :
+<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 4/8 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 5/8 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 6/8 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 7/8 (dimension 6) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+Computing the boundary of the generator 8/8 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+
+
+Homology in dimension 5 :
+
+
+Component Z/3Z
+
+Component Z/2Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 40m 26s.
+
+#+end_example
+
+Let us take now the first loop space $\Omega^1 S^3$
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar os3)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf os3 (loop-space s3))
+#+END_SRC
+
+#+RESULTS:
+: [K179 Simplicial-Group]
+
+In the following instruction, we locate in the symbol =L1= the canonical
+generator of $\pi_2 (\Omega^1S^3)$, that is the 2-simplex coming from the
+original sphere. In fact, the object created by the command
+$\texttt{(loop3 0 's3 1)}$ is the "word" $S3^1$ belonging to the Kan simplicial
+version $G(S^3)$ (a simplicial group) of the loop space $\Omega S^3$.
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar L1)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (setf L1 (loop3 0 's3 1)))
+#+END_SRC
+
+#+RESULTS:
+: <<Loop[S3]>>
+
+Let us consider also the 2-degeneracy of the base point of the loop space. In
+the printed result, the user will recognize the degeneracy $\eta_1\eta_0$ of
+the null loop, base point of $\Omega^1 S^3$: 
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar null-simp)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (setf null-simp (absm 3 +null-loop+)))
+#+END_SRC
+
+#+RESULTS:
+: 
+: <AbSm 1-0 <<Loop>>> 
+
+We may  build now a new space by pasting a disk $D3$ as indicated by the
+following call. It means that we "paste" to the space =os3= a 3-simplex named
+=D3=, the attaching map being described by the list of its faces in dimension
+2. The faces 0 and 2 of =D3= are pasted to =L1=, the faces 1 and 3 are
+collapsed on the base point; in this way the attaching map
+$(S^2 \rightarrow \Omega S^3)$ has degree 2.
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar dos3)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf dos3 (disk-pasting os3 3 '<D3> (list L1 null-simp L1 null-simp)))
+#+END_SRC
+
+#+RESULTS:
+: [K435 Simplicial-Set]
+
+Let us compute a few homology groups of the new space =dos3=:
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(homology dos3 2 4)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 2.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 14h 50m 23s.
+Computing the boundary of the generator 1/1 (dimension 2) :
+<<AlLp[2 S3]>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 3.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 14h 50m 23s.
+Computing the boundary of the generator 1/1 (dimension 3) :
+<D3> 
+End of computing.
+
+
+
+
+Homology in dimension 2 :
+
+
+Component Z/2Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 50m 23s.
+
+
+Computing boundary-matrix in dimension 3.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 14h 50m 23s.
+Computing the boundary of the generator 1/1 (dimension 3) :
+<D3> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 14h 50m 23s.
+Computing the boundary of the generator 1/1 (dimension 4) :
+<<AlLp[2 S3][2 S3]>> 
+End of computing.
+
+
+
+
+Homology in dimension 3 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-27, 14h 50m 23s.
+
+#+end_example
+
+But more interesting, let us build the loop space of the object =dos3= and let
+us compute the homology in dimension 5:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar odos3)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf odos3 (loop-space dos3))
+#+END_SRC
+
+#+RESULTS:
+: [K471 Simplicial-Group]
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(homology odos3 5)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 14.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 1/14 (dimension 5) :
+<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 2/14 (dimension 5) :
+<<AlLp[2 <D3>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 3/14 (dimension 5) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 4/14 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 5/14 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 6/14 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 7/14 (dimension 5) :
+<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 8/14 (dimension 5) :
+<<AlLp[2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 9/14 (dimension 5) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 10/14 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 11/14 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 12/14 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 13/14 (dimension 5) :
+<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 14/14 (dimension 5) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 6.
+Rank of the source-module : 26.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 1/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][5 <<AlLp[2 S3][2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 2/26 (dimension 6) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 3/26 (dimension 6) :
+<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 4/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 5/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 6/26 (dimension 6) :
+<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 7/26 (dimension 6) :
+<<AlLp[2 <D3>][2 <D3>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 8/26 (dimension 6) :
+<<AlLp[2 <D3>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 9/26 (dimension 6) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 10/26 (dimension 6) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 11/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 12/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 14s.
+Computing the boundary of the generator 13/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 14/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 15/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 16/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 17/26 (dimension 6) :
+<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 18/26 (dimension 6) :
+<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 19/26 (dimension 6) :
+<<AlLp[2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 20/26 (dimension 6) :
+<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 21/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 22/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 23/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 24/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 25/26 (dimension 6) :
+<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+Computing the boundary of the generator 26/26 (dimension 6) :
+<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> 
+End of computing.
+
+
+
+
+Homology in dimension 5 :
+
+
+Component Z/2Z
+
+Component Z/2Z
+
+Component Z/2Z
+
+Component Z/2Z
+
+Component Z/2Z
+
+Component Z/2Z
+
+Component Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 15h 2m 15s.
+
+#+end_example
+
+Let us continue with the Kan theory. First, we check that $S^3$ is not of type
+Kan and that $\Omega S^3$ is indeed of type Kan and a non-Abelian simplicial
+group.
+
+#+BEGIN_SRC lisp :session cat :exports both
+(typep s3 'kan)
+#+END_SRC
+
+#+RESULTS:
+: NIL
+
+#+BEGIN_SRC lisp :session cat :exports both
+(typep os3 'kan)
+#+END_SRC
+
+#+RESULTS:
+: T
+
+#+BEGIN_SRC lisp :session cat :exports both
+(typep os3 'simplicial-group)
+#+END_SRC
+
+#+RESULTS:
+: T
+
+#+BEGIN_SRC lisp :ession cat exports both
+(typep os3 'ab-simplicial-group)
+#+END_SRC
+
+#+RESULTS:
+: NIL
+
+Let us create the word $L2=(S3)^2$, i.e an object belonging to $\Omega S^3$ and
+let us apply the product of the underlying algebra upon $L2 \otimes L2$:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar L2)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (setf L2 (loop3 0 's3 2)))
+#+END_SRC
+
+#+RESULTS:
+: <<Loop[S3\2]>>
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar square)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (setf square (aprd os3 4 (tnpr 2 L2 2 L2))))
+#+END_SRC
+
+#+RESULTS:
+: 
+: ----------------------------------------------------------------------{CMBN 4}
+: <1 * <<Loop[1-0 S3\2][3-2 S3\2]>>>
+: <-1 * <<Loop[2-0 S3\2][3-1 S3\2]>>>
+: <1 * <<Loop[2-1 S3\2][3-0 S3\2]>>>
+: <1 * <<Loop[3-0 S3\2][2-1 S3\2]>>>
+: <-1 * <<Loop[3-1 S3\2][2-0 S3\2]>>>
+: <1 * <<Loop[3-2 S3\2][1-0 S3\2]>>>
+: ------------------------------------------------------------------------------
+
+We see that the result is a linear combination of words composed from
+degeneracies of =L2=. The following instruction selects the generator part of
+the second element of the previous combination.
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar L4)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (setf L4 (gnrt (second (cmbn-list square)))))
+#+END_SRC
+
+#+RESULTS:
+: 
+: <<Loop[2-0 S3\2][3-1 S3\2]>> 
+
+Let us use the lisp function =mapcar= (one among the various iteration
+functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object
+=L4=, this list is a "Kan horn".
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar horn)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (setf horn (mapcar #'(lambda (i) (face os3 i 4 l4)) '(1 2 3 4))))
+#+END_SRC
+
+#+RESULTS:
+: 
+: (<AbSm - <<Loop[1 S3\2][2 S3\2]>>> <AbSm - <<Loop[0 S3\2][2 S3\2]>>> <AbSm - <<Loop[0 S3\2][1 S3\2]>>> <AbSm 1 <<Loop[S3\2]>>>) 
+
+The function =kfll= tries to find a filling of this "Kan horn", and we see that
+the face 2 of the resulting simplex (which is very different from =L4=) is the
+same as the face 2 of =L4=.
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar kan-simplex)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (setf kan-simplex (kfll os3 0 4 horn)))
+#+END_SRC
+
+#+RESULTS:
+: 
+: <AbSm - <<Loop[3-1 S3\2][2-1 S3\-2][2-0 S3\2][1-0 S3\-2][2-1 S3\2][3-1 S3\-2][1-0 S3\2][3-1 S3\2][3-0 S3\-2][1-0 S3\-2][3-0 S3\2][2-0 S3\-2][1-0 S3\2][3-0 S3\-2][2-0 S3\2][3-0 S3\2]>>> 
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (face os3 2 4 kan-simplex))
+#+END_SRC
+
+#+RESULTS:
+: 
+: <AbSm - <<Loop[0 S3\2][2 S3\2]>>> 
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(princ (second horn))
+#+END_SRC
+
+#+RESULTS:
+: 
+: <AbSm - <<Loop[0 S3\2][2 S3\2]>>> 
+
+Let ${\cal G}$ be a simplicial group 0-reduced. $\Omega S^3$ is such a group.
+The program =Kenzo= allows the construction of the universal bundle
+$\mathcal{WG}$ and in particular of its base space
+$\overline{\mathcal{W}} \mathcal{G}$, i.e. the classifying space of
+${\cal G}$. In our case, as $\Omega S^3$ in non-Abelian, the result is not a
+simplicial group but only a simplicial set. We verify that the $H_4$, which in
+principle should be isomorphic to $H_4 S^3 = 0$, is really null.
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar cls-os3)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf cls-os3 (classifying-space os3))
+#+END_SRC
+
+#+RESULTS:
+: [K938 Simplicial-Set]
+
+#+BEGIN_SRC lisp :session cat :exports both
+(typep cls-os3 'simplicial-group)
+#+END_SRC
+
+#+RESULTS:
+: NIL
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(homology cls-os3 4)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 0.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 1.
+
+
+;; Clock -> 2017-12-27, 15h 17m 56s.
+Computing the boundary of the generator 1/1 (dimension 5) :
+<<Abar[5 <<AlLp[2 S3][2 S3]>>]>> 
+End of computing.
+
+
+
+
+Homology in dimension 4 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-27, 15h 17m 56s.
+
+#+end_example
+
+Let us end this short overview with an example of computation of homotopy
+groups. The method used in =Kenzo= is the Whitehead tower. An extended version
+of the Kenzo program due to Ana Romero is more general, but in this present
+version only the case where the first non-null homology group (in non-null
+dimension) is $\mathbb{Z}$ or $\mathbb{Z}/{2\mathbb{Z}}$ can be processed;
+however if this homology group is a direct sum of several copies of
+$\mathbb{Z}$ or $\mathbb{Z}/{2 \mathbb{Z}}$, then the corresponding stage of
+the Whitehead tower may also be constructed step by step.
+
+We take again $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}, 3)$ whose $H_3$ is
+$\mathbb{Z}/2\mathbb{Z}$. First the fundamental cohomology class is
+constructed:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar ch3)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf ch3 (chml-clss m23 3))
+#+END_SRC
+
+#+RESULTS:
+: [K1230 Cohomology-Class on K1 of degree 3]
+
+Then the function =z2-whitehead= is called to build a fibration over the
+simplicial set =m23= canonically associated to the cohomology class =ch3=.
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar f3)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf f3 (z2-whitehead m23 ch3))
+#+END_SRC
+
+#+RESULTS:
+: [K1231 Fibration K1 -> K64]
+
+Then the total space of the fibration is built:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar x4)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf x4 (fibration-total f3))
+#+END_SRC
+
+#+RESULTS:
+: [K1237 Simplicial-Set]
+
+The $H_4$ of this total space is the $\pi_4$ of
+$\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:
+
+#+BEGIN_SRC lisp :session cat :exports both :results output
+(homology x4 3 5)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 3.
+Rank of the source-module : 2.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 1/2 (dimension 3) :
+<TnPr * <<Abar[3 2]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 2/2 (dimension 3) :
+<TnPr M3 <<Abar>>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 3.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 1/3 (dimension 4) :
+<TnPr * <<Abar[4 3]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 2/3 (dimension 4) :
+<TnPr * <<Abar[2 1][2 1]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 3/3 (dimension 4) :
+<TnPr N4 <<Abar>>> 
+End of computing.
+
+
+
+
+Homology in dimension 3 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 3.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 1/3 (dimension 4) :
+<TnPr * <<Abar[4 3]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 2/3 (dimension 4) :
+<TnPr * <<Abar[2 1][2 1]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 3/3 (dimension 4) :
+<TnPr N4 <<Abar>>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 4.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 1/4 (dimension 5) :
+<TnPr * <<Abar[5 4]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 2/4 (dimension 5) :
+<TnPr * <<Abar[2 1][3 2]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 3/4 (dimension 5) :
+<TnPr * <<Abar[3 2][2 1]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+Computing the boundary of the generator 4/4 (dimension 5) :
+<TnPr M3 <<Abar[2 1]>>> 
+End of computing.
+
+
+
+
+Homology in dimension 4 :
+
+
+Component Z/2Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 15h 22m 48s.
+
+#+end_example
+
+We may now iterate the process, to compute the $\pi_5$ of
+$\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar ch4)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf ch4 (chml-clss x4 4))
+#+END_SRC
+
+#+RESULTS:
+: [K1318 Cohomology-Class on K1303 of degree 4]
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar f4)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session cat :exports both
+(setf f4 (z2-whitehead x4 ch4))
+#+END_SRC
+
+#+RESULTS:
+: [K1333 Fibration K1237 -> K1319]
+
+#+BEGIN_SRC lisp :session cat :exports none
+(defvar x5)
+#+END_SRC
+
+#+BEGIN_SRC lisp :session lisp :exports both
+(setf x5 (fibration-total f4))
+#+END_SRC
+
+#+RESULTS:
+: [K1339 Simplicial-Set]
+
+#+BEGIN_SRC lisp :exports both :results output
+(homology x5 4 6)
+#+END_SRC
+
+#+RESULTS:
+#+begin_example
+
+Computing boundary-matrix in dimension 4.
+Rank of the source-module : 4.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 1/4 (dimension 4) :
+<TnPr <TnPr * <<Abar>>> <<Abar[4 <<Abar[3 2]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 2/4 (dimension 4) :
+<TnPr <TnPr * <<Abar[4 3]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 3/4 (dimension 4) :
+<TnPr <TnPr * <<Abar[2 1][2 1]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 4/4 (dimension 4) :
+<TnPr <TnPr N4 <<Abar>>> <<Abar>>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 7.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 1/7 (dimension 5) :
+<TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[4 3]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 2/7 (dimension 5) :
+<TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 3/7 (dimension 5) :
+<TnPr <TnPr * <<Abar[2 1]>>> <<Abar[3 <<Abar[2 1]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 4/7 (dimension 5) :
+<TnPr <TnPr * <<Abar[5 4]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 5/7 (dimension 5) :
+<TnPr <TnPr * <<Abar[2 1][3 2]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 6/7 (dimension 5) :
+<TnPr <TnPr * <<Abar[3 2][2 1]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 7/7 (dimension 5) :
+<TnPr <TnPr M3 <<Abar[2 1]>>> <<Abar>>> 
+End of computing.
+
+
+
+
+Homology in dimension 4 :
+
+
+
+---done---
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+
+
+Computing boundary-matrix in dimension 5.
+Rank of the source-module : 7.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 1/7 (dimension 5) :
+<TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[4 3]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 2/7 (dimension 5) :
+<TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 3/7 (dimension 5) :
+<TnPr <TnPr * <<Abar[2 1]>>> <<Abar[3 <<Abar[2 1]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 4/7 (dimension 5) :
+<TnPr <TnPr * <<Abar[5 4]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 5/7 (dimension 5) :
+<TnPr <TnPr * <<Abar[2 1][3 2]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 6/7 (dimension 5) :
+<TnPr <TnPr * <<Abar[3 2][2 1]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 7/7 (dimension 5) :
+<TnPr <TnPr M3 <<Abar[2 1]>>> <<Abar>>> 
+End of computing.
+
+
+Computing boundary-matrix in dimension 6.
+Rank of the source-module : 14.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 1/14 (dimension 6) :
+<TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[5 4]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 2/14 (dimension 6) :
+<TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[2 1][3 2]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 3/14 (dimension 6) :
+<TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[3 2][2 1]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 4/14 (dimension 6) :
+<TnPr <TnPr * <<Abar>>> <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 5/14 (dimension 6) :
+<TnPr <TnPr * <<Abar[2 1]>>> <<Abar[4 <<Abar[3 2]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 6/14 (dimension 6) :
+<TnPr <TnPr * <<Abar[3 2]>>> <<Abar[3 <<Abar[2 1]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 7/14 (dimension 6) :
+<TnPr <TnPr M3 <<Abar>>> <<Abar[3 <<Abar[2 1]>>]>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 8/14 (dimension 6) :
+<TnPr <TnPr * <<Abar[6 5]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 9/14 (dimension 6) :
+<TnPr <TnPr * <<Abar[2 1][4 3]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 10/14 (dimension 6) :
+<TnPr <TnPr * <<Abar[3 2][3 2]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 11/14 (dimension 6) :
+<TnPr <TnPr * <<Abar[4 3][2 1]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 12/14 (dimension 6) :
+<TnPr <TnPr * <<Abar[2 1][2 1][2 1]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 13/14 (dimension 6) :
+<TnPr <TnPr M3 <<Abar[3 2]>>> <<Abar>>> 
+End of computing.
+
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+Computing the boundary of the generator 14/14 (dimension 6) :
+<TnPr <TnPr N4 <<Abar[2 1]>>> <<Abar>>> 
+End of computing.
+
+
+
+
+Homology in dimension 5 :
+
+
+Component Z/4Z
+
+
+---done---
+
+;; Clock -> 2017-12-27, 15h 25m 8s.
+
+#+end_example
+
+So $\pi_5(\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}))$ is $\mathbb{Z}/4\mathbb{Z}$.
diff --git a/doc/org/Foreword.org b/doc/org/Foreword.org
new file mode 100644
index 0000000..31a4d07
--- /dev/null
+++ b/doc/org/Foreword.org
@@ -0,0 +1,48 @@
+\begin{center}
+\line(1,0){450}
+\end{center}
+
+The *Kenzo* program implements the general ideas of the second author about
+/Effective Homology/[fn:1], mainly around the Serre and Eilenberg-Moore
+spectral sequences. The first author (re-)discovered the importance of the
+/Basic Perturbation Lemma/ in these questions, already noted by Victor
+Guggenheim[fn:2] and this program directly implements and directly uses this
+"lemma" which should be called the /Fundamental Theorem of Algebraic Topology/.
+The first version of the program, called *EAT*, was written in 1989-90 by the
+first and the second authors. It has been demonstrated in several universities:
+France: Grenoble and Montpellier, Belgium: Louvain-la-Neuve, Italy: Genoa and
+Pisa, Sweden: Stockholm, Japan: Sapporo, Morioka, Urawa, Tokyo, Kyoto, Nara,
+Osaka and Hiroshima.
+
+In this *Kenzo* version, numerous improvements have been integrated in
+comparison with *EAT*.
+
+1. Standard CLOS (Common Lisp Object System) techniques.
+2. Use of Zermelo theorem about ordered sets (!): any set can be well ordered
+   and this remark gives important new ideas to improve execution speed.
+3. Better memory management.
+4. Use of Szczarba's formulas instead of Shih's for implementing the twisted
+   Eilenberg-Zilber theorem.
+5. New mathematical objects:
+   - Serre spectral sequences.
+   - Differential algebras.
+   - Simplicial morphisms.
+   - Kan simplicial sets.
+   - Simplicial groups.
+   - Fibrations.
+   - Classifying spaces.
+   In particular, using these tools, the first homotopy groups of /arbitrary/
+   simplicial sets with effective homology, are now reachable.
+
+The present documentation is a joint work between the second and third authors.
+
+\begin{center}
+\line(1,0){450}
+\end{center}
+
+[fn:1] *Francis Sergeraert.* /The computability problem in algebraic topology./
+Advances in Mathematics, 1994, vol. 104, pp 1-29.
+
+[fn:2] *V.K.A.M. Guggenheim.* /On a perturbation theory for the homology of the
+loop space./ Journal of Pure and Applied Algebra, 1982, vol. 25, pp 197-205.
+
diff --git a/doc/org/Handbook.org b/doc/org/Handbook.org
new file mode 100644
index 0000000..9420d1c
--- /dev/null
+++ b/doc/org/Handbook.org
@@ -0,0 +1,11 @@
+#+TITLE: Kenzo - a Symbolic Software for Effective Homology Computation
+#+AUTHOR: Julio Rubio Garcia, Francis Sergeraert, Yvon Siret
+#+DATE: April 26, 1999
+
+#+BEGIN_SRC lisp :session cat :exports none :results silent
+(ql:quickload "kenzo")
+(use-package :cat)
+#+END_SRC
+
+#+INCLUDE: "Foreword.org"
+#+INCLUDE: "00_Overview.org"