Allegro CL Personal Edition 5.0.1 [Windows/x86] (6/29/99 16:56)
Copyright (C) 1985-1999, Franz Inc., Berkeley, CA, USA.  All Rights Reserved.
CG/IDE Version: 1.323.2.169

; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\classes.fasl
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---done---
>
---------- Comment -------------------------------
;; Let us begin by the minimum information
;;   which is necessary to attend
;;   a Lisp demonstration.
--------------------------------------------------
>
---------- Comment -------------------------------
;; A Lisp statement is always in prefixed notation,
;;   between parentheses.
--------------------------------------------------
>
---------- Statement -----------------------------
(+ 3 5)
--------------------------------------------------
>
---------- Result --------------------------------
8
--------------------------------------------------
>
---------- Comment -------------------------------
;; The Lisp assignment operator is SETF:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF X 3)
--------------------------------------------------
>
---------- Result --------------------------------
3
--------------------------------------------------
>
---------- Comment -------------------------------
;; Now the value of X is 3:
--------------------------------------------------
>
---------- Statement -----------------------------
X
--------------------------------------------------
>
---------- Result --------------------------------
3
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF Y (+ X 5))
--------------------------------------------------
>
---------- Result --------------------------------
8
--------------------------------------------------
>
---------- Comment -------------------------------
;; So that now the value of y is 8:
--------------------------------------------------
>
---------- Statement -----------------------------
Y
--------------------------------------------------
>
---------- Result --------------------------------
8
--------------------------------------------------
>
---------- Comment -------------------------------
;; And so on...
;;
;; Because of the short time for the demonstration,
;;   we go directly to the most significant parts
;;   of the Kenzo program.
--------------------------------------------------
>
---------- Comment -------------------------------
;; Firstly, let us see an example with the Serre spectral sequence.
;;   We construct the 2-sphere, assigned to the symbol S2.
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF S2 (SPHERE 2))
--------------------------------------------------
>
---------- Result --------------------------------
[K1 Simplicial-Set]
--------------------------------------------------
>
---------- Comment -------------------------------
;; The result is the Kenzo object #1,
;;   namely, a simplicial set.
--------------------------------------------------
>
---------- Comment -------------------------------
;; We intend to construct the unique non trivial fibration
;;   over S^2 where the fibre is the
;;   INFINITE real projective space,
;;     that is, the Eilenberg-MacLane space: K(Z_2,1).
;;
;;   Let us construct the associated CoHoMoLogy CLaSS,
;;     assigned to the symbol CH2:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF CH2 (CHML-CLSS S2 2))
--------------------------------------------------
>
---------- Result --------------------------------
[K12 Cohomology-Class on K1 of degree 2]
--------------------------------------------------
>
---------- Comment -------------------------------
;; and then the associated fibration with respect to
;;   the group Z_2.
;; The constructor is called WHITEHEAD,
;;   for this is in particular the process
;;   leading to the construction of the so-called
;;   Whitehead tower, the key point when
;;   computing homotopy groups.
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF FIBRATION (Z2-WHITEHEAD S2 CH2))
--------------------------------------------------
>
---------- Result --------------------------------
[K25 Fibration K1 -> K13]
--------------------------------------------------
>
---------- Comment -------------------------------
;; More precisely the twisting Kan-operator defining
;;   the simplicial fibration has been constructed.
;;
;; Then the total space of this fibration is constructed:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF TOTAL (FIBRATION-TOTAL FIBRATION))
--------------------------------------------------
>
---------- Result --------------------------------
[K31 Simplicial-Set]
--------------------------------------------------
>
---------- Comment -------------------------------
;; Our EFFECTIVE version of the Serre spectral sequence
;;   has been "secretely" applied, so that
;;   the homology groups of the total space are available.
;; For example the H_5:
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY TOTAL 5)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 5.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 7m 33s.
Computing the boundary of the generator 1 (dimension 5) :
<TnPr * 5>
End of computing.

;; Clock -> 2000-08-30, 16h 7m 33s.
Computing the boundary of the generator 2 (dimension 5) :
<TnPr S2 3>
End of computing.

Computing boundary-matrix in dimension 6.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 7m 33s.
Computing the boundary of the generator 1 (dimension 6) :
<TnPr * 6>
End of computing.

;; Clock -> 2000-08-30, 16h 7m 33s.
Computing the boundary of the generator 2 (dimension 6) :
<TnPr S2 4>
End of computing.


Homology in dimension 5 :

Component Z/4Z

---done---
;; Clock -> 2000-08-30, 16h 7m 33s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; The H_5 group is Z/4Z.
;;
;; And the H_6?
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY TOTAL 6)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 6.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 7m 37s.
Computing the boundary of the generator 1 (dimension 6) :
<TnPr * 6>
End of computing.

;; Clock -> 2000-08-30, 16h 7m 37s.
Computing the boundary of the generator 2 (dimension 6) :
<TnPr S2 4>
End of computing.

Computing boundary-matrix in dimension 7.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 7m 37s.
Computing the boundary of the generator 1 (dimension 7) :
<TnPr * 7>
End of computing.

;; Clock -> 2000-08-30, 16h 7m 37s.
Computing the boundary of the generator 2 (dimension 7) :
<TnPr S2 5>
End of computing.


Homology in dimension 6 :

---done---
;; Clock -> 2000-08-30, 16h 7m 37s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; The H_6 is null.
;;
;; Processing in the same way the other small dimensions,
;;   you obtain the homology groups:
;;
;;     H_0 = Z,   H_1 = 0,   H_2 = Z,
;;
;;     H_3 = H_5 = H_7 = ... = Z/4Z,
;;
;;     H_4 = H_6 = H_8 = ... = 0.
;;
;;  If you look at the ordinary Serre spectral sequence,
;;    you obtain at the abutment the extension problem:
;;  0 -> Z_2 -> H_{2n-1} -> Z_2 -> 0 which is
;;    an interesting student exercice to be solved.
--------------------------------------------------
>
---------- Comment -------------------------------
;; A little more difficult.
;;
;;   Let us use again our total space as the
;;   BASE SPACE of a NEW analogous fibration.
;;
;;   What about the homology groups of the NEW total space?
;;
;;   The necessary cohomology-class:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF CH2-2 (CHML-CLSS TOTAL 2))
--------------------------------------------------
>
---------- Result --------------------------------
[K87 Cohomology-Class on K60 of degree 2]
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF FIBRATION-2 (Z2-WHITEHEAD TOTAL CH2-2))
--------------------------------------------------
>
---------- Result --------------------------------
[K90 Fibration K31 -> K13]
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF TOTAL-2 (FIBRATION-TOTAL FIBRATION-2))
--------------------------------------------------
>
---------- Result --------------------------------
[K96 Simplicial-Set]
--------------------------------------------------
>
---------- Comment -------------------------------
;; The H_4 group:
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY TOTAL-2 4)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 4.
Rank of the source-module : 8.

;; Clock -> 2000-08-30, 16h 8m 0s.
Computing the boundary of the generator 1 (dimension 4) :
<TnPr <TnPr * 0> 4>
End of computing.
[... ... Lines deleted ... ...]
;; Clock -> 2000-08-30, 16h 8m 0s.
Computing the boundary of the generator 8 (dimension 4) :
<TnPr <TnPr S2 2> 0>
End of computing.

Computing boundary-matrix in dimension 5.
Rank of the source-module : 10.

;; Clock -> 2000-08-30, 16h 8m 1s.
Computing the boundary of the generator 1 (dimension 5) :
<TnPr <TnPr * 0> 5>
End of computing.
[... ... Lines deleted ... ...]
;; Clock -> 2000-08-30, 16h 8m 1s.
Computing the boundary of the generator 10 (dimension 5) :
<TnPr <TnPr S2 3> 0>
End of computing.


Homology in dimension 4 :

---done---
;; Clock -> 2000-08-30, 16h 8m 2s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; You see H_4 = 0.
;;
;; The H_5 group:
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY TOTAL-2 5)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 5.
Rank of the source-module : 10.

;; Clock -> 2000-08-30, 16h 8m 9s.
Computing the boundary of the generator 1 (dimension 5) :
<TnPr <TnPr * 0> 5>
End of computing.
[... ... Lines deleted ... ...]
;; Clock -> 2000-08-30, 16h 8m 9s.
Computing the boundary of the generator 10 (dimension 5) :
<TnPr <TnPr S2 3> 0>
End of computing.

Computing boundary-matrix in dimension 6.
Rank of the source-module : 12.

;; Clock -> 2000-08-30, 16h 8m 9s.
Computing the boundary of the generator 1 (dimension 6) :
<TnPr <TnPr * 0> 6>
End of computing.
[... ... Lines deleted ... ...]
;; Clock -> 2000-08-30, 16h 8m 11s.
Computing the boundary of the generator 12 (dimension 6) :
<TnPr <TnPr S2 4> 0>
End of computing.


Homology in dimension 5 :

Component Z/16Z

---done---
;; Clock -> 2000-08-30, 16h 8m 12s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; In the same way, you can obtain the EXPERIMENTAL results:
;;
;;     H_0 = Z,   H_1 = 0,   H_2 = Z,
;;
;;     H_3 = H_5 = H_7 = Z/16Z,
;;
;;     H_4 = H_6 = H_8 = 0.
;;
;; Try to prove it. Not easy!
;;
;; Only the good "students" will succeed.
;;
;; Hint: the iterated fibration is "associative", so that...
--------------------------------------------------
>
---------- Comment -------------------------------
;; We show a result
;;   which can be obtained now
;;   ONLY through this machine program.
--------------------------------------------------
>
---------- Comment -------------------------------
;; We start with the infinite real projective space
;;   truncated in dimension 4, that is, the canonical quotient:
;;
;;   P^\infty(R) / P^3(R).
;;
;; It is constructed as follows:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF P4 (R-PROJ-SPACE 4))
--------------------------------------------------
>
---------- Result --------------------------------
[K176 Simplicial-Set]
--------------------------------------------------
>
---------- Comment -------------------------------
;; Its homology begins in dimension 4 with Z:
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY P4 4)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 4.
Rank of the source-module : 1.

;; Clock -> 2000-08-30, 16h 8m 33s.
Computing the boundary of the generator 1 (dimension 4) :
4
End of computing.

Computing boundary-matrix in dimension 5.
Rank of the source-module : 1.

;; Clock -> 2000-08-30, 16h 8m 33s.
Computing the boundary of the generator 1 (dimension 5) :
5
End of computing.


Homology in dimension 4 :

Component Z

---done---
;; Clock -> 2000-08-30, 16h 8m 33s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; The second loop space is constructed:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF OOP4 (LOOP-SPACE P4 2))
--------------------------------------------------
>
---------- Result --------------------------------
[K197 Simplicial-Group]
--------------------------------------------------
>
---------- Comment -------------------------------
;; You see the second loop space is a simplicial GROUP,
;;   namely the Kan model of it, a "highly" infinite
;;   simplicial set.
;;
;; According to the classical desuspension result,
;;   the homology of OOP4 begins in dimension 2 by Z:
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY OOP4 2)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 2.
Rank of the source-module : 1.

;; Clock -> 2000-08-30, 16h 8m 39s.
Computing the boundary of the generator 1 (dimension 2) :
<<AlLp[2 <<AlLp[3 4]>>]>>
End of computing.

Computing boundary-matrix in dimension 3.
Rank of the source-module : 1.

;; Clock -> 2000-08-30, 16h 8m 39s.
Computing the boundary of the generator 1 (dimension 3) :
<<AlLp[3 <<AlLp[4 5]>>]>>
End of computing.


Homology in dimension 2 :

Component Z

---done---
;; Clock -> 2000-08-30, 16h 8m 39s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; And the \pi_2 is also Z,
;;
;;   so that attaching a 3-cell to OOP4
;;           by a map of degree 2
;;   makes sense.
;;
;; This can be done as follows:
;;   (we don't have enough time to explain the details,
;;       but they are easy)
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF DOOP4
      (DISK-PASTING OOP4 3 'NEW
                    (LIST
                     (LOOP3 0 (LOOP3 0 4 1) 2)
                     (LOOP3)
                     (LOOP3)
                     (LOOP3))))
--------------------------------------------------
>
---------- Result --------------------------------
[K437 Simplicial-Set]
--------------------------------------------------
>
---------- Comment -------------------------------
;; We verify the H_2 group:
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY DOOP4 2)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 2.
Rank of the source-module : 1.

;; Clock -> 2000-08-30, 16h 8m 52s.
Computing the boundary of the generator 1 (dimension 2) :
<<AlLp[2 <<AlLp[3 4]>>]>>
End of computing.

Computing boundary-matrix in dimension 3.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 8m 53s.
Computing the boundary of the generator 1 (dimension 3) :
NEW
End of computing.

;; Clock -> 2000-08-30, 16h 8m 53s.
Computing the boundary of the generator 2 (dimension 3) :
<<AlLp[3 <<AlLp[4 5]>>]>>
End of computing.


Homology in dimension 2 :

Component Z/2Z

---done---
;; Clock -> 2000-08-30, 16h 8m 53s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; We take now the loop-space of the last space:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF ODOOP4 (LOOP-SPACE DOOP4))
--------------------------------------------------
>
---------- Result --------------------------------
[K455 Simplicial-Group]
--------------------------------------------------
>
---------- Comment -------------------------------
;; Now this machine program is the only object, human or not,
;;   able to compute the H_4 of this space.
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY ODOOP4 4)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 4.
Rank of the source-module : 19.

;; Clock -> 2000-08-30, 16h 9m 1s.
Computing the boundary of the generator 1 (dimension 4) :
<<AlLp[4 <<AlLp[5 <<AlLp[6 7]>>]>>]>>
End of computing.
[... ... Lines deleted ... ...]
;; Clock -> 2000-08-30, 16h 9m 4s.
Computing the boundary of the generator 19 (dimension 4) :
<<AlLp[1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>]>>
End of computing.

Computing boundary-matrix in dimension 5.
Rank of the source-module : 50.

;; Clock -> 2000-08-30, 16h 9m 5s.
Computing the boundary of the generator 1 (dimension 5) :
<<AlLp[5 <<AlLp[6 <<AlLp[7 8]>>]>>]>>
End of computing.
[... ... Lines deleted ... ...]
;; Clock -> 2000-08-30, 16h 9m 41s.
Computing the boundary of the generator 50 (dimension 5) :
<<AlLp[1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>]>>
End of computing.


Homology in dimension 4 :

Component Z/2Z
Component Z/2Z
Component Z/2Z
Component Z/2Z
Component Z/2Z
Component Z/2Z
Component Z/2Z
Component Z/2Z
Component Z

---done---
;; Clock -> 2000-08-30, 16h 9m 42s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; The H_4 of ODOOP4 is therefore Z + (Z/2Z)^8
--------------------------------------------------
>
---------- Comment -------------------------------
;; The simplest example of
;;   a significant computation of homotopy group.
;;
;; What about the homotopy groups of
;;   P2 = P^\infty(R) / P^1(R) ??
;;
;; Construction of the space P2:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF P2 (R-PROJ-SPACE 2))
--------------------------------------------------
>
---------- Result --------------------------------
[K598 Simplicial-Set]
--------------------------------------------------
>
---------- Comment -------------------------------
;; The homology and the homotopy begins
;;   with Z in dimension 2:
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY P2 2)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 2.
Rank of the source-module : 1.

;; Clock -> 2000-08-30, 16h 9m 58s.
Computing the boundary of the generator 1 (dimension 2) :
2
End of computing.

Computing boundary-matrix in dimension 3.
Rank of the source-module : 1.

;; Clock -> 2000-08-30, 16h 9m 58s.
Computing the boundary of the generator 1 (dimension 3) :
3
End of computing.


Homology in dimension 2 :

Component Z

---done---
;; Clock -> 2000-08-30, 16h 9m 58s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; The fundamental cohomology class:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF CH2 (CHML-CLSS P2 2))
--------------------------------------------------
>
---------- Result --------------------------------
[K607 Cohomology-Class on K598 of degree 2]
--------------------------------------------------
>
---------- Comment -------------------------------
;; The associated fibration. Fibre=S^1.
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF F2 (Z-WHITEHEAD P2 CH2))
--------------------------------------------------
>
---------- Result --------------------------------
[K620 Fibration K598 -> K608]
--------------------------------------------------
>
---------- Comment -------------------------------
;; The total space:
--------------------------------------------------
>
---------- Statement -----------------------------
(SETF TOTAL (FIBRATION-TOTAL F2))
--------------------------------------------------
>
---------- Result --------------------------------
[K626 Simplicial-Set]
--------------------------------------------------
>
---------- Comment -------------------------------
;; The total space is the first stage of
;;   the Whitehead tower, so that
;;   \pi_3(P2) = H_3(total):
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY TOTAL 3)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 3.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 10m 15s.
Computing the boundary of the generator 1 (dimension 3) :
<TnPr 2 S1>
End of computing.

;; Clock -> 2000-08-30, 16h 10m 15s.
Computing the boundary of the generator 2 (dimension 3) :
<TnPr 3 *>
End of computing.

Computing boundary-matrix in dimension 4.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 10m 15s.
Computing the boundary of the generator 1 (dimension 4) :
<TnPr 3 S1>
End of computing.

;; Clock -> 2000-08-30, 16h 10m 15s.
Computing the boundary of the generator 2 (dimension 4) :
<TnPr 4 *>
End of computing.


Homology in dimension 3 :

Component Z

---done---
;; Clock -> 2000-08-30, 16h 10m 15s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; So that \pi_3(P2) = Z.
;;
;; Let us compute H_4(total) and H_5(total):
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY TOTAL 4)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 4.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 10m 19s.
Computing the boundary of the generator 1 (dimension 4) :
<TnPr 3 S1>
End of computing.

;; Clock -> 2000-08-30, 16h 10m 19s.
Computing the boundary of the generator 2 (dimension 4) :
<TnPr 4 *>
End of computing.

Computing boundary-matrix in dimension 5.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 10m 19s.
Computing the boundary of the generator 1 (dimension 5) :
<TnPr 4 S1>
End of computing.

;; Clock -> 2000-08-30, 16h 10m 19s.
Computing the boundary of the generator 2 (dimension 5) :
<TnPr 5 *>
End of computing.


Homology in dimension 4 :

---done---
;; Clock -> 2000-08-30, 16h 10m 20s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; H_4(total) = 0.
--------------------------------------------------
>
---------- Statement -----------------------------
(HOMOLOGY TOTAL 5)
--------------------------------------------------
>
---------- Result --------------------------------

Computing boundary-matrix in dimension 5.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 10m 22s.
Computing the boundary of the generator 1 (dimension 5) :
<TnPr 4 S1>
End of computing.

;; Clock -> 2000-08-30, 16h 10m 22s.
Computing the boundary of the generator 2 (dimension 5) :
<TnPr 5 *>
End of computing.

Computing boundary-matrix in dimension 6.
Rank of the source-module : 2.

;; Clock -> 2000-08-30, 16h 10m 23s.
Computing the boundary of the generator 1 (dimension 6) :
<TnPr 5 S1>
End of computing.

;; Clock -> 2000-08-30, 16h 10m 23s.
Computing the boundary of the generator 2 (dimension 6) :
<TnPr 6 *>
End of computing.


Homology in dimension 5 :

---done---
;; Clock -> 2000-08-30, 16h 10m 23s.
NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; You can continue and all the following H_i are null.
;;
;; Again you have experimentally discovered something:
;; The total space is in fact a version of the 3-sphere.
;;   A strange fibration:
;;
;;   S^1 -> S^3 -> P\^infty(R)/P^1(R)
;;
;; can be defined; it is not hard to prove
;;   it CONTAINS the Hopf fibration.
;;
;; Exercise: try to define it directly.
--------------------------------------------------
>
---------- Comment -------------------------------
;; Finally we outline the general organization of the program.
--------------------------------------------------
>
---------- Comment -------------------------------
;; When an object is constructed,
;;   if, given the data, this is possible,
;;   the constructed object is
;;   an OBJECT WITH EFFECTIVE HOMOLOGY.
--------------------------------------------------
>
---------- Comment -------------------------------
;; Let us consider for example the space
;;   constructed by attaching a 3-disk:
--------------------------------------------------
>
---------- Statement -----------------------------
DOOP4
--------------------------------------------------
>
---------- Result --------------------------------
[K437 Simplicial-Set]
--------------------------------------------------
>
---------- Comment -------------------------------
;; It is a simplicial set with effective homology.
;;
;; This means an Effective CHain CoMplex (= ECHCM) is available.
--------------------------------------------------
>
---------- Statement -----------------------------
(ECHCM DOOP4)
--------------------------------------------------
>
---------- Result --------------------------------
[K444 Chain-Complex]
--------------------------------------------------
>
---------- Comment -------------------------------
;; Effective means of finite type in any dimension,
;;   so that its homology is computable.
;;
;; And this chain complex is related to the
;;   chain complex of DOOP4 (highly infinite!)
;;   through a (very) STRONG chain equivalence:
;; (EFHM = EFfective HoMology)
--------------------------------------------------
>
---------- Statement -----------------------------
(EFHM DOOP4)
--------------------------------------------------
>
---------- Result --------------------------------
[K454 Homotopy-Equivalence K437 <= K442 => K444]
--------------------------------------------------
>
---------- Comment -------------------------------
;; Now if a "reasonable" construction is made
;;   using this version with effective homology
;;   of DOOP4, our program uses our EFFECTIVE versions
;;   of the classical spectral sequences
;;   to obtain again a version with effective homology
;;   of the new space.
--------------------------------------------------
>
---------- Comment -------------------------------
;; For example when the loop space ODOOP4 has been asked for,
;;   our effective version
;;     of the EILENBERG-MOORE SPECTRAL SEQUENCE
;;   has been used, constructing the analogous objects
;;   for the new space.
--------------------------------------------------
>
---------- Statement -----------------------------
ODOOP4
--------------------------------------------------
>
---------- Result --------------------------------
[K455 Simplicial-Group]
--------------------------------------------------
>
---------- Statement -----------------------------
(EFHM ODOOP4)
--------------------------------------------------
>
---------- Result --------------------------------
[K597 Homotopy-Equivalence K455 <= K587 => K583]
--------------------------------------------------
>
---------- Comment -------------------------------
;; A basis for example in dimension 3 is
;;   available for the effective chain
;;   complex associated to ODOOP4:
--------------------------------------------------
>
---------- Statement -----------------------------
(LOOP FOR ITEM IN (BASIS (ECHCM ODOOP4) 3) DO
      (PRINT ITEM))
--------------------------------------------------
>
---------- Result --------------------------------

<<AlLp[3 <<AlLp[4 <<AlLp[5 6]>>]>>]>>
<<AlLp[3 <<AlLp[2 <<AlLp[3 4]>>][2 <<AlLp[3 4]>>]>>]>>
<<AlLp[1 <<AlLp[2 <<AlLp[3 4]>>]>>][2 NEW]>>
<<AlLp[1 <<AlLp[2 <<AlLp[3 4]>>]>>][2 <<AlLp[3 <<AlLp[4 5]>>]>>]>>
<<AlLp[2 NEW][1 <<AlLp[2 <<AlLp[3 4]>>]>>]>>
<<AlLp[2 <<AlLp[3 <<AlLp[4 5]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>]>>
<<AlLp[1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>][1 <<AlLp[2 <<AlLp[3 4]>>]>>]>> NIL
--------------------------------------------------
>
---------- Comment -------------------------------
;; But this is not possible for the chain complex of ODOOP4 itself.
--------------------------------------------------
>
---------- Statement -----------------------------
(BASIS ODOOP4 3)
--------------------------------------------------
>
---------- Result --------------------------------
Error: The object [K455 Simplicial-Group] is locally-effective.
>
---------- Comment -------------------------------
;; and in fact this basis is highly infinite!!!
--------------------------------------------------
>
---------- Comment -------------------------------
;; When the homology of ODOOP4 is asked for,
;;   it's in fact the homology of (ECHCM ODOOP4)
;;   which is computed.
--------------------------------------------------
>
---------- Comment -------------------------------
;; More details at:
;;
;;    www-fourier.univ-grenoble-alpes.fr/~sergerar/Kenzo
;;
;; In particular, a rich documentation (Yvon Siret, 340pp) ;;   is
available.
;;
;; Thanks you for your attention.
--------------------------------------------------
>