> 
--------------------------------------------------<Comment>
;; Loading the Kenzo program.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(LOAD-CFILES)
--------------------------------------------------
> 
--------------------------------------------------<Result>
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\classes.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\macros.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\various.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\combinations.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\chain-complexes.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\chcm-elementary-op.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\effective-homology.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\homology-groups.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\searching-homology.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\cones.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\tensor-products.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\coalgebras.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\cobar.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\algebras.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\bar.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\simplicial-sets.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\simplicial-mrphs.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\delta.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\special-smsts.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\suspensions.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\disk-pasting.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\cartesian-products.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\eilenberg-zilber.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\kan.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\simplicial-groups.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\fibrations.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\loop-spaces.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\ls-twisted-products.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\lp-space-efhm.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\classifying-spaces.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\k-pi-n.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\serre.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\cs-twisted-products.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\cl-space-efhm.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\whitehead.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-2\smith.fasl
--- done ---
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Constructing S2PR = S_2 P^infty R.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF S2PR (R-PROJ-SPACE 2))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K1 Simplicial-Set]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  The wished fibration is associated with
;;    a canonical cohomology class
;;    constructed by CHML-CLSS.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF CH2 (CHML-CLSS S2PR 2))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K9 Cohomology-Class on K1 of degree 2]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Corresponding fibration.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF TAU (Z-WHITEHEAD S2PR CH2))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K15 Fibration K1 -> K10]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Corresponding total space :
;;
;;  X_3 = K(Z,1) times_tau S2PR
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF X3 (FIBRATION-TOTAL TAU))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K18 Simplicial-Set]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Computing H_*(X_3) up to dimension 8.
;;
;; The EFFECTIVE Serre spectral sequence
;; is automatically applied.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(HOMOLOGY X3 0 9)
--------------------------------------------------
> 
--------------------------------------------------<Result>

Computing boundary-matrix in dimension 0.
Rank of the source-module : 1.


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 0) :
<TnPr 0 *> 
End of computing.


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


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 1) :
<TnPr 0 S1> 
End of computing.




Homology in dimension 0 :


Component Z


---done---

;; Clock -> 2006-08-25, 17h 53m 36s.


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


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 1) :
<TnPr 0 S1> 
End of computing.


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


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 2) :
<TnPr 2 *> 
End of computing.




Homology in dimension 1 :



---done---

;; Clock -> 2006-08-25, 17h 53m 36s.


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


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 2) :
<TnPr 2 *> 
End of computing.


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


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 3) :
<TnPr 2 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 2 (dimension 3) :
<TnPr 3 *> 
End of computing.




Homology in dimension 2 :



---done---

;; Clock -> 2006-08-25, 17h 53m 36s.


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


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 3) :
<TnPr 2 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 36s.
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 -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 4) :
<TnPr 3 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 2 (dimension 4) :
<TnPr 4 *> 
End of computing.




Homology in dimension 3 :


Component Z


---done---

;; Clock -> 2006-08-25, 17h 53m 36s.


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


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 4) :
<TnPr 3 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 36s.
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 -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 5) :
<TnPr 4 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 2 (dimension 5) :
<TnPr 5 *> 
End of computing.




Homology in dimension 4 :



---done---

;; Clock -> 2006-08-25, 17h 53m 36s.


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


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 5) :
<TnPr 4 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 36s.
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 -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 1 (dimension 6) :
<TnPr 5 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 36s.
Computing the boundary of the generator 2 (dimension 6) :
<TnPr 6 *> 
End of computing.




Homology in dimension 5 :



---done---

;; Clock -> 2006-08-25, 17h 53m 37s.


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


;; Clock -> 2006-08-25, 17h 53m 37s.
Computing the boundary of the generator 1 (dimension 6) :
<TnPr 5 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 37s.
Computing the boundary of the generator 2 (dimension 6) :
<TnPr 6 *> 
End of computing.


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


;; Clock -> 2006-08-25, 17h 53m 37s.
Computing the boundary of the generator 1 (dimension 7) :
<TnPr 6 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 37s.
Computing the boundary of the generator 2 (dimension 7) :
<TnPr 7 *> 
End of computing.




Homology in dimension 6 :



---done---

;; Clock -> 2006-08-25, 17h 53m 37s.


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


;; Clock -> 2006-08-25, 17h 53m 37s.
Computing the boundary of the generator 1 (dimension 7) :
<TnPr 6 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 37s.
Computing the boundary of the generator 2 (dimension 7) :
<TnPr 7 *> 
End of computing.


Computing boundary-matrix in dimension 8.
Rank of the source-module : 2.


;; Clock -> 2006-08-25, 17h 53m 37s.
Computing the boundary of the generator 1 (dimension 8) :
<TnPr 7 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 37s.
Computing the boundary of the generator 2 (dimension 8) :
<TnPr 8 *> 
End of computing.




Homology in dimension 7 :



---done---

;; Clock -> 2006-08-25, 17h 53m 38s.


Computing boundary-matrix in dimension 8.
Rank of the source-module : 2.


;; Clock -> 2006-08-25, 17h 53m 38s.
Computing the boundary of the generator 1 (dimension 8) :
<TnPr 7 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 38s.
Computing the boundary of the generator 2 (dimension 8) :
<TnPr 8 *> 
End of computing.


Computing boundary-matrix in dimension 9.
Rank of the source-module : 2.


;; Clock -> 2006-08-25, 17h 53m 38s.
Computing the boundary of the generator 1 (dimension 9) :
<TnPr 8 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 53m 42s.
Computing the boundary of the generator 2 (dimension 9) :
<TnPr 9 *> 
End of computing.




Homology in dimension 8 :



---done---

;; Clock -> 2006-08-25, 17h 53m 47s.

NIL
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  There is a strong similarity between X_3 and S^3 !!
;;
;;  How to prove X_3 = S^3 (up to homotopy equivalence) ??
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  Looking for an EXPLICIT map S^3 -> X_3.
;;
;;  Effective homology of X_3.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF X3EH (EFHM X3))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K97 Equivalence K18 <= K87 => K82]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The chain complex K82 is effective.
;;
;; Generator of the 3-homology.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF G3 (FIRST (HOMOLOGY-GEN (K 82) 3)))
--------------------------------------------------
> 
--------------------------------------------------<Result>

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


;; Clock -> 2006-08-25, 17h 54m 4s.
Computing the boundary of the generator 1 (dimension 3) :
<TnPr 2 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 54m 4s.
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 -> 2006-08-25, 17h 54m 4s.
Computing the boundary of the generator 1 (dimension 4) :
<TnPr 3 S1> 
End of computing.


;; Clock -> 2006-08-25, 17h 54m 4s.
Computing the boundary of the generator 2 (dimension 4) :
<TnPr 4 *> 
End of computing.




Homology in dimension 3 :

Component Z


----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr 3 *>>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Image of this generator in X_3.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF S3? (LF X3EH (RG X3EH G3)))
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 3}
<-1 * <CrPr - 3 2-1-0 NIL>>
<-1 * <CrPr 1 2 2 (-1 1)>>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;;  It is a combination of two 3-simplices.
;;
;;  Extracting the 3-simplices.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF S31 (GNRT (FIRST (CMBN-LIST S3?))))
--------------------------------------------------
> 
--------------------------------------------------<Result>
<CrPr - 3 2-1-0 NIL>
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF S32 (GNRT (SECOND (CMBN-LIST S3?))))
--------------------------------------------------
> 
--------------------------------------------------<Result>
<CrPr 1 2 2 (-1 1)>
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Faces of s31 and s32.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(PROGN (DOTIMES (I 4) (PRINT (FACE X3 I 3 S31)))
       (TERPRI)
       (DOTIMES (I 4) (PRINT (FACE X3 I 3 S32))))
--------------------------------------------------
> 
--------------------------------------------------<Result>

<AbSm - <CrPr - 2 1-0 NIL>> 
<AbSm 1-0 <CrPr - 0 - NIL>> 
<AbSm 1-0 <CrPr - 0 - NIL>> 
<AbSm - <CrPr - 2 - (-1 1)>> 

<AbSm 1 <CrPr 0 0 - (1)>> 
<AbSm - <CrPr - 2 1-0 NIL>> 
<AbSm - <CrPr - 2 - (-1 1)>> 
<AbSm 0 <CrPr 0 0 - (1)>> 
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Two interesting 2-faces s21 and s22.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF S21 (FACE X3 0 3 S31))
--------------------------------------------------
> 
--------------------------------------------------<Result>
<AbSm - <CrPr - 2 1-0 NIL>>
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF S22 (FACE X3 3 3 S31))
--------------------------------------------------
> 
--------------------------------------------------<Result>
<AbSm - <CrPr - 2 - (-1 1)>>
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Faces of s21 and s22.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(PROGN (DOTIMES (I 3) (PRINT (FACE X3 I 2 S21)))
       (TERPRI)
       (DOTIMES (I 3) (PRINT (FACE X3 I 2 S22))))
--------------------------------------------------
> 
--------------------------------------------------<Result>

<AbSm 0 <CrPr - 0 - NIL>> 
<AbSm 0 <CrPr - 0 - NIL>> 
<AbSm - <CrPr 0 0 - (1)>> 

<AbSm - <CrPr 0 0 - (1)>> 
<AbSm 0 <CrPr - 0 - NIL>> 
<AbSm 0 <CrPr - 0 - NIL>> NIL
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Only one edge.
;;
;; => The sub-simplicial set defined by s31 and s32 is a 3-sphere.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; +-----------------+
;; +                 |
;; +     The END     |
;; +                 |
;; +-----------------+
--------------------------------------------------
>