> 
--------------------------------------------------<Comment>
;; Loading the Kenzo program.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(LOAD-CFILES)
--------------------------------------------------
> 
--------------------------------------------------<Result>
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\classes.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\macros.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\various.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\combinations.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\chain-complexes.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\chcm-elementary-op.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\effective-homology.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\searching-homology.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\homology-groups.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\cones.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\tensor-products.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\koszul.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\groebner.fasl
; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\smithQ.fasl
--- done ---
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Kenzo expression of the Macaulay ideal:
;; | t5-x t3y-x2 t2y2-xz t3z-y2 t2x-y tx2-z x3-ty2 y3-x2z xy-tz |
;; = the Groebner basis of
;; ideal(x-t^5, y-t^7, z-t^11)
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF IDEAL
 (LIST
  (CMBN 0 1 '(5 0 0 0) -1 '(0 1 0 0))  ;; t5 - x
  (CMBN 0 1 '(3 0 1 0) -1 '(0 2 0 0))  ;; t3y - x2
  (CMBN 0 1 '(2 0 2 0) -1 '(0 1 0 1))  ;; t2y2 - xz
  (CMBN 0 1 '(3 0 0 1) -1 '(0 0 2 0))  ;; t3z - y2
  (CMBN 0 1 '(2 1 0 0) -1 '(0 0 1 0))  ;; t2x - y
  (CMBN 0 1 '(1 2 0 0) -1 '(0 0 0 1))  ;; tx2 - z
  (CMBN 0 1 '(0 3 0 0) -1 '(1 0 2 0))  ;; x3 - ty2
  (CMBN 0 1 '(0 0 3 0) -1 '(0 2 0 1))  ;; y3 - x2z
  (CMBN 0 1 '(0 1 1 0) -1 '(1 0 0 1))));; xy - tz
--------------------------------------------------
> 
--------------------------------------------------<Result>
(
----------------------------------------------------------------------{CMBN 0}
<1 * (5 0 0 0)>
<-1 * (0 1 0 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (3 0 1 0)>
<-1 * (0 2 0 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (2 0 2 0)>
<-1 * (0 1 0 1)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (3 0 0 1)>
<-1 * (0 0 2 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (2 1 0 0)>
<-1 * (0 0 1 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (1 2 0 0)>
<-1 * (0 0 0 1)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (0 3 0 0)>
<-1 * (1 0 2 0)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (0 0 3 0)>
<-1 * (0 2 0 1)>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 0}
<1 * (0 1 1 0)>
<-1 * (1 0 0 1)>
------------------------------------------------------------------------------
)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The corresponding Koszul complex.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF KSZ (K-COMPLEX/GI 4 IDEAL))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K5 Chain-Complex]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The illustration is easier with MINIMAL effective homology.
;; The procedure KOSZUL-MIN-RDCT does this work.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF MMM-RDCT (KOSZUL-MIN-RDCT IDEAL "H"))
--------------------------------------------------
> 
--------------------------------------------------<Result>
[K778 Reduction K5 => K763]
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The result is a REDUCTION K5 => K763.
;; The chain-complex K763 is EFFECTIVE.
;; Q-Dimensions in each degree?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(MAPCAR
   #'(LAMBDA (I) (LENGTH (BASIS (K 763) I)))
   '(0 1 2 3 4))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(1 3 3 1 0)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Basis of K763 in degree 2 ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(BASIS (BCC MMM-RDCT) 2)
--------------------------------------------------
> 
--------------------------------------------------<Result>
(H-2-1 H-2-2 H-2-3)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Three symbols H-2-1, H-2-2 and H-2-3.
;; Boundary of H-2-1 ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? (BCC MMM-RDCT) 2 'H-2-1)
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The boundary is null.
;; In fact the differential of the small chain complex
;; is ENTIRELY NULL.
;; It is the MINIMAL effective homology.
;;
;; Every generator of K763 is an "abstract" homology class.
;;
;; Representant of the homology-class H-2-1 ?
;; We apply the g-component of the reduction MMM-RDCT = (f,g,h).
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF Z-2-1 (G MMM-RDCT 2 'H-2-1))
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 2}
<-1 * ((0 2 0 0) (1 1 0 0))>
<1 * ((4 0 0 0) (1 0 0 1))>
<-1 * ((0 0 0 0) (0 1 0 1))>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; => Representant = - x^2 dt.dx + t^4 dt.dz - dx.dz
;; Is it a cycle ?!
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? KSZ Z-2-1)
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; OK!
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The HOMOLOGICAL PROBLEM in K5 is solved.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Is t^2 dt.dx a cycle ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? KSZ 2 '((2 0 0 0) (1 1 0 0)))
--------------------------------------------------
> 
--------------------------------------------------<Result>

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

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; It is not a cycle.
;; Verifying d o d = 0 (!)
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? KSZ (? KSZ 2 '((2 0 0 0) (1 1 0 0))))
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; This object z1:
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF Z1
  (CMBN 2
         1 '((1 0 1 9) (1 1 0 0))
        -1 '((0 2 0 0) (1 1 0 0))
        -1 '((1 1 0 9) (1 0 1 0))
         1 '((4 0 0 0) (1 0 0 1))
         1 '((2 0 0 9) (0 1 1 0))
        -2 '((1 1 0 0) (0 1 1 0))
         2 '((2 0 0 0) (0 1 0 1))
        -1 '((0 0 0 0) (0 1 0 1))
        -2 '((0 0 0 0) (0 0 1 1))))
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 2}
<1 * ((1 0 1 9) (1 1 0 0))>
<-1 * ((0 2 0 0) (1 1 0 0))>
<-1 * ((1 1 0 9) (1 0 1 0))>
<1 * ((4 0 0 0) (1 0 0 1))>
<1 * ((2 0 0 9) (0 1 1 0))>
<-2 * ((1 1 0 0) (0 1 1 0))>
<2 * ((2 0 0 0) (0 1 0 1))>
<-1 * ((0 0 0 0) (0 1 0 1))>
<-2 * ((0 0 0 0) (0 0 1 1))>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Is a cycle:
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? KSZ Z1)
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; What about its homology class ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(F MMM-RDCT Z1)
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 2}
<1 * H-2-1>
<-2 * H-2-3>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The homology class is H-2-1 - 2 H-2-3.
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; This object z2 also is a cycle.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(setf z2
  (cmbn 2
         1 '((1 0 1 9) (1 1 0 0))
        -1 '((1 1 0 9) (1 0 1 0))
         1 '((2 0 0 9) (0 1 1 0))))
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 2}
<1 * ((1 0 1 9) (1 1 0 0))>
<-1 * ((1 1 0 9) (1 0 1 0))>
<1 * ((2 0 0 9) (0 1 1 0))>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(? KSZ Z2)
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Homology class ?
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(F MMM-RDCT Z2)
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; The homology class is null and
;; we must be able to find a boundary preimage.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(SETF PRE-IMAGE (H MMM-RDCT Z2))
--------------------------------------------------
> 
--------------------------------------------------<Result>

----------------------------------------------------------------------{CMBN 3}
<1 * ((1 0 1 8) (1 1 0 1))>
<-1 * ((1 1 0 8) (1 0 1 1))>
<1 * ((2 0 0 8) (0 1 1 1))>
------------------------------------------------------------------------------

--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; Verification.
--------------------------------------------------
> 
--------------------------------------------------<Lisp Statement>
(LIST Z2 (? KSZ PRE-IMAGE))
--------------------------------------------------
> 
--------------------------------------------------<Result>
(
----------------------------------------------------------------------{CMBN 2}
<1 * ((1 0 1 9) (1 1 0 0))>
<-1 * ((1 1 0 9) (1 0 1 0))>
<1 * ((2 0 0 9) (0 1 1 0))>
------------------------------------------------------------------------------
 
----------------------------------------------------------------------{CMBN 2}
<1 * ((1 0 1 9) (1 1 0 0))>
<-1 * ((1 1 0 9) (1 0 1 0))>
<1 * ((2 0 0 9) (0 1 1 0))>
------------------------------------------------------------------------------
)
--------------------------------------------------
> 
--------------------------------------------------<Comment>
;; +-----------------+
;; +                 |
;; +     The END     |
;; +                 |
;; +-----------------+
--------------------------------------------------
>