> -------------------------------------------------- ;; Loading the Kenzo program. -------------------------------------------------- > -------------------------------------------------- (LOAD-CFILES) -------------------------------------------------- > -------------------------------------------------- ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\macros.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\various.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\classes.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\combinations.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\chain-complexes.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\chcm-elementary-op.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\effective-homology.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\homology-groups.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\searching-homology.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\cones.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\tensor-products.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\coalgebras.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\cobar.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\algebras.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\bar.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\simplicial-sets.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\simplicial-mrphs.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\delta.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\special-smsts.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\suspensions.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\disk-pasting.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\cartesian-products.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\eilenberg-zilber.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\kan.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\simplicial-groups.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\fibrations.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\loop-spaces.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\ls-twisted-products.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\lp-space-efhm.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\classifying-spaces.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\k-pi-n.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\serre.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\cs-twisted-products.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\cl-space-efhm.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\whitehead.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\Kenzo-1-Acl5\smith.fasl ("macros" "various" "classes" "combinations" "chain-complexes" "chcm-elementary-op" "effective-homology" "homology-groups" "searching-homology" "cones" ...) -------------------------------------------------- > -------------------------------------------------- ;; I intend to compute 2+2. -------------------------------------------------- > -------------------------------------------------- (+ 2 2) -------------------------------------------------- > -------------------------------------------------- 4 -------------------------------------------------- > -------------------------------------------------- ;; Constructing P^infty R. -------------------------------------------------- > -------------------------------------------------- (SETF P-INFTY-R (K-Z2 1)) -------------------------------------------------- > -------------------------------------------------- [K1 Abelian-Simplicial-Group] -------------------------------------------------- > -------------------------------------------------- ;; Simplices in dimension < 6. -------------------------------------------------- > -------------------------------------------------- (DOTIMES (I 6) (PRINT (BASIS P-INFTY-R I))) -------------------------------------------------- > -------------------------------------------------- (0) (1) (2) (3) (4) (5) NIL -------------------------------------------------- > -------------------------------------------------- ;; Examining the simplicial structure. -------------------------------------------------- > -------------------------------------------------- (DOTIMES (I 6) (FORMAT T "~%Dimension ~A" I) (UNLESS (ZEROP I) (DOTIMES (J (1+ I)) (FORMAT T "~% Face ~A = ~A" J (FACE P-INFTY-R J I I))))) -------------------------------------------------- > -------------------------------------------------- Dimension 0 Dimension 1 Face 0 = Face 1 = Dimension 2 Face 0 = Face 1 = Face 2 = Dimension 3 Face 0 = Face 1 = Face 2 = Face 3 = Dimension 4 Face 0 = Face 1 = Face 2 = Face 3 = Face 4 = Dimension 5 Face 0 = Face 1 = Face 2 = Face 3 = Face 4 = Face 5 = NIL -------------------------------------------------- > -------------------------------------------------- ;; Checking the coherence of the simplicial structure. -------------------------------------------------- > -------------------------------------------------- (CHECK-SMST P-INFTY-R 0 6) -------------------------------------------------- > -------------------------------------------------- Checking the 0-simplices... Checking the 1-simplices... Checking the 2-simplices... Checking the 3-simplices... Checking the 4-simplices... Checking the 5-simplices...T -------------------------------------------------- > -------------------------------------------------- ;; Computing H_i(P-infty-R) for i < 6. -------------------------------------------------- > -------------------------------------------------- (HOMOLOGY P-INFTY-R 0 6) -------------------------------------------------- > -------------------------------------------------- Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 0) : 0 End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 1) : 1 End of computing. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2008-09-19, 18h 39m 49s. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 1) : 1 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 2) : 2 End of computing. Homology in dimension 1 : Component Z/2Z ---done--- ;; Clock -> 2008-09-19, 18h 39m 49s. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. 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 -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 3) : 3 End of computing. Homology in dimension 2 : ---done--- ;; Clock -> 2008-09-19, 18h 39m 49s. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 3) : 3 End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 4) : 4 End of computing. Homology in dimension 3 : Component Z/2Z ---done--- ;; Clock -> 2008-09-19, 18h 39m 49s. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. 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 -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 5) : 5 End of computing. Homology in dimension 4 : ---done--- ;; Clock -> 2008-09-19, 18h 39m 49s. Computing boundary-matrix in dimension 5. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 5) : 5 End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 39m 49s. Computing the boundary of the generator 1 (dimension 6) : 6 End of computing. Homology in dimension 5 : Component Z/2Z ---done--- ;; Clock -> 2008-09-19, 18h 39m 49s. NIL -------------------------------------------------- > -------------------------------------------------- ;; Constructing K(Z,2). -------------------------------------------------- > -------------------------------------------------- (SETF K-Z-2 (K-Z 2)) -------------------------------------------------- > -------------------------------------------------- [K29 Abelian-Simplicial-Group] -------------------------------------------------- > -------------------------------------------------- ;; Examining the 4-simplices. -------------------------------------------------- > -------------------------------------------------- (BASIS K-Z-2 4) -------------------------------------------------- > -------------------------------------------------- Error: The object [K29 Abelian-Simplicial-Group] is locally-effective. > -------------------------------------------------- ;; Example of 4-simplex. -------------------------------------------------- > -------------------------------------------------- (SETF 4-SIMPLEX (GBAR 4 0 '(1234 2345 3456) 0 '(12345 23456) 0 '(9999999999) 0 NIL)) -------------------------------------------------- > -------------------------------------------------- <<- (12345 23456)><- (9999999999)><- NIL>>> -------------------------------------------------- > -------------------------------------------------- ;; Computing its boundary. -------------------------------------------------- > -------------------------------------------------- (? K-Z-2 4 4-SIMPLEX) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 3} <1 * <<- (10000012344)><- NIL>>>> <1 * <<- (23456)><- NIL>>>> <-1 * <<- (35801)><- NIL>>>> <1 * <<- (9999999999)><- NIL>>>> <-1 * <<- (9999999999)><- NIL>>>> ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; Constructing ;; X = P^infty R / P^3 R. -------------------------------------------------- > -------------------------------------------------- (SETF X (R-PROJ-SPACE :INFINITY 4)) -------------------------------------------------- > -------------------------------------------------- [K41 Simplicial-Set] -------------------------------------------------- > -------------------------------------------------- ;; Constructing the third loop space: ;; Omega^3 (X). -------------------------------------------------- > -------------------------------------------------- (SETF OOOX (LOOP-SPACE X 3)) -------------------------------------------------- > -------------------------------------------------- [K70 Simplicial-Group] -------------------------------------------------- > -------------------------------------------------- ;; Constructing the Adams chain complex. -------------------------------------------------- > -------------------------------------------------- (SETF ADAMS (RBCC (EFHM OOOX))) -------------------------------------------------- > -------------------------------------------------- [K430 Chain-Complex] -------------------------------------------------- > -------------------------------------------------- ;; Computing the boundary matrix ;; between dimensions 5 and 4. -------------------------------------------------- > -------------------------------------------------- (CHCM-MAT ADAMS 5) -------------------------------------------------- > -------------------------------------------------- Computing boundary-matrix in dimension 5. Rank of the source-module : 33. ;; Clock -> 2008-09-19, 18h 40m 26s. Computing the boundary of the generator 1 (dimension 5) : <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 29s. Computing the boundary of the generator 2 (dimension 5) : <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 29s. Computing the boundary of the generator 3 (dimension 5) : <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 30s. Computing the boundary of the generator 4 (dimension 5) : <>][4 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 30s. Computing the boundary of the generator 5 (dimension 5) : <>][3 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 30s. Computing the boundary of the generator 6 (dimension 5) : <>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 30s. Computing the boundary of the generator 7 (dimension 5) : <>][2 <>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 8 (dimension 5) : <>]>>][4 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 9 (dimension 5) : <>]>>][4 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 10 (dimension 5) : <>]>>][4 <>][3 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 11 (dimension 5) : <>]>>][4 <>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 12 (dimension 5) : <>]>>][3 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 13 (dimension 5) : <>]>>][3 <>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 14 (dimension 5) : <>]>>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 15 (dimension 5) : <>][2 <>]>>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 31s. Computing the boundary of the generator 16 (dimension 5) : <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 32s. Computing the boundary of the generator 17 (dimension 5) : <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 32s. Computing the boundary of the generator 18 (dimension 5) : <>][3 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 32s. Computing the boundary of the generator 19 (dimension 5) : <>][2 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 32s. Computing the boundary of the generator 20 (dimension 5) : <>]>>][1 <>]>>][3 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 32s. Computing the boundary of the generator 21 (dimension 5) : <>]>>][1 <>]>>][3 <>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 32s. Computing the boundary of the generator 22 (dimension 5) : <>]>>][2 <>]>>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 32s. Computing the boundary of the generator 23 (dimension 5) : <>]>>][3 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 33s. Computing the boundary of the generator 24 (dimension 5) : <>]>>][3 <>][2 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 33s. Computing the boundary of the generator 25 (dimension 5) : <>]>>][1 <>]>>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 33s. Computing the boundary of the generator 26 (dimension 5) : <>]>>][2 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 33s. Computing the boundary of the generator 27 (dimension 5) : <>]>>][1 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 33s. Computing the boundary of the generator 28 (dimension 5) : <>][2 <>]>>][1 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 34s. Computing the boundary of the generator 29 (dimension 5) : <>]>>][1 <>]>>][1 <>]>>][2 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 34s. Computing the boundary of the generator 30 (dimension 5) : <>]>>][1 <>]>>][2 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 34s. Computing the boundary of the generator 31 (dimension 5) : <>]>>][2 <>]>>][1 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 34s. Computing the boundary of the generator 32 (dimension 5) : <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> End of computing. ;; Clock -> 2008-09-19, 18h 40m 34s. Computing the boundary of the generator 33 (dimension 5) : <>]>>][1 <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> End of computing. ========== MATRIX 13 lines + 33 columns ===== L1=[C1=-2] L2=[C1=-1] L3=[C1=-4][C2=1][C3=-1][C4=-2] L4=[C2=1][C3=-1][C6=2] L5=[C1=6][C4=1][C6=1] L6=[C1=4][C4=4][C6=4][C7=3] L7=[C1=4][C12=-2][C14=2] L8=[C1=6][C4=1][C6=1] L9=[C1=4][C4=4][C6=4][C7=3] L10=[C8=4][C10=1][C11=-1][C14=-4][C15=-2][C20=-2] L11=[C1=4][C8=4][C10=1][C11=-1][C16=-4][C18=-1][C19=1][C23=-2] L12=[C12=4][C13=2][C16=-4][C18=-1][C19=1][C27=-2] L13=[C1=-1][C20=4][C21=2][C23=-4][C24=-2][C27=4][C28=2] ========== END-MATRIX -------------------------------------------------- > -------------------------------------------------- ;; Equivalence of K-Z-2 with a chain-complex of finite type. -------------------------------------------------- > -------------------------------------------------- (SETF EFHM (EFHM K-Z-2)) -------------------------------------------------- > -------------------------------------------------- [K573 Homotopy-Equivalence K29 <= K563 => K559] -------------------------------------------------- > -------------------------------------------------- ;; This chain complex. -------------------------------------------------- > -------------------------------------------------- (SETF FINITE-CC (RBCC EFHM)) -------------------------------------------------- > -------------------------------------------------- [K559 Chain-Complex] -------------------------------------------------- > -------------------------------------------------- ;; Basis up to dimension 5. -------------------------------------------------- > -------------------------------------------------- (DOTIMES (I 6) (PRINT (BASIS FINITE-CC I))) -------------------------------------------------- > -------------------------------------------------- (<>) NIL (<>) NIL (<>) NIL NIL -------------------------------------------------- > -------------------------------------------------- ;; Generator of H_4(K-Z-2). -------------------------------------------------- > -------------------------------------------------- (SETF HCYCLE (LF EFHM (RG EFHM 4 (FIRST (BASIS FINITE-CC 4))))) -------------------------------------------------- > -------------------------------------------------- ----------------------------------------------------------------------{CMBN 4} <-1 * <<1-0 NIL><- (1)><- NIL>>>> <1 * <<1 (1)><0 NIL><- NIL>>>> <-1 * <<0 (1)><0 NIL><- NIL>>>> ------------------------------------------------------------------------------ -------------------------------------------------- > -------------------------------------------------- ;; Extracting the simplices. -------------------------------------------------- > -------------------------------------------------- (PROGN (SETF B4 (MAPCAR #'CDR (CMBN-LIST HCYCLE))) (MAP NIL #'PRINT B4)) -------------------------------------------------- > -------------------------------------------------- <<1-0 NIL><- (1)><- NIL>>> <<1 (1)><0 NIL><- NIL>>> <<0 (1)><0 NIL><- NIL>>> NIL -------------------------------------------------- > -------------------------------------------------- ;; Examining their faces. -------------------------------------------------- > -------------------------------------------------- (DOLIST (ITEM B4) (TERPRI) (DOTIMES (I 5) (PRINT (FACE K-Z-2 I 4 ITEM)))) -------------------------------------------------- > -------------------------------------------------- <- NIL>>>> <- NIL>>>> <- (1)><- NIL>>>> <- NIL>>>> <- NIL>>>> <- NIL>>>> <- (1)><- NIL>>>> <- (1)><- NIL>>>> <0 NIL><- NIL>>>> <- NIL>>>> <- NIL>>>> <- (1)><- NIL>>>> <- NIL>>>> <0 NIL><- NIL>>>> <- NIL>>>> NIL -------------------------------------------------- > -------------------------------------------------- ;; Constituting the list of interesting 3-simplices. -------------------------------------------------- > -------------------------------------------------- (PROGN (SETF B3 (MAPCAR #'GMSM (LIST (FACE K-Z-2 2 4 (NTH 0 B4)) (FACE K-Z-2 1 4 (NTH 1 B4)) (FACE K-Z-2 3 4 (NTH 1 B4))))) (MAP NIL #'PRINT B3)) -------------------------------------------------- > -------------------------------------------------- <<- (1)><- NIL>>> <<- (1)><- NIL>>> <<0 NIL><- NIL>>> NIL -------------------------------------------------- > -------------------------------------------------- ;; Examining their faces. -------------------------------------------------- > -------------------------------------------------- (DOLIST (ITEM B3) (TERPRI) (DOTIMES (I 4) (PRINT (FACE K-Z-2 I 3 ITEM)))) -------------------------------------------------- > -------------------------------------------------- <- NIL>>>> <- NIL>>>> <- NIL>>>> <- NIL>>>> >> <- NIL>>>> <- NIL>>>> <- NIL>>>> <- NIL>>>> <- NIL>>>> <- NIL>>>> >> NIL -------------------------------------------------- > -------------------------------------------------- ;; Constituting the list of interesting 2-simplices. -------------------------------------------------- > -------------------------------------------------- (PROGN (SETF B2 (MAPCAR #'GMSM (LIST (FACE K-Z-2 0 3 (NTH 0 B3)) (FACE K-Z-2 2 3 (NTH 1 B3))))) (MAP NIL #'PRINT B2)) -------------------------------------------------- > -------------------------------------------------- <<- NIL>>> <<- NIL>>> NIL -------------------------------------------------- > -------------------------------------------------- ;; Examining their faces. -------------------------------------------------- > -------------------------------------------------- (DOLIST (ITEM B2) (TERPRI) (DOTIMES (I 3) (PRINT (FACE K-Z-2 I 2 ITEM)))) -------------------------------------------------- > -------------------------------------------------- >> >> >> >> >> >> NIL -------------------------------------------------- > -------------------------------------------------- ;; We intend to replace the infinite basis of K-Z-2 ;; by a finite one, in order to construct ;; a SUB-SIMPLICIAL SET of FINITE TYPE ;; with the SAME HOMOTOPY TYPE. ;; ;; Constructing the appropriate BASIS function. -------------------------------------------------- > -------------------------------------------------- (SETF NEW-BASIS #'(LAMBDA (N) (CASE N (0 (LIST (BSGN K-Z-2))) (2 B2) (3 B3) (4 B4) (OTHERWISE NIL)))) -------------------------------------------------- > -------------------------------------------------- # -------------------------------------------------- > -------------------------------------------------- ;; Constructing an interesting simplicial set. -------------------------------------------------- > -------------------------------------------------- (SETF TENTATIVE (BUILD-SMST :CMPR (CMPR K-Z-2) :BASIS NEW-BASIS :BSPN (BSPN K-Z-2) :FACE (FACE K-Z-2))) -------------------------------------------------- > -------------------------------------------------- [K574 Simplicial-Set] -------------------------------------------------- > -------------------------------------------------- ;; Verifying the construction is simplicially coherent. -------------------------------------------------- > -------------------------------------------------- (CHECK-SMST TENTATIVE 0 5) -------------------------------------------------- > -------------------------------------------------- Checking the 0-simplices... Checking the 1-simplices... Checking the 2-simplices... Checking the 3-simplices... Checking the 4-simplices...T -------------------------------------------------- > -------------------------------------------------- ;; Homology groups of "TENTATIVE". -------------------------------------------------- > -------------------------------------------------- (HOMOLOGY TENTATIVE 0 5) -------------------------------------------------- > -------------------------------------------------- Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2008-09-19, 18h 41m 52s. Computing the boundary of the generator 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 -> 2008-09-19, 18h 41m 52s. Computing boundary-matrix in dimension 1. Rank of the source-module : 0. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2008-09-19, 18h 41m 52s. Computing the boundary of the generator 1 (dimension 2) : <<- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 52s. Computing the boundary of the generator 2 (dimension 2) : <<- NIL>>> End of computing. Homology in dimension 1 : ---done--- ;; Clock -> 2008-09-19, 18h 41m 52s. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2008-09-19, 18h 41m 52s. Computing the boundary of the generator 1 (dimension 2) : <<- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 52s. Computing the boundary of the generator 2 (dimension 2) : <<- NIL>>> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 3. ;; Clock -> 2008-09-19, 18h 41m 52s. Computing the boundary of the generator 1 (dimension 3) : <<- (1)><- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 2 (dimension 3) : <<- (1)><- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 3 (dimension 3) : <<0 NIL><- NIL>>> End of computing. Homology in dimension 2 : Component Z ---done--- ;; Clock -> 2008-09-19, 18h 41m 53s. Computing boundary-matrix in dimension 3. Rank of the source-module : 3. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 1 (dimension 3) : <<- (1)><- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 2 (dimension 3) : <<- (1)><- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 3 (dimension 3) : <<0 NIL><- NIL>>> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 3. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 1 (dimension 4) : <<1-0 NIL><- (1)><- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 2 (dimension 4) : <<1 (1)><0 NIL><- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 3 (dimension 4) : <<0 (1)><0 NIL><- NIL>>> End of computing. Homology in dimension 3 : ---done--- ;; Clock -> 2008-09-19, 18h 41m 53s. Computing boundary-matrix in dimension 4. Rank of the source-module : 3. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 1 (dimension 4) : <<1-0 NIL><- (1)><- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 2 (dimension 4) : <<1 (1)><0 NIL><- NIL>>> End of computing. ;; Clock -> 2008-09-19, 18h 41m 53s. Computing the boundary of the generator 3 (dimension 4) : <<0 (1)><0 NIL><- NIL>>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Homology in dimension 4 : Component Z ---done--- ;; Clock -> 2008-09-19, 18h 41m 53s. NIL -------------------------------------------------- > -------------------------------------------------- ;; The homology groups of "TENTATIVE" are: ;; {Z 0 Z 0 Z} = Those of K-Z-2. ;; ;; A little more elemetary work proves ;; the canonical inclusion: ;; TENTATIVE -> K-Z-2 ;; actually is a homotopy equivalence. ;; ;; => Model of (P^2 C) with ;; (1 0 2 3 3) simplices. -------------------------------------------------- > -------------------------------------------------- ;; +-----------+ ;; | | ;; | The END. | ;; | | ;; +-----------+ -------------------------------------------------- >