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 ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\macros.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\various.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\combinations.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\chain-complexes.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\chcm-elementary-op.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\effective-homology.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\homology-groups.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\searching-homology.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\cones.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\tensor-products.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\coalgebras.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\cobar.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\algebras.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\bar.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\simplicial-sets.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\simplicial-mrphs.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\delta.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\special-smsts.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\suspensions.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\disk-pasting.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\cartesian-products.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\eilenberg-zilber.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\kan.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\simplicial-groups.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\fibrations.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\loop-spaces.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\ls-twisted-products.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\lp-space-efhm.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\classifying-spaces.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\k-pi-n.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\serre.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\cs-twisted-products.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\cl-space-efhm.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\whitehead.fasl ; Fast loading C:\Mes documents\Lisp\Kenzo\Officiel\smith.fasl ---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) : End of computing. ;; Clock -> 2000-08-30, 16h 7m 33s. Computing the boundary of the generator 2 (dimension 5) : 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) : End of computing. ;; Clock -> 2000-08-30, 16h 7m 33s. Computing the boundary of the generator 2 (dimension 6) : 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) : End of computing. ;; Clock -> 2000-08-30, 16h 7m 37s. Computing the boundary of the generator 2 (dimension 6) : 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) : End of computing. ;; Clock -> 2000-08-30, 16h 7m 37s. Computing the boundary of the generator 2 (dimension 7) : 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) : 4> End of computing. [... ... Lines deleted ... ...] ;; Clock -> 2000-08-30, 16h 8m 0s. Computing the boundary of the generator 8 (dimension 4) : 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) : 5> End of computing. [... ... Lines deleted ... ...] ;; Clock -> 2000-08-30, 16h 8m 1s. Computing the boundary of the generator 10 (dimension 5) : 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) : 5> End of computing. [... ... Lines deleted ... ...] ;; Clock -> 2000-08-30, 16h 8m 9s. Computing the boundary of the generator 10 (dimension 5) : 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) : 6> End of computing. [... ... Lines deleted ... ...] ;; Clock -> 2000-08-30, 16h 8m 11s. Computing the boundary of the generator 12 (dimension 6) : 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) : <>]>> 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) : <>]>> 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) : <>]>> 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) : <>]>> 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) : <>]>>]>> End of computing. [... ... Lines deleted ... ...] ;; Clock -> 2000-08-30, 16h 9m 4s. Computing the boundary of the generator 19 (dimension 4) : <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> 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) : <>]>>]>> End of computing. [... ... Lines deleted ... ...] ;; Clock -> 2000-08-30, 16h 9m 41s. Computing the boundary of the generator 50 (dimension 5) : <>]>>][1 <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> 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) : End of computing. ;; Clock -> 2000-08-30, 16h 10m 15s. Computing the boundary of the generator 2 (dimension 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) : End of computing. ;; Clock -> 2000-08-30, 16h 10m 15s. Computing the boundary of the generator 2 (dimension 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) : End of computing. ;; Clock -> 2000-08-30, 16h 10m 19s. Computing the boundary of the generator 2 (dimension 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) : End of computing. ;; Clock -> 2000-08-30, 16h 10m 19s. Computing the boundary of the generator 2 (dimension 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) : End of computing. ;; Clock -> 2000-08-30, 16h 10m 22s. Computing the boundary of the generator 2 (dimension 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) : End of computing. ;; Clock -> 2000-08-30, 16h 10m 23s. Computing the boundary of the generator 2 (dimension 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 -------------------------------- <>]>>]>> <>][2 <>]>>]>> <>]>>][2 NEW]>> <>]>>][2 <>]>>]>> <>]>>]>> <>]>>][1 <>]>>]>> <>]>>][1 <>]>>][1 <>]>>]>> 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. -------------------------------------------------- >