544 lines
18 KiB
Common Lisp
544 lines
18 KiB
Common Lisp
;;; EFFECTIVE-HOMOLOGY EFFECTIVE-HOMOLOGY EFFECTIVE-HOMOLOGY
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;;; EFFECTIVE-HOMOLOGY EFFECTIVE-HOMOLOGY EFFECTIVE-HOMOLOGY
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;;; EFFECTIVE-HOMOLOGY EFFECTIVE-HOMOLOGY EFFECTIVE-HOMOLOGY
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(IN-PACKAGE #:cat)
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(PROVIDE "effective-homology")
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;;;
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;;; REDUCTIONS
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;;;
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#+clisp(eval-when (:compile-toplevel :load-toplevel :execute)
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(setf (ext:package-lock :clos) nil))
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(DEFMETHOD PRINT-OBJECT ((rdct reduction) stream)
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(the reduction
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(progn
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(format stream "[K~D Reduction K~D => K~D]"
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(idnm rdct) (idnm (tcc rdct)) (idnm (bcc rdct)))
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rdct)))
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#+clisp(eval-when (:compile-toplevel :load-toplevel :execute)
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(setf (ext:package-lock :clos) t))
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(DEFUN RDCT (n)
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(declare (fixnum n))
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"--------------------------------------------------------------[function-doc]
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RDCT
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Args: (n)
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Returns the N-th user-created reduction from the list *RDCT-LIST*. Returns NIL
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if it doesn't exist.
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------------------------------------------------------------------------------"
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(the (or null reduction)
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(find n *rdct-list* :key #'idnm)))
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(DEFUN BUILD-RDCT (&key f g h orgn)
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(declare
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(type morphism f g h)
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(list orgn))
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"--------------------------------------------------------------[function-doc]
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BUILD-RDCT
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Args: (&key f g h orgn)
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Returns an instance of the class REDUCTION. The keyword arguments are as
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follows:
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:F F, an object of type morphism representing the morphism f of a reduction
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:G G, an object of type morphism representing the morphism g of a reduction
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:H H, an object of type morphism representing the morphism h of a reduction
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:ORGN ORGN, a list containing a relevant and carefully chosen comment about
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the origin of the chain complex. This comment should be unique
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for a Kenzo session (between calls of CAT-INIT), as it is used
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for caching purposes.
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Use this function instead of creating instances via the standard constructor
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MAKE-INSTANCE.
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------------------------------------------------------------------------------"
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(the reduction
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(progn
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(unless orgn
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(setf orgn `(build-rdct ,f ,g ,h)))
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(let ((already (find orgn *rdct-list* :test #'equal :key #'orgn)))
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(declare (type (or reduction null) already))
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(when already
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(return-from build-rdct already)))
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(with-slots ((fsorc sorc) (ftrgt trgt) (fdegr degr)) f
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(declare
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(type chain-complex fsorc ftrgt)
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(fixnum fdegr))
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(with-slots ((gsorc sorc) (gtrgt trgt) (gdegr degr)) g
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(declare
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(type chain-complex gsorc gtrgt)
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(fixnum gdegr))
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(with-slots ((hsorc sorc) (htrgt trgt) (hdegr degr)) h
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(declare
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(type chain-complex hsorc htrgt)
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(fixnum hdegr))
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(unless (and (eq gsorc ftrgt)
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(eq fsorc gtrgt)
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(eq hsorc fsorc)
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(eq htrgt fsorc)
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(zerop fdegr)
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(zerop gdegr)
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(= +1 hdegr))
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(error "In BUILD-RDCT, the data are non coherent."))
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(let ((rdct (make-instance 'reduction
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:tcc fsorc :bcc ftrgt
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:f f :g g :h h
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:orgn orgn)))
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(declare (type reduction rdct))
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(push rdct *rdct-list*)
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rdct)))))))
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(DEFUN TRIVIAL-RDCT (chcm)
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(declare (type chain-complex chcm))
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"--------------------------------------------------------------[function-doc]
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TRIVIAL-RDCT
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Args: (chcm)
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Builds the trivial reduction according to the following diagram, which
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involves the chain complex CHCM only:
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Zero s
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C -------> C
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^
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Id | | Id
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v
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C
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Zero is the zero morphism of degree 1 of C and Id is its identity morphism.
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See also: ZERO-MRPH, IDNT-MRPH.
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------------------------------------------------------------------------------"
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(build-rdct
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:f (idnt-mrph chcm)
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:g (idnt-mrph chcm)
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:h (zero-mrph chcm chcm +1)
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:orgn `(trivial-rdct ,chcm)))
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;;;
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;;; HOMOTOPY-EQUIVALENCES
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;;;
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;;(defgeneric build-hmeq (keyword lrdct &key &allow-other-keys)
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(DEFGENERIC BUILD-HMEQ (keyword lrdct &key))
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(DEFMETHOD BUILD-HMEQ ((keyword1 (eql :lrdct)) lrdct &key rrdct orgn)
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(declare
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(type reduction lrdct rrdct)
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(list orgn))
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(the homotopy-equivalence
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(progn
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(with-slots ((lf f) (lg g) (lh h) (ltcc tcc) (lbcc bcc)) lrdct
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(declare
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(type morphism lf lg lh)
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(type chain-complex ltcc lbcc))
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(with-slots ((rf f) (rg g) (rh h) (rtcc tcc) (rbcc bcc)) rrdct
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(declare
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(type morphism rf rg rh)
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(type chain-complex rtcc rbcc))
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(unless (eq ltcc rtcc)
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(error "In BUILD-HMEQ (version from rdct), the tcc's are not the same."))
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(unless orgn
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(setf orgn `(build-hmeq ,lrdct ,rrdct)))
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(let ((already (find orgn *hmeq-list* :test #'equal :key #'orgn)))
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(declare (type (or homotopy-equivalence null) already))
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(when already
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(return-from build-hmeq already)))
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(let ((hmeq (make-instance 'homotopy-equivalence
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:lbcc lbcc :tcc ltcc :rbcc rbcc
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:lf lf :lg lg :lh lh
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:rf rf :rg rg :rh rh
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:lrdct lrdct :rrdct rrdct
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:orgn orgn)))
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(declare (type homotopy-equivalence hmeq))
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(push hmeq *hmeq-list*)
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hmeq))))))
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#+clisp(eval-when (:compile-toplevel :load-toplevel :execute)
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(setf (ext:package-lock :clos) nil))
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(DEFMETHOD PRINT-OBJECT ((hmeq homotopy-equivalence) stream)
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(the homotopy-equivalence
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(progn
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(format stream "[K~D Homotopy-Equivalence K~D <= K~D => K~D]"
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(idnm hmeq)
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(idnm (lbcc hmeq)) (idnm (tcc hmeq)) (idnm (rbcc hmeq)))
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hmeq)))
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#+clisp(eval-when (:compile-toplevel :load-toplevel :execute)
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(setf (ext:package-lock :clos) t))
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(DEFGENERIC HMEQ (integer))
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(DEFMETHOD HMEQ ((n integer))
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(declare (fixnum n))
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(the (or homotopy-equivalence null)
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(find n *hmeq-list* :key #'idnm)))
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(DEFMETHOD BUILD-HMEQ ((keyword1 (eql :lf)) lf &key lg lh rf rg rh orgn)
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(declare
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(type morphism lf lg lh rf rg rh)
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(list orgn))
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(the homotopy-equivalence
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(progn
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(unless orgn
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(setf orgn `(build-hmeq ,lf ,lg ,lh ,rf ,rg ,rh)))
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(let ((already (find orgn *hmeq-list* :test #'equal :key #'orgn)))
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(declare (type (or null homotopy-equivalence) already))
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(when already
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(return-from build-hmeq already)))
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(build-hmeq
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:lrdct (build-rdct :f lf :g lg :h lh
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:orgn `(build-hmeq ,lf ,lg ,lh ,rf ,rg ,rh lrdct))
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:rrdct (build-rdct :f rf :g rg :h rh
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:orgn `(build-hmeq ,lf ,lg ,lh ,rf ,rg ,rh rrdct))
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:orgn orgn))))
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(DEFUN TRIVIAL-HMEQ (chcm)
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(declare (type chain-complex chcm))
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(build-hmeq
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:lrdct (trivial-rdct chcm)
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:rrdct (trivial-rdct chcm)
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:orgn `(trivial-hmeq ,chcm)))
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;; Functions.
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(DEFUN PRE-CHECK-RDCT (rdct)
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"--------------------------------------------------------------[function-doc]
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PRE-CHECK-RDCT
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Args: (rdct)
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Assigns the following Lisp global variables, using morphisms from RDCT and the
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differentials and identity morphisms of the underlying chain complexes,
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according to these formulas:
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*TDD* = d ° d
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^ ^
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C C
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*BDD* = d ° d
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C C
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*ID-FG* = Id - f ° g
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C
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*ID-GF-DH-HD* = Id - g ° f - (d ° h + h ° d )
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^ ^ ^
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C C C
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*HH* = h ° h
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*FH* = f ° h
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*HG* = h ° g
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*DF-FD* = d ° f - f ° d
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C ^
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C
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*DG-GD* = d ° g - g ° d
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^ C
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C
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------------------------------------------------------------------------------"
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(declare (type reduction rdct))
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(with-slots (bcc tcc f g h) rdct
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(declare
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(type chain-complex bcc tcc)
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(type morphism f g h))
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(setf *tdd* (cmps tcc tcc)
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*bdd* (cmps bcc bcc)
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*df-fd* (sbtr
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(cmps bcc f)
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(cmps f tcc))
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*dg-gd* (sbtr
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(cmps tcc g)
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(cmps g bcc))
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*id-fg* (sbtr (idnt-mrph bcc) (cmps f g))
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*id-gf-dh-hd* (i-sbtr (idnt-mrph tcc)
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(cmps g f)
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(cmps tcc h)
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(cmps h tcc))
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*hh* (cmps h h)
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*fh* (cmps f h)
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*hg* (cmps h g)))
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(done))
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(DEFUN CHECK-RDCT ()
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"--------------------------------------------------------------[function-doc]
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CHECK-RDCT
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Args: ()
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Maps combinations of the top and bottom chain complexes of a reduction using
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the morphisms created by PRE-CHECK-RDCT. Having no parameters, the
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combinations must be provided via the Lisp global variables *TC* and *BC*,
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for the top and bottom chain complex, respectively. If the morphisms are
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coherent, the result of each mapping is a null combination.
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Note: This function is intended for interactive use. It pauses after each
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evaluation for the user to inspect the result. Resume execution, by pressing
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the <Enter> key.
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------------------------------------------------------------------------------"
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(dolist (cmbn '(*tc* *bc*))
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(declare (type symbol cmbn))
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(format t "~%~A => ~A" cmbn (eval cmbn)))
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(dolist (phi '(*tdd* *bdd* *df-fd* *dg-gd* *id-fg* *id-gf-dh-hd*
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*hh* *fh* *hg*))
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(declare (type symbol phi))
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(format t "~%Checking ~A = 0" phi)
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(format t "~%Result: ")
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(princ (cmbn-? (eval phi)
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(if (member phi '(*bdd* *dg-gd* *id-fg* *dg-gd* *hg*))
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*bc* *tc*)))
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(read-line))
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(done))
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(DEFUN CHECK-RDCT-NO-WAIT ()
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"--------------------------------------------------------------[function-doc]
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CHECK-RDCT-NO-WAIT
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Args: ()
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Maps combinations of the top and bottom chain complexes of a reduction using
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the morphisms created by PRE-CHECK-RDCT. Having no parameters, the
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combinations must be provided via the Lisp global variables *TC* and *BC*,
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for the top and bottom chain complex, respectively. If the morphisms are
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coherent, the result of each mapping is a null combination.
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------------------------------------------------------------------------------"
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(dolist (cmbn '(*tc* *bc*))
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(declare (type symbol cmbn))
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(format t "~%~A => ~A" cmbn (eval cmbn)))
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(dolist (phi '(*tdd* *bdd* *df-fd* *dg-gd* *id-fg* *id-gf-dh-hd*
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*hh* *fh* *hg*))
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(declare (type symbol phi))
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(format t "~%Checking ~A = 0" phi)
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(format t "~%Result: ")
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(princ (cmbn-? (eval phi)
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(if (member phi '(*bdd* *dg-gd* *id-fg* *dg-gd* *hg*))
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*bc* *tc*))))
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(done))
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(DEFMETHOD CMPS ((brdct reduction) (trdct reduction) &optional dummy)
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(declare (ignore dummy))
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(when (eq (first (orgn brdct)) 'trivial-rdct)
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(return-from cmps trdct))
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(when (eq (first (orgn trdct)) 'trivial-rdct)
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(return-from cmps brdct))
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(with-slots ((tf f) (tg g) (th h)) trdct
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(declare (type morphism tf tg th))
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(with-slots ((bf f) (bg g) (bh h)) brdct
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(declare (type morphism bf bg bh))
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(build-rdct
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:f (cmps bf tf)
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:g (cmps tg bg)
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:h (add th (i-cmps tg bh tf))
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:orgn `(cmps ,brdct ,trdct)))))
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;;
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;; BASIC PERTURBATION LEMMA.
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;;
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(DEFMETHOD ADD ((rdct reduction) (perturbation morphism) &optional dummy)
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(declare (type null dummy))
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(when dummy
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(error "Why a third argument in (ADD REDUCTION MORPHISM)?"))
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(when (eq (grmd (tcc rdct)) (grmd (sorc perturbation)))
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(return-from add
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(basic-perturbation-lemma rdct perturbation)))
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(when (eq (grmd (bcc rdct)) (grmd (sorc perturbation)))
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(return-from add
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(easy-perturbation-lemma rdct perturbation)))
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(error "In (METHOD ADD REDUCTION MORPHISM),~@
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the data do not sound like coherent."))
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(DEFUN BASIC-PERTURBATION-LEMMA (reduction top-perturbation)
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(declare
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(type reduction reduction)
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(type morphism top-perturbation))
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(the (values reduction morphism)
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(with-slots ((old-tcc tcc) (old-bcc bcc)
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(old-f f) (old-g g) (old-h h)) reduction
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(declare
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(type chain-complex old-tcc old-bcc)
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(type morphism old-f old-g old-h))
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(when (eq (first (orgn top-perturbation)) 'zero-mrph)
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(return-from basic-perturbation-lemma
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(values reduction (zero-mrph old-bcc))))
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(when (eq (first (orgn (h reduction))) 'zero-mrph)
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(return-from basic-perturbation-lemma
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(values
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(trivial-rdct (add old-tcc top-perturbation))
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(i-cmps old-f top-perturbation old-g))))
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(let* ((sigma (bpl-*-sigma old-h top-perturbation))
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(new-f (cmps old-f
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(sbtr (idnt-mrph old-tcc)
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(i-cmps
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top-perturbation
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sigma
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old-h))))
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(new-g (cmps sigma old-g))
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(new-h (cmps sigma old-h))
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(bottom-perturbation (i-cmps
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old-f
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top-perturbation
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new-g))
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(new-tcc (add old-tcc top-perturbation))
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(new-bcc (add old-bcc bottom-perturbation)))
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(declare
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(type chain-complex new-tcc new-bcc)
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(type morphism sigma new-f new-g new-h bottom-perturbation))
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(setf new-f (dstr-change-sorc-trgt new-f :new-sorc new-tcc
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:new-trgt new-bcc)
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new-g (dstr-change-sorc-trgt new-g :new-sorc new-bcc
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:new-trgt new-tcc)
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new-h (dstr-change-sorc-trgt new-h :new-sorc new-tcc
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:new-trgt new-tcc))
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(values
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(build-rdct :f new-f :g new-g :h new-h
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:orgn `(basic-perturbation-lemma ,reduction
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,top-perturbation))
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bottom-perturbation)))))
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(DEFUN EASY-PERTURBATION-LEMMA (reduction bottom-perturbation)
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(declare
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(type reduction reduction)
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(type morphism bottom-perturbation))
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(the (values reduction morphism)
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(with-slots ((old-tcc tcc) (old-bcc bcc)
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(old-f f) (old-g g) (old-h h)) reduction
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(declare
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(type chain-complex old-tcc old-bcc)
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(type morphism old-f old-g old-h))
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(when (eq 'zero-mrph (first (orgn bottom-perturbation)))
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(return-from easy-perturbation-lemma
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(values reduction (zero-mrph old-tcc))))
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(when (eq 'trivial-rdct (first (orgn reduction)))
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(return-from easy-perturbation-lemma
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(trivial-rdct (add (bcc reduction) bottom-perturbation))))
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(let ((top-perturbation (i-cmps old-g bottom-perturbation old-f)))
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(declare (type morphism top-perturbation))
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(let ((new-bcc (add old-bcc bottom-perturbation))
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(new-tcc (add old-tcc top-perturbation)))
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(declare (type chain-complex new-bcc new-tcc))
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(values
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(build-rdct
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:f (dstr-change-sorc-trgt old-f :new-sorc new-tcc
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:new-trgt new-bcc)
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:g (dstr-change-sorc-trgt old-g :new-sorc new-bcc
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:new-trgt new-tcc)
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:h (dstr-change-sorc-trgt old-h :new-sorc new-tcc
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:new-trgt new-tcc)
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:orgn `(easy-perturbation-lemma ,reduction
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,bottom-perturbation))
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top-perturbation))))))
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(DEFUN SPECIAL-BPL (reduction top-perturbation)
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(declare
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(type reduction reduction)
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(type morphism top-perturbation))
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(when (eq (first (orgn top-perturbation)) 'zero-mrph)
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(return-from special-bpl reduction))
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(the reduction
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(with-slots ((old-tcc tcc) (old-bcc bcc)
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(old-f f) (old-g g) (old-h h)) reduction
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(declare
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(type chain-complex old-tcc old-bcc)
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(type morphism old-f old-g old-h))
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(let* ((sigma (bpl-*-sigma old-h top-perturbation))
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(new-f (cmps old-f
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(sbtr (idnt-mrph old-tcc)
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(i-cmps
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top-perturbation
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sigma
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old-h))))
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(new-h (cmps sigma old-h))
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(new-tcc (add old-tcc top-perturbation)))
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(declare
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(type chain-complex new-tcc)
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(type morphism sigma new-f new-h))
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(setf new-f (dstr-change-sorc-trgt new-f :new-sorc new-tcc)
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old-g (dstr-change-sorc-trgt old-g :new-trgt new-tcc)
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new-h (dstr-change-sorc-trgt new-h :new-sorc new-tcc
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:new-trgt new-tcc))
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(build-rdct :f new-f :g old-g :h new-h
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:orgn `(special-bpl ,reduction ,top-perturbation))))))
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|
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(DEFUN SPECIAL-BPL-2 (reduction top-perturbation)
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(declare
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|
(type reduction reduction)
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|
(type morphism top-perturbation))
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|
(the reduction
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|
(with-slots ((old-tcc tcc) (old-bcc bcc)
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|
(old-f f) (old-g g) (old-h h)) reduction
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|
(declare
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|
(type chain-complex old-tcc old-bcc)
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|
(type morphism old-f old-g old-h))
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|
(when (eq (first (orgn top-perturbation)) 'zero-mrph)
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|
(return-from special-bpl-2
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|
(values reduction (zero-mrph old-bcc))))
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|
(when (eq (first (orgn (h reduction))) 'zero-mrph)
|
|
(return-from special-bpl-2
|
|
(values
|
|
(trivial-rdct (add old-tcc top-perturbation))
|
|
(i-cmps old-f top-perturbation old-g))))
|
|
(let* ((sigma (bpl-*-sigma old-h top-perturbation))
|
|
(new-g (cmps sigma old-g))
|
|
(new-h (cmps sigma old-h))
|
|
(new-tcc (add old-tcc top-perturbation)))
|
|
(declare
|
|
(type chain-complex new-tcc)
|
|
(type morphism sigma new-g new-h))
|
|
(setf old-f (dstr-change-sorc-trgt old-f :new-sorc new-tcc)
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|
new-g (dstr-change-sorc-trgt new-g :new-trgt new-tcc)
|
|
new-h (dstr-change-sorc-trgt new-h :new-sorc new-tcc
|
|
:new-trgt new-tcc))
|
|
(values
|
|
(build-rdct :f old-f :g new-g :h new-h
|
|
:orgn `(special-bpl-2 ,reduction ,top-perturbation)))))))
|
|
|
|
(DEFUN BPL-*-sigma (homotopy perturbation)
|
|
(declare (type morphism homotopy perturbation))
|
|
(the morphism
|
|
(let ((cmpr (cmpr (sorc perturbation)))
|
|
(h-delta (cmps homotopy perturbation)))
|
|
(declare
|
|
(type cmprf cmpr)
|
|
(type morphism h-delta))
|
|
(flet
|
|
((sigma-* (degr gnrt)
|
|
(declare
|
|
(fixnum degr)
|
|
(type gnrt gnrt))
|
|
(do ((rslt (zero-cmbn degr) (2cmbn-add cmpr rslt iterated))
|
|
(iterated (term-cmbn degr 1 gnrt)
|
|
(cmbn-opps (cmbn-? h-delta iterated))))
|
|
((cmbn-zero-p iterated) rslt)
|
|
(declare (type cmbn rslt iterated)))))
|
|
(build-mrph
|
|
:sorc (sorc homotopy) :trgt (sorc homotopy) :degr 0
|
|
:intr #'sigma-*
|
|
:strt :gnrt
|
|
:orgn `(bpl-*-sigma ,homotopy ,perturbation))))))
|
|
|
|
(DEFMETHOD ADD ((hmeq homotopy-equivalence) (lb-perturbation morphism)
|
|
&optional dummy)
|
|
(declare (ignore dummy))
|
|
(the homotopy-equivalence
|
|
(with-slots (lrdct rrdct) hmeq
|
|
(declare (type reduction lrdct rrdct))
|
|
(multiple-value-bind (new-lrdct top-perturbation)
|
|
(add lrdct lb-perturbation)
|
|
(declare
|
|
(type reduction new-lrdct)
|
|
(type morphism lb-perturbation))
|
|
(build-hmeq
|
|
:lrdct new-lrdct
|
|
:rrdct (add rrdct top-perturbation)
|
|
:orgn `(add ,hmeq ,lb-perturbation))))))
|