Functions added for computing spectral sequences groups and differential matrices.
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@ -182,7 +182,7 @@
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#|
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(setq gnrt-list (list '(2 4 6 8) '(0 1 0 0) '(0 3 3 3)))
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(Gnrt-List-to-Mtrx gnrt-list)
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(gnrt-list-to-mtrx gnrt-list)
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|#
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;; Returns the list of columns of the matrix mtrx.
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@ -357,9 +357,7 @@
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(type filtered-chain-complex fltrcm)
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(type fixnum r p q))
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(the matrix
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(let* ((cmpr (cmpr1 fltrcm))
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(dffr (dffr fltrcm))
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(degr (+ p q))
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(let* ((degr (+ p q))
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(sbasis (fltrd-basis fltrcm degr p))
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(tbasis
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(mapcan
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@ -372,8 +370,6 @@
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;; consider (<a-b> (1+ (-p r)) t(degr),
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;; where t(degr)=max{flin(x), x \in X_degr}.
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(declare
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(type cmprf cmpr)
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(type morphism dffr)
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(type fixnum degr)
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(list sbasis tbasis))
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(let ((srank (length sbasis))
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@ -401,8 +397,8 @@
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(setq kz2 (k-z2-1))
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(setq fcc (bar kz2))
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(Change-Chcm-TO-Flcc fcc abar-flin `(abar-flin))
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(FLTR-CHCM-Z-MTRX fcc 2 3 2)
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(FLTR-CHCM-Z-MTRX fcc 1 2 4)
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(fltr-chcm-z-mtrx fcc 2 3 2)
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(fltr-chcm-z-mtrx fcc 1 2 4)
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|#
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@ -430,8 +426,8 @@
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all-basis-gnrt-list
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(kernel mat))))))))
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#|
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(Fltr-Chcm-Z-Gnrt-List fcc 2 3 2)
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(Fltr-Chcm-Z-Gnrt-List fcc 1 2 4)
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(fltr-chcm-z-gnrt-list fcc 2 3 2)
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(fltr-chcm-z-gnrt-list fcc 1 2 4)
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|#
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;; Returns a list of lists with the coordinates of the elements of
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@ -814,7 +810,7 @@
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;; on each of the components of E^r_{p-r,q+r-1} of the differential
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;; d: E^r_{p,q} --> E^r_{p-r,q+r-1} applied to the element of E^r_{p,q}
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;; which has as coordinates the list int-list. (Only for finitely generated complexes)
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(DEFUN EFF-SPSQ-DFFR (fltrcm r p q int-list)
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(DEFUN EFF-SPSQ-DFFR-OF-ONE-ELEMENT (fltrcm r p q int-list)
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(declare
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(type filtered-chain-complex fltrcm)
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(type fixnum r p q)
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@ -826,11 +822,11 @@
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(tbasis (fltrd-basis fltrcm (1- degr) (- p r)))
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(num-mtrx (spsq-num-mtrx fltrcm r p q))
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(den-mtrx (spsq-den-mtrx fltrcm r p q))
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(Z-gnrt-list (Fltr-Chcm-Z-Gnrt-List fltrcm r p q))
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(z-gnrt-list (fltr-chcm-z-gnrt-list fltrcm r p q))
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(s-basis-dvs (mtrx-quotient (copy-mtrx num-mtrx) den-mtrx))
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(s-gnrt-list (first s-basis-dvs))
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(s-dvs (second s-basis-dvs))
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(t-basis-dvs (eff-spsq-basis-dvs fltrcm r (- p r) (1- (+ q r))))
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(t-basis-dvs (spsq-basis-dvs fltrcm r (- p r) (1- (+ q r))))
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(t-gnrt-list (first t-basis-dvs))
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(t-dvs (second t-basis-dvs))
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(selement (make-list (length sbasis) :initial-element 0))
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@ -905,13 +901,123 @@
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rslt)))))
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#|
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(eff-spsq-dffr fcc 1 2 3 '(1))
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(eff-spsq-dffr fcc 1 2 5 '(0 1))
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(eff-spsq-dffr fcc 1 2 5 '(1 0))
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(eff-spsq-dffr fcc 1 3 4 '(1 0))
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(eff-spsq-dffr fcc 1 3 4 '(0 1))
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(eff-spsq-dffr-of-one-element fcc 1 2 3 '(1))
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(eff-spsq-dffr-of-one-element fcc 1 2 5 '(0 1))
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(eff-spsq-dffr-of-one-element fcc 1 2 5 '(1 0))
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(eff-spsq-dffr-of-one-element fcc 1 3 4 '(1 0))
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(eff-spsq-dffr-of-one-element fcc 1 3 4 '(0 1))
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|#
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;; Function that returns the matrix (as a list of lists) of the differential
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;; d: E^r_{p,q} --> E^r_{p-r,q+r-1} (Only for finitely generated complexes)
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(DEFUN EFF-SPSQ-DFFR-MTRX (fltrcm r p q )
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(declare
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(type filtered-chain-complex fltrcm)
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(type fixnum r p q))
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(the list
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(let* ((cmpr (cmpr1 fltrcm))
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(degr (+ p q))
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(sbasis (fltrd-basis fltrcm degr p))
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(tbasis (fltrd-basis fltrcm (1- degr) (- p r)))
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(num-mtrx (spsq-num-mtrx fltrcm r p q))
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(den-mtrx (spsq-den-mtrx fltrcm r p q))
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(Z-gnrt-list (Fltr-Chcm-Z-Gnrt-List fltrcm r p q))
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(s-basis-dvs (mtrx-quotient (copy-mtrx num-mtrx) den-mtrx))
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(s-gnrt-list (first s-basis-dvs))
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(s-dvs (second s-basis-dvs))
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(t-basis-dvs (eff-spsq-basis-dvs fltrcm r (- p r) (1- (+ q r))))
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(t-gnrt-list (first t-basis-dvs))
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(t-dvs (second t-basis-dvs))
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(s-length (length
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(mapcan #'(lambda (i)
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(if (not (eql 1 i)) (list i) nil))
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s-dvs)))
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(base-list nil))
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(declare
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(type cmprf cmpr)
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(type fixnum degr s-length)
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(type list sbasis tbasis z-gnrt-list s-basis-dvs s-gnrt-list s-dvs
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t-basis-dvs t-gnrt-list t-dvs)
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(type matrix num-mtrx den-mtrx))
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(progn
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(dotimes (i s-length)
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(push (nconc (make-list (1- (- s-length i)) :initial-element 0) (list 1)
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(make-list i :initial-element 0))
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base-list ))
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(mapcar #'(lambda (int-list)
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(let* ((selement (make-list (length sbasis) :initial-element 0))
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(cmbn (cmbn degr))
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)
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(declare
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(type list selement)
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(type cmbn cmbn))
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(progn
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(labels ((2list-add (list1 list2)
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(declare (type list list1 list2))
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(the list
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(mapcar #'+ list1 list2)))
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(n-list (n list)
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(declare
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(type list list)
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(type fixnum n))
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(the list
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(mapcar #'(lambda (i)
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(* n i))
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list))))
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(do ((mark1 s-gnrt-list (cdr mark1))
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(mark2 s-dvs (cdr mark2))
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(mark3 int-list))
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((endp mark1))
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(declare (list mark1 mark2 mark3))
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(if (not (eql 1 (car mark2)))
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(progn
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(setq selement (2list-add selement (n-list (car mark3) (car mark1))))
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(pop mark3)))))
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(let* ((selt-coord (vctr-coordinates selement num-mtrx)))
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(declare (list selt-coord))
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(do ((mark1 z-gnrt-list (cdr mark1))
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(mark2 selt-coord (cdr mark2)))
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((endp mark1))
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(declare (list mark1 mark2))
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(if (not (eql 0 (car mark2)))
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(setq cmbn (2cmbn-add cmpr cmbn (n-cmbn (car mark2)
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(funcall
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#'(lambda (int-list)
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(declare (list int-list))
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(the cmbn
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(let* ((cmbn2 (cmbn degr)))
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(declare
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(type cmbn cmbn))
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(do ((mark3 int-list (cdr mark3))
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(mark4 sbasis (cdr mark4)))
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((or (endp mark3) (endp mark4)))
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(declare (type list mark3 mark4))
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(if (not (= 0 (car mark3)))
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(setq cmbn2 (2cmbn-add cmpr cmbn2
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(cmbn degr (car mark3) (car mark4))))))
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cmbn2)))
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(car mark1))))))))
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(let* ((dffr-cmbn (dffr fltrcm cmbn))
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(crdnt (cmbn-coordinates dffr-cmbn t-gnrt-list tbasis cmpr))
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(rslt nil))
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(declare
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(type cmbn dffr-cmbn)
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(type list crdnt rslt))
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(do ((mark1 t-dvs (cdr mark1))
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(mark2 crdnt (cdr mark2)))
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((endp mark1))
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(declare (list mark1 mark2))
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(if (not (eql 1 (car mark1)))
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(if (eql 0 (car mark1))
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(setq rslt (nconc rslt (list (car mark2))))
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(setq rslt (nconc rslt (list (mod (car mark2) (car mark1))))))))
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rslt))))
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base-list)))))
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;; CONVERGENCE OF THE SPECTRAL SEQUENCE
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;; Function that determines if at level r and for degree n the E^r_{p,q} of the
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@ -1009,7 +1115,7 @@
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;; when the top complex is effective (we need to obtain the elements in the basis of this complex).
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(DEFUN HMTP-EQ-FLTR-ORDER (hmtp-eq degr)
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(declare
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(type HOMOTOPY-EQUIVALENCE hmtp-eq)
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(type homotopy-equivalence hmtp-eq)
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(type fixnum degr))
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(let* ((lcc (lbcc hmtp-eq))
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(tcc (tcc hmtp-eq))
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@ -1035,11 +1141,11 @@
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(setq k3 (k-z2 3))
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(Setq hmtp-eq3 (efhm k3))
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(setq ek3 (rbcc hmtp-eq3))
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(Change-Chcm-TO-Flcc ek3 :flin abar-flin :orgn `(filtered-chain-complex ,ek3))
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(change-chcm-to-flcc ek3 :flin abar-flin :orgn `(filtered-chain-complex ,ek3))
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(setf k3-flin
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#'(lambda (degr gnrt)
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(flin ek3 (rf hmtp-eq3 (lg hmtp-eq3 degr gnrt)))))
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(Change-Chcm-TO-Flcc k3 :flin k3-flin :orgn `(filtered-chain-complex ,k3))
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(change-chcm-to-flcc k3 :flin k3-flin :orgn `(filtered-chain-complex ,k3))
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(hmtp-eq-fltr-order hmtp-eq3 3)
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(hmtp-eq-fltr-order hmtp-eq3 5)
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|#
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@ -1069,9 +1175,9 @@
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;; complex translating the filtration in the original filtered complex FltrCm, suposing that the
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;; homotopy in the right side is compatible with the filtration in the left complex for such r (we
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;; supose that flin(rh(lf(y)))-flin(lf(y))<r for every y in the top complex).
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(DEFMETHOD SPSQ-BASIS-DVS ((FltrCm filtered-chain-complex) r p q)
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(DEFMETHOD SPSQ-BASIS-DVS ((fltrcm FILTERED-CHAIN-COMPLEX) r p q)
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(declare
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(type fixnum r p q))
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(type fixnum r p q))
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(let* ((hmtp-eq (efhm fltrcm))
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(eff-chcm (rbcc hmtp-eq))
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)
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@ -1126,8 +1232,7 @@
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(type filtered-chain-complex fltrcm)
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(type fixnum r p q))
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(let* ((hmtp-eq (efhm fltrcm))
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(eff-chcm (rbcc hmtp-eq))
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)
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(eff-chcm (rbcc hmtp-eq)))
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(declare
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(type homotopy-equivalence hmtp-eq)
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(type chain-complex eff-chcm)
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@ -1140,23 +1245,42 @@
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;; Method that returns the list of integers which correspond to the coordinates
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;; on each of the components of E^r_{p-r,q+r-1} of the differential
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;; d: E^r_{p,q} --> E^r_{p-r,q+r-1} applied to the element of E^r_{p,q}
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;; which has as coordinates the list int-list.
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(DEFMETHOD SPSQ-DFFR ((fltrcm FILTERED-CHAIN-COMPLEX) r p q int-list)
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;; which has as coordinates the list int-list, using the effective homology
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;; when the complex is not of finite type.
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(DEFMETHOD SPSQ-DFFR-OF-ONE-ELEMENT ((fltrcm FILTERED-CHAIN-COMPLEX) r p q int-list)
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(declare
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(type filtered-chain-complex fltrcm)
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(type fixnum p q r)
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(type list int-list))
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(let* ((hmtp-eq (efhm fltrcm))
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(eff-chcm (rbcc hmtp-eq))
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)
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(eff-chcm (rbcc hmtp-eq)))
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(declare
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(type homotopy-equivalence hmtp-eq)
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(type chain-complex eff-chcm)
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)
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(type chain-complex eff-chcm))
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(progn
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(if (not (typep eff-chcm 'filtered-chain-complex))
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(translate-filtration hmtp-eq))
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(eff-spsq-dffr eff-chcm r p q int-list))))
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(eff-spsq-dffr-of-one-element eff-chcm r p q int-list))))
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;; Function that returns the matrix (as a list of lists) of the differential
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;; d: E^r_{p,q} --> E^r_{p-r,q+r-1}, using the effective homology
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;; when the complex is not of finite type.
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(DEFMETHOD SPSQ-DFFR-MTRX ((fltrcm FILTERED-CHAIN-COMPLEX) r p q)
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(declare
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(type filtered-chain-complex fltrcm)
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(type fixnum p q r))
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(let* ((hmtp-eq (efhm fltrcm))
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(eff-chcm (rbcc hmtp-eq)))
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(declare
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(type homotopy-equivalence hmtp-eq)
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(type chain-complex eff-chcm))
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(progn
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(if (not (typep eff-chcm 'filtered-chain-complex))
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(translate-filtration hmtp-eq))
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(eff-spsq-dffr-mtrx eff-chcm r p q))))
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;; Function that determines the level r at which for degree n the E^r_{p,q} (with p+q=degr)
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;; of the filtered complex fltrcm (with effective homology) are equal to E^\infty_{p,q} (i.e., it
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@ -1178,9 +1302,6 @@
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(eff-spsq-cnvg eff-chcm degr))))
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;; Homology filtration
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@ -1322,11 +1443,6 @@
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(eff-hmlg-fltr eff-chcm degr p))))
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;;(DEFTYPE ss-type () '(member :serre :eilenberg-moore :filtration))
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(DEFCLASS SPECTRAL-SEQUENCE ()
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(;; FiLTeRedchainCoMplex
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(fltrcm :type filtered-chain-complex :initarg :fltrcm :reader fltrcm)
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@ -1358,11 +1474,12 @@
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(DEFUN SS (idnm)
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(declare (type fixnum idnm))
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(the (or SPECTRAL-SEQUENCE null)
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(the (or spectral-sequence null)
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(find idnm *ss-list* :key #'idnm)))
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(DEFUN BUILD-ss (fltrcm orgn)
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;;; Function to build a spectral sequences from a filtered chain complex and an origin
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(DEFUN BUILD-SS (fltrcm orgn)
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(declare
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(type filtered-chain-complex fltrcm)
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(type list orgn))
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@ -1373,7 +1490,7 @@
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(when already
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(return-from build-ss already)))
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(let ((ss (make-instance 'spectral-sequence
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:fltrcm fltrcm
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:fltrcm fltrcm
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:orgn orgn
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:group-rslts +empty-list+
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:dffr-rslts +empty-list+)))
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@ -1382,16 +1499,17 @@
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ss))))
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;; Method that returns the representation basis-divisors of E^r_{p,q} of the spectral sequence
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(DEFMETHOD SPSQ-BASIS-DVS ((ss SPECTRAL-SEQUENCE) r p q)
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(declare
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(type fixnum r p q))
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(let* ((fltrcm (fltrcm ss))
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(rslts (group-rslts ss))
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(pos (position (list r p q nil) rslts :test #'(lambda (l1 l2)
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(and
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(eq (first l1) (first l2))
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(eq (second l1) (second l2))
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(eq (third l1) (third l2)))))))
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(and
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(eq (first l1) (first l2))
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(eq (second l1) (second l2))
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(eq (third l1) (third l2)))))))
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(the list
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(if pos (fourth (nth pos rslts))
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(let ((basis-dvs (spsq-basis-dvs fltrcm r p q)))
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@ -1402,7 +1520,22 @@
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basis-dvs))))))
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(DEFMETHOD SPSQ-GROUP ((ss SPECTRAL-SEQUENCE) r p q)
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;; Method that returns the list of abelian invariants of the group E^r_{p,q} of the spectral sequence
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(DEFMETHOD SPECTRAL-SEQUENCE-GROUP ((ss SPECTRAL-SEQUENCE) r p q)
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(declare
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(type spectral-sequence ss)
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(type fixnum r p q))
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(let* ((basis-dvs (SPSQ-BASIS-DVS SS r p q))
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(divs (second basis-dvs)))
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(declare (list basis-dvs divs))
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(mapcan #'(lambda (i)
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(if (eq 1 i) nil
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(list i)))
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divs)))
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;; Method that prints the spectral sequence group
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(DEFMETHOD PRINT-SPSQ-GROUP ((ss SPECTRAL-SEQUENCE) r p q)
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(declare
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(type spectral-sequence ss)
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(type fixnum r p q))
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@ -1425,7 +1558,7 @@
|
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;; on each of the components of E^r_{p-r,q+r-1} of the differential
|
||||
;; d: E^r_{p,q} --> E^r_{p-r,q+r-1} applied to the element of E^r_{p,q}
|
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;; which has as coordinates the list int-list.
|
||||
(DEFMETHOD SPSQ-DFFR ((ss SPECTRAL-SEQUENCE) r p q int-list)
|
||||
(DEFMETHOD SPECTRAL-SEQUENCE-DIFFERENTIAL-OF-ONE-ELEMENT ((ss SPECTRAL-SEQUENCE) r p q int-list)
|
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(declare
|
||||
(type spectral-sequence ss)
|
||||
(type fixnum p q r)
|
||||
@ -1433,14 +1566,14 @@
|
||||
(let* ((fltrcm (fltrcm ss))
|
||||
(rslts (dffr-rslts ss))
|
||||
(pos (position (list r p q int-list nil) rslts :test #'(lambda (l1 l2)
|
||||
(and
|
||||
(eq (first l1) (first l2))
|
||||
(eq (second l1) (second l2))
|
||||
(eq (third l1) (third l2))
|
||||
(eq (fourth l1) (fourth l2)))))))
|
||||
(and
|
||||
(eq (first l1) (first l2))
|
||||
(eq (second l1) (second l2))
|
||||
(eq (third l1) (third l2))
|
||||
(eq (fourth l1) (fourth l2)))))))
|
||||
(the list
|
||||
(if pos (fifth (nth pos rslts))
|
||||
(let ((l (spsq-dffr fltrcm r p q int-list)))
|
||||
(let ((l (spsq-dffr-of-one-element fltrcm r p q int-list)))
|
||||
(declare (type list l))
|
||||
(progn
|
||||
(push (list r p q int-list l) rslts)
|
||||
@ -1448,7 +1581,34 @@
|
||||
l))))))
|
||||
|
||||
|
||||
(DEFUN SERRE-SS (f)
|
||||
;; Function that returns the matrix (as a list of lists) of the differential
|
||||
;; d: E^r_{p,q} --> E^r_{p-r,q+r-1}, using the effective homology
|
||||
;; when the complex is not of finite type.
|
||||
(DEFMETHOD SPECTRAL-SEQUENCE-DIFFERENTIAL-MATRIX ((ss SPECTRAL-SEQUENCE) r p q)
|
||||
(declare
|
||||
(type spectral-sequence ss)
|
||||
(type fixnum p q r))
|
||||
(let* ((fltrcm (fltrcm ss))
|
||||
(rslts (dffr-rslts ss))
|
||||
(pos (position (list r p q nil) rslts :test #'(lambda (l1 l2)
|
||||
(and
|
||||
(eq (first l1) (first l2))
|
||||
(eq (second l1) (second l2))
|
||||
(eq (third l1) (third l2))
|
||||
)))))
|
||||
(the list
|
||||
(if pos (fourth (nth pos rslts))
|
||||
(let ((l (spsq-dffr-mtrx fltrcm r p q )))
|
||||
(declare (type list l))
|
||||
(progn
|
||||
(push (list r p q l) rslts)
|
||||
(setf (slot-value ss 'dffr-rslts) rslts)
|
||||
l))))))
|
||||
|
||||
|
||||
;; Funcion that constructs the Serre spectral sequence associated with a fibration f
|
||||
;; Returns an object of the class Spectral-sequence
|
||||
(DEFUN SERRE-SPECTRAL-SEQUENCE (f)
|
||||
(declare (type fibration f))
|
||||
(let* ((fltrcm (fibration-total f))
|
||||
(ecc (rbcc (efhm fltrcm))))
|
||||
@ -1458,7 +1618,10 @@
|
||||
(build-ss fltrcm `(Serre-Spectral-Sequence ,f)))))
|
||||
|
||||
|
||||
(DEFUN EILENBERG-MOORE-SS (x)
|
||||
;; Funcion that constructs the Eilenberg-Moore spectral sequence associated with a (1-reduced)
|
||||
;; simplicial set X
|
||||
;; Returns an object of the class Spectral-sequence
|
||||
(DEFUN EILENBERG-MOORE-SPECTRAL-SEQUENCE (x)
|
||||
(declare (type simplicial-set x))
|
||||
(let* ((ox (loop-space x))
|
||||
(ecc (rbcc (efhm OX))))
|
||||
@ -1475,29 +1638,45 @@
|
||||
(setf F3 (z-whitehead s3 k3)))
|
||||
|
||||
|
||||
(setf sss (serre-ss f3))
|
||||
(setf ss1 (serre-spectral-sequence f3))
|
||||
|
||||
|
||||
(setf r 2)
|
||||
|
||||
(dotimes (n 8)
|
||||
(dotimes (p (1+ n))
|
||||
(let ((q (- n p)))
|
||||
(spsq-group sss r p q))))
|
||||
|
||||
(dotimes (n 8)
|
||||
(dotimes (p (1+ n))
|
||||
(let ((q (- n p)))
|
||||
;;(print-spsq-group ss1 r p q)
|
||||
(format t "Spectral sequence E^~D_{~D,~D}" r p q)
|
||||
(print (spectral-sequence-group ss1 r p q))
|
||||
(terpri))))
|
||||
|
||||
(spectral-sequence-differential-matrix ss1 3 3 0)
|
||||
(spectral-sequence-differential-matrix ss1 3 3 2)
|
||||
(spectral-sequence-differential-matrix ss1 3 3 4)
|
||||
|
||||
|
||||
(cat-init)
|
||||
(setf ss2 (eilenberg-moore-ss (loop-space (sphere 3))))
|
||||
(setf effox (rbcc (efhm (fltrcm ss2))))
|
||||
(setf x (loop-space (sphere 3)))
|
||||
(setf ss2 (eilenberg-moore-spectral-sequence x))
|
||||
|
||||
(dotimes (n 6)
|
||||
(dotimes (p (1+ n))
|
||||
(let ((q (+ n p)))
|
||||
(print-spsq-group ss2 1 (- p) q))))
|
||||
|
||||
(spectral-sequence-differential-matrix ss2 1 -2 8)
|
||||
(spectral-sequence-differential-of-one-element ss2 1 -2 8 '(1 0 0))
|
||||
(spectral-sequence-differential-of-one-element ss2 1 -2 8 '(0 1 0))
|
||||
(spectral-sequence-differential-of-one-element ss2 1 -2 8 '(0 0 1))
|
||||
|
||||
|
||||
(spectral-sequence-differential-of-one-element ss2 1 -1 6 '(1))
|
||||
(spectral-sequence-differential-matrix ss2 1 -1 6)
|
||||
(spectral-sequence-differential-matrix ss2 1 -2 6)
|
||||
|
||||
(spsq-group effOX 1 -2 6)
|
||||
|
||||
(dotimes (n 9)
|
||||
(dotimes (p (1+ n))
|
||||
(let ((q (+ n p)))
|
||||
(spsq-group ss2 1 (- p) q))))
|
||||
|
||||
|#
|
||||
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user