From 086bbc82bcb1cf31c1a54902e2e68f5f5bcb00e8 Mon Sep 17 00:00:00 2001 From: Ana Romero Date: Thu, 11 Jul 2019 13:52:38 +0200 Subject: [PATCH] Serre spectral sequence of a product added Serre-Whitehead spectral sequence added Problems solved in file homotopy (reductions are now compatible with filtrations) --- src/anromero/homotopy.lisp | 12 ++-- src/anromero/spectral-sequences.lisp | 97 ++++++++++++++++++++++++---- 2 files changed, 94 insertions(+), 15 deletions(-) diff --git a/src/anromero/homotopy.lisp b/src/anromero/homotopy.lisp index 42a3e90..7c8bc22 100644 --- a/src/anromero/homotopy.lisp +++ b/src/anromero/homotopy.lisp @@ -119,21 +119,24 @@ (cond ((equal (car list) 1) (let* ((ch (chml-clss chcm indx)) (fib (z-whitehead chcm ch)) (ft (fibration-total fib))) + (if (endp (cdr list)) ft (progn (kill-epis ft 1 indx) - (construct-space-iterative ft (cdr list) indx)))) + (construct-space-iterative ft (cdr list) indx))))) ((equal (car list) 2) (let* ((ch (chml-clss chcm indx)) (fib (z2-whitehead chcm ch)) (ft (fibration-total fib))) + (if (endp (cdr list)) ft (progn (kill-epis ft 1 indx) - (construct-space-iterative ft (cdr list) indx)))) + (construct-space-iterative ft (cdr list) indx))))) (t (let* ((ch (chml-clss chcm indx)) (fib (zp-whitehead (car list) chcm ch)) (ft (fibration-total fib))) + (if (endp (cdr list)) ft (progn - (kill-epis ft 1 indx) - (construct-space-iterative ft (cdr list) indx)))) + (kill-epis ft 1 indx) + (construct-space-iterative ft (cdr list) indx))))) ))) @@ -205,3 +208,4 @@ + diff --git a/src/anromero/spectral-sequences.lisp b/src/anromero/spectral-sequences.lisp index 2356592..ef80e41 100644 --- a/src/anromero/spectral-sequences.lisp +++ b/src/anromero/spectral-sequences.lisp @@ -1581,7 +1581,7 @@ l)))))) -;; Function that returns the matrix (as a list of lists) of the differential +;; Method 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) @@ -1606,28 +1606,65 @@ l)))))) -;; Funcion that constructs the Serre spectral sequence associated with a fibration f +;; Function that constructs the Serre spectral sequence associated with a fibration f ;; Returns an object of the class Spectral-sequence -(DEFUN SERRE-SPECTRAL-SEQUENCE (f) +(DEFUN SERRE-SPECTRAL-SEQUENCE-FIBRATION (f) (declare (type fibration f)) (let* ((fltrcm (fibration-total f)) (ecc (rbcc (efhm fltrcm)))) + (declare (type simplicial-set fltrcm) + (type chain-complex ecc)) (progn (change-chcm-to-flcc fltrcm crpr-flin 'crpr-flin) (change-chcm-to-flcc ecc tnpr-flin 'tnpr-flin) - (build-ss fltrcm `(Serre-Spectral-Sequence ,f))))) + (the spectral-sequence + (build-ss fltrcm `(Serre-Spectral-Sequence ,f)))))) + +;; Function that constructs the Serre spectral sequence associated with a simplicial set +;; which is given as a (twisted) cartesian product. Returns an object of the class Spectral-sequence +(DEFUN SERRE-SPECTRAL-SEQUENCE-PRODUCT (x) + (declare (type simplicial-set x)) + (let ((ecc (rbcc (efhm x)))) + (declare (type chain-complex ecc)) + (progn + (change-chcm-to-flcc x crpr-flin 'crpr-flin) + (change-chcm-to-flcc ecc tnpr-flin 'tnpr-flin) + (the spectral-sequence + (build-ss x `(Serre-Spectral-Sequence ,x)))))) -;; Funcion that constructs the Eilenberg-Moore spectral sequence associated with a (1-reduced) +;; Function that constructs the Serre spectral sequence of the first fibration of the +;; Whitehead tower of fibrations of a 1-reduced simplicial set +(DEFUN SERRE-WHITEHEAD-SPECTRAL-SEQUENCE (x) + (let* (;; we obtain the first non null homology group + (first-non-null (first-non-null-homology-group x 20)) + (degr (1+ first-non-null)) + (hom (homology-format x degr)) + (ft (construct-space-iterative x (split-components hom) degr))) + (declare (fixnum first-non-null degr) + (type string hom) + (type simplicial-set ft)) + (the spectral-sequence + (serre-spectral-sequence-product ft)))) + + +;; Function 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)))) + (declare + (type simplicial-set ox) + (type chain-complex ecc)) (progn (change-chcm-to-flcc ecc cobar-flin 'cobar-flin) - (build-ss ecc `(Eilenberg-Moore-Spectral-Sequence ,x))))) + (the spectral-sequence + (build-ss ecc `(Eilenberg-Moore-Spectral-Sequence ,x)))))) + + + #| @@ -1638,7 +1675,9 @@ (setf F3 (z-whitehead s3 k3))) -(setf ss1 (serre-spectral-sequence f3)) +(setf ss1 (serre-spectral-sequence-fibration f3)) + +(spectral-sequence-group ss1 2 0 2) (setf r 2) @@ -1660,10 +1699,10 @@ (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)))) +;;(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)) @@ -1676,6 +1715,42 @@ (spectral-sequence-differential-matrix ss2 1 -2 6) +(cat-init) +(setf s3 (sphere 3)) +(setf ss3 (serre-whitehead-spectral-sequence s3)) + +(spectral-sequence-group ss3 2 0 2) + + +(dotimes (n 8) + (dotimes (p (1+ n)) + (let ((q (- n p))) + ;;(print-spsq-group ss3 r p q) + (format t "Spectral sequence E^~D_{~D,~D}" r p q) + (print (spectral-sequence-group ss3 r p q)) + (terpri)))) + + +(spectral-sequence-differential-matrix ss3 3 3 0) +(spectral-sequence-differential-matrix ss3 3 3 2) +(spectral-sequence-differential-matrix ss3 3 3 4) + + + +(cat-init) +(setf s3 (sphere 3) s2 (sphere 2)) +(setf ss4 (serre-spectral-sequence-product (crts-prdc s3 s2))) + +(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 ss4 r p q)) + (terpri)))) + + + |#