54 KiB
(ql:quickload :kenzo)
(in-package "CAT")
(cat-init)
Objects with effective homology¶
Introduction¶
The present chapter describes the set of programming tools for handling objects with effective homology. The theoretical material may be found in the paper Constructive Algebraic Topology by Julio Rubio Garcia and Francis Sergeraert (Available at the web site https://www-fourier.ujf.grenoble.fr/~sergerar/). The terminology used in this chapter is compatible with this reference.). The terminology used in this chapter is compatible with this reference.
Reduction¶
A reduction in Kenzo is a 5-tuple $(\hat C, C, f, g, h)$:
$$ \begin{align} \hat{C}\quad & \qquad\stackrel{h}{\longrightarrow} & \mbox{$^s\hat{C}$} \\ {\scriptstyle f} \downarrow \uparrow {\scriptstyle g} & & \\ C\quad & & \end{align} $$
where $\hat C$ and $C$ are chain complexes, $f$ and $g$ chain complex morphisms and $h$ a homotopy operator. Hereafter, $\hat C$ is called the top chain complex and $C$ the bottom chain complex. ${}^s\hat C$ is $\hat C$ shifted, i.e. $h$ has degree $1$. The mappings $f$, $g$, $h$, together with the differential operator $d$ on $\hat C$, must verify the following relations:
$$ \begin{eqnarray*} f \circ g & = & 1_C \\ h \circ d + d \circ h & = & 1_{\hat C} - g \circ f \\ f \circ h & = & 0 \\ h \circ g & = & 0 \\ h \circ h & = & 0 \end{eqnarray*} $$
The morphisms $f$ and $g$ and the homotopy operator $h$ describe the (big) chain complex $\hat C$ as the direct sum
$$ \hat C= \hat C_1 \oplus \hat C_2$$
where $\hat C_1 = Im(g) \simeq C$ and $\hat C_2= Ker(f)$ ($\hat C_2$ is acyclic).
Representation of a reduction¶
A reduction\index{reduction!representation} is implemented as an instance of the CLOS class REDUCTION, whose definition is:
(DEFCLASS REDUCTION () ;; Top Chain Complex ((tcc :type chain-complex :initarg :tcc :reader tcc1) ;; Bottom Chain Complex (bcc :type chain-complex :initarg :bcc :reader bcc1) (f :type morphism :initarg :f :reader f1) (g :type morphism :initarg :g :reader g1) (h :type morphism :initarg :h :reader h1) ;; IDentification NuMber (idnm :type fixnum :initform (incf *idnm-counter*) :reader idnm) ;; ORiGiN (orgn :type list :initarg :orgn :reader orgn)))
This class has 7 slots:
tcc, the object of typechain-complexrepresenting the chain complex $\hat C$ (top chain complex).bcc, the object of typechain-complexrepresenting the chain complex $C$ (bottom chain complex).f, the object of typemorphismrepresenting the morphism $f$.g, the object of typemorphismrepresenting the morphism $g$.h, the object of typemorphismrepresenting the morphism $h$.idmn, an integer, number plate for the object.orgn, a comment list carefully chosen.
The accessors of the slots are the functions whose name appears after the specifier :reader in the class definition. A printing method has been associated to the class REDUCTION and the external representation of an instance is a string like [Kn Reduction], where n is the number plate of the Kenzo object.
The function build-rdct¶
To facilitate the construction of instances of the REDUCTION class, the
software provides the function build-rdct.
build-rdct :f f :g g :h h :orgn orgn
defined with keyword parameters. The returned value is an instance of the class REDUCTION.
The keyword arguments are:
- f, the object of type
morphismrepresenting the morphism $f$. - g, the object of type
morphismrepresenting the morphism $g$. - h, the object of type
morphismrepresenting the morphism $h$. - orgn, the comment list carefully chosen since the system does not build a new instance of the class if it finds in the list of already built reductions,
*rdct-list*, a reduction with the same comment list.
The tcc slot and the bcc slot of the instance are taken respectively from the sorc slot (source slot) and the trgt slot (target slot) of the morphism $f$. The function build-rdc controls the validity of the degrees of the morphisms and pushes the new created instance on the list *rdct-list*.
Useful macros and functions¶
(cat-init) [Function]
Clear among others, the list
*rdct-list*, list of user created reductions and reset the global counter to 1. See also the description of this function in chapter 1.
(rdct n) [Function]
Retrieve in the list
*rdct-list*the reduction instance whose identification (asKenzoobject) isn. If it does not exist, returnNIL.
(bcc rdct &rest args) [Macro]
With only one argument (a reduction rdct) this macro selects the bottom chain complex of the reduction. Otherwise, it applies the differential of the bottom chain complex of the reduction rdc on the arguments args, (either degree generator or cmb).
(tcc rdct &rest args) [Macro]
With only one argument (a reduction rdct) this macro selects the top chain complex of the reduction. Otherwise, it applies the differential of the top chain complex of the reduction rdc on the arguments args, (either degree generator or cmb).
(f rdct &rest args) [Macro]
With only one argument (a reduction rdct) this macro selects the morphism $f$ of the reduction. Otherwise, it applies the morphism $f$ of the reduction rdc on the arguments args, (either degree generator or cmb).
(g rdct &rest args) [Macro]
With only one argument (a reduction rdct this macro selects the morphism $g$ of the reduction. Otherwise, it applies the morphism $g$ of the reduction rdc on the arguments args, (either degree generator or cmb).
(h rdct &rest args) [Macro]
With only one argument (a reduction rdct) this macro selects the morphism $h$ of the reduction. Otherwise, it applies the morphism $h$ of the reduction rdc on the arguments args, (either degree generator or cmb).
(trivial-rdct chcm) [Function]
Build the trivial reduction involving only the chain complex chcm, as shown in the following diagram, $$ \begin{align} C \quad & \quad\stackrel{Zero} {\longrightarrow} & {}^s C \\ {\scriptstyle {Id}} \downarrow \uparrow {\scriptstyle {Id}} & & \\ C \quad & & \end{align} $$ where the morphism Zero is the zero morphism of degree 1 in the chain complex $C$ and Id is the identity morphism in that chain complex (see the functions
zero-mrphandidnt-mrphin the chain complex chapter).
(cmps brdct trdct) [Method]
Build a new reduction from the two reductions $r1$ and $r2$ (here, respectively brdct and trdct). This is done by a call to
build-rdctwith the following parameters: $$ \begin{eqnarray*} f & = & f_{r1}\circ f_{r2}, \\ g & = & g_{r2}\circ g_{r1}, \\ h & = & h_{r2} + g_{r2} \circ h_{r1} \circ f_{r2}. \end{eqnarray*} $$ The compositions and additions of morphisms are realized respectively by the methodscmps,i-cmpsandadd(see chapter 1). We recall that thetccand thebccslots of this new created reduction are respectively the source and target slots of the new morphim $f$.
Verification functions¶
The two following functions are very helpful to verify the coherence of the the various mappings involved in a reduction. Let us recall the diagram of a reduction: $$ \begin{align} \hat{C} \quad & \quad\stackrel{h}{\longrightarrow} & \mbox{$^s\hat{C}$} \\ {\scriptstyle f} \downarrow \uparrow {\scriptstyle g} & & \\ C \quad & & \end{align} $$
(pre-check-rdct rdct) [Function]
Assign to the following lisp global variables, the morphism instances computed from the morphisms of the reduction rdct, according to the formulas: $$ \begin{align} \mathtt{*tdd*} & = & d_{\hat C} \circ d_{\hat C},\\ \mathtt{*bdd*} & = & d_C \circ d_C , \\ \mathtt{*id-fg*} & = & f \circ g - Id_C, \mathtt{*id-gf-dh-hd*} & = & Id_{\hat C} - g \circ f - (d_{\hat C} \circ h + h \circ d_{\hat C}),\\ \mathtt{*hh*} & = & h \circ h, \\ \mathtt{*fh*} & = & f \circ h, \\ \mathtt{*hg*} & = & h \circ g, \\ \mathtt{*df-fd*} & = & d_C \circ f - f \circ d_{\hat C}, \\ \mathtt{*dg-gd*} & = & d_{\hat C} \circ g - g \circ d_C. \end{align} $$ In these formulas $d_C$ and $d_{\hat C}$ are the respective boundary operators of the chain complexes $C$ and $\hat C$. $Id_C$ and $Id_{\hat C}$ are the identity morphisms on the chain complexes $C$ and $\hat C$ built by the function
idnt-mrph
(check-rdct) [Function]
Map all the morphisms prepared by the function pre-check-rdct on chosen combinations. This function having no parameters, the user must assign to the two lisp global variables:
*tc*and*bc*, valid combinations belonging respectively to $\hat C$ and $C$. The call to this function is simply(check-rdct). This function knows what morphism to apply to the combinations and pauses after each evaluation to allow the user to inspect each result. To resume the execution, the user must enter a blank character. If the morphisms are coherent, the result of each mapping is a null combination.
Example¶
To show an example about the verification functions, we first define a locally effective version
of the standard simplex $\Delta^n$. The following function cdelta builds the standard simplex
in dimension dmns. In using the created chain complex, the user must bear in mind that
the only valid vertices are the vertices numbered from $(0)$ to $(dmns)$.
(defun cdelta (dmns) (build-chcm :cmpr #'l-cmpr :basis :locally-effective :bsgn '(0) :intr-dffr #'(lambda (degr gmsm) (make-cmbn :degr (1- degr) :list (do ((rslt +empty-list+ (cons (cons sign (append (subseq gmsm 0 nark) (subseq gmsm (1+ nark)))) rslt)) (sign 1 (- sign)) (nark 0 (1+ nark))) ((> nark degr) rslt)))) :strt :gnrt :orgn `(locally effective version of C_* delta ,dmns)))
Now, let us define 3 functions make-f, make-g and make-h which build the following respective morphisms between $\Delta^m$ and $\Delta^n$:
(make-f tdms bdms)builds a projection morphism $f$ from $\Delta^{tdms}$ to $\Delta^{bdms}$, $tdms>=bdms$, where the vertices $(0)$ to $(dms)$ of $\Delta^{tdms}$ are applied on the vertices of the same number in $\Delta^{bdms}$ and the vertices $(bdms+1)$ to $(tdms)$ are applied on the vertex $(bdms)$ of $\Delta^{bdms}$.(make-g tdms bdms)builds the injection morphism $g$ from $\Delta^{bdms}$ to $\Delta^{tdms}$ (the slot {\tt :intr} is the identity function).(make-h tdms bdms)builds a homotopy morphism of degree 1, $h$, from $\Delta^{tdms}$ to itself, connecting w.r.t. the homotopy relation, the chain map $g \circ f$ with the identity morphism $Id$.
(defun make-f (tdmns bdmns) (build-mrph :sorc (cdelta tdmns) :trgt (cdelta bdmns) :degr 0 :intr #'(lambda (degr gmsm) (let ((pos (position-if #'(lambda (vertex) (>= vertex bdmns)) gmsm))) (if pos (if (< pos degr) (zero-cmbn degr) (cmbn degr 1 (nconc (butlast gmsm) (list bdmns)))) (cmbn degr 1 gmsm)))) :strt :gnrt :orgn `(projection delta ,tdmns => delta ,bdmns)))
(defun make-g (tdmns bdmns) (build-mrph :sorc (cdelta bdmns) :trgt (cdelta tdmns) :degr 0 :intr #'identity :strt :cmbn :orgn `(injection delta ,bdmns => delta ,tdmns)))
(defun make-h (tdmns bdmns) (build-mrph :sorc (cdelta tdmns) :trgt (cdelta tdmns) :degr +1 :intr #'(lambda (degr gmsm) (let ((pos (position-if #'(lambda (vertex) (>= vertex bdmns)) gmsm))) (if pos (if (member bdmns gmsm) (zero-cmbn (1+ degr)) (cmbn (1+ degr) (-1-expt-n pos) (append (subseq gmsm 0 pos) (list bdmns) (subseq gmsm pos)))) (zero-cmbn (1+ degr))))) :strt :gnrt :orgn `(homotopy for delta ,tdmns => ,bdmns)))
We may now define a function to build a reduction. One has not to define a priori the standard simplices: this is done in all the functions and we know that there is no duplication, because before creation, the system checks, owing to the comment list, if the instance of a class (here a chain complex) already exists.
(defun make-rdct (tdmns bdmns) (let ((rdct (build-rdct :f (make-f tdmns bdmns) :g (make-g tdmns bdmns) :h (make-h tdmns bdmns) :orgn `(reduction delta ,tdmns ,bdmns)))) rdct))
We now build with a simple call to the function make-rdct the chain complexes corresponding
to $\Delta^6$ and $\Delta^3$, the 3 morphisms $f$, $g$, $h$ and the reduction. The lisp utility function inspect gives an idea of the organisation of the class instance. Then we verify the coherency of the reduction with the functions pre-check-rdct and check-rdct.
(def rdct (make-rdct 6 3))
(inspect rdct)
(orgn rdct)
(pre-check-rdct rdct)
(setf *tc* (cmbn 2 1 '(0 1 2) 10 '(1 2 3) 100 '(1 2 4) 1000 '(2 3 4)))
(setf *bc* (cmbn 3 4 '(0 1 2 3)))
(check-rdct-no-wait)
We do now the same thing with a more complicated reduction which is the composition of the two following reductions:
(def trdct (make-rdct 6 4))
(def brdct (make-rdct 4 3))
(setf rdct (cmps brdct trdct))
(pre-check-rdct rdct)
(check-rdct-no-wait)
Homotopy equivalence¶
A homotopy equivalence between two chain complexes $C$ and $EC$ is a pair of reductions:
$$ \begin{align} & \qquad\hat{C} & \\ \rho_1\swarrow\nearrow & & \nwarrow\searrow \rho_2 \\ C \quad & & EC\quad \end{align} $$
If $C$ and $EC$ are free $\mathbb{Z}$-chain complexes, a usual chain equivalence between them can be organized in this way. Frequently the chain complexes $C$ and $\hat C$ are locally effective and on the contrary, the chain complex $EC$ is effective, so that $EC$ can be understood as a description of the homology of $C$. More precisely, $EC$ is a tool allowing one to compute the homology of $C$. The chain complexe $\hat C$ is only an intermediate object.
Representation of a homotopy equivalence¶
A homotopy equivalence is implemented as an instance of the CLOS class HOMOTOPY-EQUIVALENCE,
whose definition is
(DEFCLASS HOMOTOPY-EQUIVALENCE () ;; Left Bottom Chain Complex ((lbcc :type chain-complex :initarg :lbcc :reader lbcc1) ;; Top Chain Complex (tcc :type chain-complex :initarg :tcc :reader tcc1) ;; Bottom Right Chain Complex (rbcc :type chain-complex :initarg :rbcc :reader rbcc1) ;; Left f (lf :type morphism :initarg :lf :reader lf1) ;; Left g (lg :type morphism :initarg :lg :reader lg1) ;; Left h (lh :type morphism :initarg :lh :reader lh1) ;; Right f (rf :type morphism :initarg :rf :reader rf1) ;; Right g (rg :type morphism :initarg :rg :reader rg1) ;; Right h (rh :type morphism :initarg :rh :reader rh1) ;; Left ReDuCTion (lrdct :type reduction :initarg :lrdct :reader lrdct) ;; Right ReDuCTion (rrdct :type reduction :initarg :rrdct :reader rrdct) ;; IDentification NuMber (idnm :type fixnum :initform (incf *idnm-counter*) :reader idnm) (orgn :type list :initarg :orgn :reader orgn)))
This class has 13 slots:
lbcc, the object of typechain complexrepresenting the chain complex $C$ (left bottom chain complex on the diagram).tcc, the object of typechain complexrepresenting the chain complex $\hat C$ (top chain complex).rbcc, the object of typechain complexrepresenting the chain complex $EC$ (right bottom chain complex on the diagram).lf, the object of typemorphismrepresenting the morphism $f$ of the (left) reduction $\rho_1$.lg, the object of typemorphismrepresenting the morphism $g$ of the (left) reduction $\rho_1$.lh, the object of typemorphismrepresenting the morphism $h$ of the (left) reduction $\rho_1$.rf, the object of typemorphismrepresenting the morphism $f$ of the (right) reduction $\rho_2$.rg, the object of typemorphismrepresenting the morphism $g$ of the (right) reduction $\rho_2$.rh, the object of typemorphismrepresenting the morphism $h$ of the (right) reduction $\rho_2$.lrdc, the object of typereductionrepresenting the (left) reduction $\rho_1$.rrdc, the object of typereductionrepresenting the (right) reduction $\rho_2$.idn, an integer, number plate for the object, set by the system.orgn, an adequate comment list.
When an instance is created, the printing method associated to the class HOMOTOPY EQUIVALENCE prints a string like [Kn Homotopy-Equivalence], where n is the number plate of the Kenzo object.
The function build-hmeq¶
To facilitate the construction of instances of the HOMOTOPY-EQUIVALENCE class, the software provides the function, (in fact a method), build-hmeq which may be used in the following ways:
either with the two reductions $\rho_1$ and $\rho_2$ or explicitly with the morphisms.
The selection of the adequate method is done by inspecting the first keyword, which is
:lrdct in the first case and :lf in the second one. In both cases the returned value is an instance of the class HOMOTOPY-EQUIVALENCE.
(build-hmeq :lrdct lrdct :rrdct rrdc :orgn orgn)[Method]
The keyword arguments are:
- lrdct, the object of type
reductionrepresenting the reduction $\rho_1$. - rrdct, the object of type
reductionrepresenting the reduction $\rho_2$. - org, the comment list.
The function
build-heq-from-rdccalls internally the standard constructormake-instance. All the needed information is contained in the two reductions.
(build-hmeq :lf lf :lg lg :lh lh :rf rf :rg rg :rh rh :orgn orgn)[Method]
The keyword arguments are:
- lf, the object of type
morphismrepresenting the morphism $f$ of the reduction $\rho_1$. - lg, the object of type
morphismrepresenting the morphism $g$ of the reduction $\rho_1$. - lh, the object of type
morphismrepresenting the morphism $h$ of the reduction $\rho_1$. - rf, the object of type
morphismrepresenting the morphism $f$ of the reduction $\rho_2$. - rg, the object of type
morphismrepresenting the morphism $g$ of the reduction $\rho_2$. - rh, the object of type
morphismrepresenting the morphism $h$ of the reduction $\rho_2$. - org, the comment list.
All the needed information is contained in the morphism structures.
In both cases, the method pushes the created instance onto the list *hmeq-list*.
Useful macros and functions¶
(cat-init) [Function]
Clear among others, the list
*hmeq-list*, list of user created homotopy equivalences and reset the global counter to 1.
(hmeq n) [Function]
Retrieve in the list
*hmeq-list*the homotopy equivalence instance whose identification is $n$. If it does not exist, returnNIL.
(lbcc hmeq &rest args) [Macro]
With only one argument (a homotopy equivalence hmeq) this macro selects the left bottom chain complex of the homotopy equivalence. Otherwise, it applies the differential of the left bottom chain complex of hmeq on the arguments args, (either degree generator or cmb).
(rbcc hmeq &rest args) [Macro]
With only one argument (a homotopy equivalence hmeq) this macro selects the right bottom chain complex of the homotopy equivalence. Otherwise, it applies the differential of the right bottom chain complex of hmeq on the arguments args, (either degree generator or cmb).
(lf hmeq &rest args) [Macro]
With only one argument (a homotopy equivalence {\em hmeq}) this macro selects the morphism $f$ of the left reduction of hmeq. Otherwise, it applies the morphism $f$ of the left reduction of hmeq on the arguments args, (either degree generator or cmb).
(lg hmeq &rest args) [Macro]
With only one argument (a homotopy equivalence hmeq) this macro selects the morphism $g$ of the left reduction of hmeq. Otherwise, it applies the morphism $g$ of the left reduction of hmeq on the arguments args, (either degree generator or cmb).
(lh hmeq &rest args) [Macro]
With only one argument (a homotopy equivalence hmeq) this macro selects the morphism $h$ of the left reduction of hmeq. Otherwise, it applies the morphism $h$ of the left reduction of hmeq on the arguments args, (either degree generator or cmb).
(rf hmeq &rest args) [Macro]
With only one argument (a homotopy equivalence hmeq) this macro selects the morphism $f$ of the right reduction of hmeq. Otherwise, it applies the morphism $f$ of the right reduction of hmeq on the arguments args, (either degree generator or cmb).
(rg hmeq &rest args) [Macro]
With only one argument (a homotopy equivalence hmeq) this macro selects the morphism $g$ of the right reduction of hmeq. Otherwise, it applies the morphism $g$ of the right reduction of hmeq on the arguments args, (either degree generator or cmb).
(rh hmeq &rest args) [Macro]
With only one argument (a homotopy equivalence hmeq) this macro selects the morphism $h$ of the right reduction of hmeq. Otherwise, it applies the morphism $h$ of the right reduction of hmeq on the arguments args, (either degree generator or cmb).
Build the trivial homotopy equivalence involving only the chain complex chcm. In that case,
(trivial-hmeq chcm) [Function]
the 3 chain complexes $C$, $EC$ and $\hat C$ are the chain complex chcm itself and the reductions $\rho_1$ and $\rho_2$ are, of course, the trivial reductions on chcm built by the statement (trivial-rdct chcm).
The perturbation lemma machinery¶
For a good understanding of the lisp functions involved in the machinery in question, we recall the perturbation lemma (Ronald Brown. The twisted Eilenberg--Zilber theorem. Celebrazioni Arch. Secolo XX Simp. Top., 1967, pp 34-37.). In the basic (resp. trivial) perturbation lemma, the given perturbation concerns the top (resp. bottom) chain complex.
Theorem 1 (Basic perturbation lemma) - Let $\rho=(\hat{C},C,f,g,h)$ a reduction and $\hat{\delta}$ a perturbation of $d_{\hat{C}}$, that is, an operator defined on $\hat{C}$ of degree -1 satisfying the relation $(d_{\hat{C}}+\hat{\delta})\circ(d_{\hat{C}}+\hat{\delta})=0$. Furthermore, the composite function $h\circ\hat{\delta}$ is assumed locally nilpotent, that is, $\forall x \in \hat{C}$, $(h\circ\hat{\delta})^nx=0$, for $n$ sufficiently large. Then a new reduction $\rho'=(\hat{C}',C',f',g',h')$ can be constructed where: the convergence being ensured by the locally nilpotency of $h \circ \hat{\delta}$ and $\hat{\delta} \circ h$.
- $\hat{C}'$ is the chain complex obtained from $C$ by replacing the old differential $d_{\hat{C}}$ by $(d_{\hat{C}}+\hat{\delta}),$
- the new chain complex $C'$ is obtained from the chain complex $C$ only by replacing the old differential $d_C$ by $(d_C + \delta),$ with $\delta = f \circ {\hat \delta} \circ \phi \circ g = f \circ \psi \circ {\hat \delta} \circ g,$
- $f'= f \circ \psi = f \circ (Id - {\hat \delta} \circ \phi \circ h),$
- $g'= \phi \circ g,$
- $h'= \phi \circ h = h \circ \psi$, with $$ \phi=\sum_{i=0}^{\infty}{(-1)^i(h \circ \hat{\delta})^i} $$ and $$ \psi=\sum_{i=0}^{\infty}{(-1)^i(\hat{\delta}\circ h)^i} = Id - {\hat \delta} \circ \phi \circ h, $$
Theorem 2 (Trivial perturbation lemma) - Let $\rho=(\hat{C},C,f,g,h)$ a reduction and ${\check\delta}$ a perturbation of $d_C$, that is, an operator defined on $C$ of degree -1 satisfying the relation $(d_C+{\check\delta})\circ (d_C+{\check\delta})=0$. Then a new reduction $\rho'=(\hat{C}',C',f',g',h')$ can be constructed where:
- $\hat{C}'$ is the chain complex obtained from $C$ by replacing the old differential $d_{\hat{C}}$ by $(d_{\hat{C}}+ g \circ {\check\delta} \circ f),$
- the new chain complex $C'$ is obtained from the chain complex $C$ only by replacing the old differential $d_C$ by $(d_C +{\check \delta}),$
- $f'= f,$
- $g'= g,$
- $h'= h.$
Useful functions related to the perturbation lemma¶
The functions implementing the perturbation-lemma are actually the heart of the program. They allow us to implement the effective versions of the classical spectral sequences (Serre, Eilenberg-Moore, ...).
Build a new reduction from the object of type
reduction, rdct and
(add rdct perturbation) [Method]
the object of type morphism, perturbation, applying the formulas given in the perturbation lemma. In fact, this method calls one or the other of the two following functions according as the perturbation is on the top or on the bottom chain complex and returns the values of the called function.
(basic-perturbation-lemma reduction top-perturbation) [Function]
Return a double value: the perturbed reduction and the computed bottom-perturbation (a morphism) $f \circ {\hat \delta}\circ \phi \circ g$. Of course, to avoid unuseful computations when the morphisms are effectively applied, the function takes in account the simplifications involved for instance whensome morphisms are the null morphism.
(easy-perturbation-lemma reduction bottom-perturbation) [Function]
Return a double value: the perturbed reduction and the computed top-perturbation (a morphism) $g \circ {\check \delta} \circ f$. Of course, to avoid unuseful computation when the morphisms will be applied, the function takes in account the simplifications involved for instance when some morphisms are the null morphism.
(special-bpl reduction top-perturbation) [Function]
This function is analogous to
basic-perturbation-lemmaand is used in some special cases when the morphism $g$ is invariant.
bpl-*-sigma homotopy perturbation) [Function]
Construct the principal series $\phi=\sum_{i=0}^{\infty}{(-1)^i(h \circ \hat{\delta})^i}$ of the basic perturbation lemma.
Build a homotopy equivalence from the
homotopy-equivalencehmeq and themorphismlb-perturbation, a perturbation of the differential of the left bottom chain complex of hmeq. From hmeq: $$ \begin{align} & \qquad\hat{C} & \\ \rho_1\swarrow\nearrow & & \nwarrow\searrow \rho_2 \\ C\quad & & EC \end{align} $$
(add hmeq lb-perturbation) [Method]
the method add builds the new homotopy equivalence:
$$
\begin{align}
& \qquad\hat{C}' & \\
\rho'_1\swarrow\nearrow & & \nwarrow\searrow \rho'_2 \\
C'\quad & & EC'
\end{align}
$$
where the construction steps are the following:
- The reduction $\rho'_1$ is computed with the method
addfor a reduction with arguments: the left reduction of hmeq and the perturbation lb-perturbation. The trivial perturbation lemma is applied. As a bonus, this returns also the top perturbation, say $\delta$ to be applied to $\hat{C}$. - The reduction $\rho'_2$ is computed with the previous method
addwith arguments: the right reduction of hmeq and the top perturbation $\delta$ just computed. That time, the basic perturbation lemma is applied. - The function
build-hmeqis called with parameters $\rho'_1$ and $\rho'_2$.
Example¶
We shall apply these methods and functions in the following chapters. Let us show on a trivial example how this machinery works. First, we build a trivial homotopy equivalence on $\Delta^4$. Then we build a new one by perturbing the differential of the left bottom chain complex with its opposite. The last call shows how this perturbation has been propagated.
(def hmeq (trivial-hmeq (cdelta 4)))
(setf hmeq (add hmeq (opps (dffr (cdelta 4)))))
(gnrt-? (dffr (rbcc hmeq)) 3 '(0 1 2 3))
Bicones¶
Given 3 chain complexes $B$, $C$ and $D$, together with 2 chain homomorphisms $f_1$ and $f_2$ of degree $0$ (commuting with the differentials), as shown in the following diagram, a bicone on these chain complexes is a chain complex, $BCN(f_1,f_2)$ where the $n$-th chain group $[BCN(f_1,f_2)]$ is $B_n \oplus C_{n+1} \oplus D_n$.
$$ \begin{align} B & & & \qquad D & \\ & f_1 \searrow & & \swarrow f_2 & \\ & & C \qquad& & \end{align} $$
Representation of a combination in a bicone¶
To distinguish to which chain complex belongs a generator in a combination of a bicone, the following convention has been adopted: if gb is a generator of any degree of $B$, it will be represented in the bicone by the list (:BcnB gb) and printed as <BcnB gb>. The symbols for $C$ and $D$ are respectively :BcnC and :BcnD. The 3 macros bcnb, bcnc and bcnd may be used to build such a generator. The building function for combinations, cmbn, may be used in the following way:
(def comb-bic (cmbn 3 2 (bcnb 'b1) 4 (bcnb 'b1) 6 (bcnb 'b3) 3 (bcnb 'c1) 5 (bcnb 'c1) 7 (bcnb 'd1)))
(cmbn-list comb-bic)
Useful functions and macros for bicones¶
(bcnb gnrt) [Macro]
Build the representation of the generator gnrt belonging to the chain complex $B$.
(bcnc gnrt) [Macro]
Build the representation of the generator gnrt belonging to the chain complex $C$.
(bcnd gnrt) [Macro]
Build the representation of the generator gnrt belonging to the chain complex $D$.
(make-bicn-cmbn cmbnb cmbnc cmbnd) [Function]
Build a bicone combination from the 3 combinations cmbnb, cmbnc and cmbnd belonging respectively and in that order to $B$, $C$ and $D$. The degrees of the combinations cmbnb and cmbnd must be the same, say $n$. The degree of cmbnc must be $n+1$. The degree of the new created combination is $n$.
(bicn-cmbn-cmbnb cmbn) [Function]
Extract from the bicone combination of degree $n$, cmbn, the combination relative to $B$ as a legal combination of degree $n$ in $B$. If there is no $B$-component, return the null combination of degree $n$.
(bicn-cmbn-cmbnc cmbn) [Function]
Extract from the bicone combination of degree $n$, cmbn, the combination relative to $C$ as a legal combination of degree $n+1$ in $C$. If there is no $C$-component, return the null combination of degree $n+1$.
(bicn-cmbn-cmbnd cmbn) [Function]
Extract from the bicone combination of degree $n$, cmbn, the combination relative to $D$ as a legal combination of degree $n$ in $D$. If there is no $D$-component, return the null combination of degree $n$
(dispatch-bicn-cmbn cmbn) [Function]
Give the 3-values result constitued by the 3 combinations components of the bicone combination cmbn. These combinations are valid combinations in their respective chain complexes.
Examples¶
(def comb-b (cmbn 3 2 'b1 4 'b2 6 'b3))
(def comb-c (cmbn 4 3 'c1 5 'c2))
(def comb-d (cmbn 3 7 'd1))
(def comb-bic (make-bicn-cmbn comb-b comb-c comb-d))
(bicn-cmbn-cmbnb comb-bic)
(bicn-cmbn-cmbnc comb-bic)
(bicn-cmbn-cmbnd comb-bic)
(multiple-value-bind (b c d) (dispatch-bicn-cmbn comb-bic) (list b c d))
Construction of a bicone from 2 reductions¶
From the slots of the respective chain complexes $B$, $C$ and $D$, it is possible to build the bicone chain complex. The 3 essential functions are the following:
(bicone-cmpr cmprb cmprc cmprd) [Function]
From the 3 comparison functions cmprb, cmprc and cmprd, build a comparison function adequate to compare the generators as represented in the bicone.
(bicone-basis basisb basisc basisd) [Function]
From the 3 basis function basisb, basisc and basisd, build a basis function for the bicone. If at least one of the chain complex component of the bicone is locally effective, the function returns the symbol
:locally-effective.
(bicone-intr-dffr cmprc dffrb dffrc dffrd f1 f2) [Function]
Define the differential in the bicone according to the formula:
$$
d(cb, cc, cd)= (d_B(cb), f_1(cb) + f_2(cd) - d_C(cc), d_D(cd)),
$$
where the notation is self explanatory. One sees that one needs the comparison operator cmprc of $C$, to order correctly the second combination of the resulting triple.
Let us consider now 2 reductions $(\hat C_1, C_1, f1, g1, h1)$ and $(\hat C_2, C_2, f2, g2, h2)$ in which $C_1 = C_2$. It is possible to define a bicone, with the identification $\hat C_1=B, C_1=C_2=C, \hat C_2=D$. This is realised by the following function:
(bicone rdct1 rdct2) [Function]
Build the bicone chain complex from the objects ot type
reductionrdct1 and rdct2. The bicone is built with the following identification: $B$ is the top chain complex of rdct1, $D$ is the top chain complex of rdct2. The bottom chain complexes of both rdct1 and rdct2 must be the same and $C$ is that chain complex. Of course, both chain morphisms $f_1$ and $f_2$ are respectively the morphism $f$ of the reductions. The construction of the bicone is realised by a call to the building functionbuild-chcmusing the 3 functionsbicone-cmpr,bicone-basisandbicone-intr-dffrwith arguments coming from rdct1 an rdct2. The slotbsgn(base point) is let undefined and the strategy is by combination.
Example¶
Let us take again the example with the standard simplex $\Delta^n$. The following function defines an effective version of ${\cal C}_*(\Delta^n)$, so we may ask for the basis in some dimensions.
(cat-init)
(defun cdelta (dmns) (build-chcm :cmpr #'l-cmpr :basis #'(lambda (n) (mapcar #'dlop-int-ext (funcall (delta-n-basis dmns) n))) :bsgn '(0) :intr-dffr #'(lambda (degr gmsm) (make-cmbn :degr (1- degr) :list (do ((rslt +empty-list+ (cons (cons sign (append (subseq gmsm 0 nark) (subseq gmsm (1+ nark)))) rslt)) (sign 1 (- sign)) (nark 0 (1+ nark))) ((> nark degr) rslt)))) :strt :gnrt :orgn `(Effective version of C_* delta ,dmns)))
(def delta3 (cdelta 3))
(basis delta3 0)
(basis delta3 1)
(basis delta3 2)
(basis delta3 3)
(basis delta3 4)
Using the function make-rdct defined in previous example which builds a reduction between
$\Delta^m$ and $\Delta^n$, we may create a bicone chain complex with the function bicone and ask for basis in some dimensions.
(def bic (bicone (make-rdct 3 2) (make-rdct 4 2)))
(basis bic 0)
(basis bic 1)
(basis bic 4)
We verify the validity of the differential in the bicone bic.
(? bic (cmbn 2 3 (bcnb '(0 1 3)) 4 (bcnc '(0 1 2 3)) 5 (bcnd '(0 1 4))))
(? bic *)
Composition of homotopy equivalences¶
Let us consider 2 homotopy equivalences, as in the following diagram.
$$ \begin{align} & \qquad B & & & & \qquad D \\ \rho_1 \swarrow\nearrow & & \nwarrow\searrow \rho_2 & & \rho'_1\swarrow\nearrow & & \nwarrow\searrow\rho'_2 \\ A \quad & & C & & C \quad & & E \quad \end{align} $$
where the right bottom complex of the first one is the same as the left bottom reduction of the second. We may use then the bicone concept to build a bicone chain complex with $B$, $C$ and $D$, say $BCN$, and finally build a new homotopy equivalence between $A$, $BCN$ and $E$.
$$ \begin{align} & \qquad BCN & \\ \rho''_1 \swarrow\nearrow & & \nwarrow\searrow \rho''_2 \\ A \quad & & E\quad \end{align} $$
This is realized by the method cmps.
(cmps hmeq1 hmeq2) [Method]
Build a homotopy equivalence by composition of both homotopy equivalences hmeq1, hmeq2. The system controls the validity of the composition.
Example¶
Starting with a very simple chain complex and the trivial homotopy equivalence on the chain complex, we verify the correctness of various compositions. The chain complex c has only one vertex (a) in every dimension and the differential is the null morphism.
(cat-init)
(def c (build-chcm :cmpr #'s-cmpr :basis #'(lambda (dmns) (declare (ignore dmns)) '(a)) :bsgn 'a :intr-dffr #'zero-intr-dffr :strt :cmbn :orgn '(c)))
We build the trivial homotopy equivalence on c and we compose it with itself.
(def h1 (trivial-hmeq c))
(def h2 (cmps h1 h1))
We verify the coherency of the morphisms in the left and right reduction of h2. The combination *tc* must belong to a bicone whereas *bc* belongs to c.
(pre-check-rdct (lrdct h2))
(setf *tc* (cmbn 3 1 (bcnb 'a) 10 (bcnc 'a) 100 (bcnd 'a)))
(setf *bc* (cmbn 3 1 'a))
(check-rdct-no-wait)
(pre-check-rdct (rrdct h2))
(check-rdct-no-wait)
We compose now h2 with itself. We let *bc* unchanged, but *tc* is a little more complicated because the chain complexes $B$ and $D$ are themselves bicones.
(def h3 (cmps h2 h2))
(setf *tc* (cmbn 3 1 (bcnb (bcnb 'a)) 10 (bcnb (bcnc 'a)) 100 (bcnb (bcnd 'a)) 1000 (bcnc 'a) 10000 (bcnd (bcnb 'a)) 5234 (bcnd (bcnc 'a)) 223 (bcnd (bcnd 'a))))
(pre-check-rdct (lrdct h3))
(check-rdct-no-wait)
(pre-check-rdct (rrdct h3))
(check-rdct-no-wait)
Lisp files concerned in this chapter¶
effective-homology.lisp, cones.lisp.
[classes.lisp, macros.lisp, various.lisp].