A computational approach of A ∞ -(co)algebras

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A computational approach of A ∞-(co)algebrasAinhoa Berciano-Alcaraz aa Dept. Matemática Aplicada , Estadística e InvestigaciónOperativa Euskal Herriko Unibertsitatea , SpainPublished online: 16 Oct 2008.

To cite this article: Ainhoa Berciano-Alcaraz (2010) A computational approach of A ∞-(co)algebras,International Journal of Computer Mathematics, 87:4, 935-953, DOI: 10.1080/00207160802247612

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International Journal of Computer MathematicsVol. 87, No. 4, March 2010, 935–953

A computational approach of A∞-(co)algebras

Ainhoa Berciano-Alcaraz*

Dept. Matemática Aplicada, Estadística e Investigación Operativa Euskal Herriko Unibertsitatea, Spain

(Received 07 August 2007; revised version received 15 May 2008; accepted 17 May 2008)

We present a computer program to compute the explicit structural components of an A∞-(co)algebradeduced from a contraction, which is a special type of homotopy equivalence. The input is a contractionfrom a dg-(co)algebra to a simple dg-module and the output is a functional object that defines the operations{μi}i≥1 (resp. {�i}i≥1) in the deduced A∞-structure on the simple dg-module. We conclude with someconcrete applications.

Keywords: homological perturbation theory; contraction; basic perturbation lemma; symboliccomputation; software

2000 AMS Subject Classification: 55P43; 68W30; 55-04; 55U05

1. Introduction

Many concepts in algebraic topology involve complicated formulae. Consequently, applicationsoften require sophisticated computations and explicit results are often difficult to obtain. Computerprograms that create an environment for experimentation can assist the researcher in makingconjectures. This process can be very fruitful.

Two important tools in homological algebra are: (1) the notion of contraction, which is a specialtype of homotopy equivalence between differential graded modules and (2) the homologicalperturbation lemma (see [4] or [20]). Given the right conditions, these tools give an explicitalgorithm for computing the structural components of an A∞-(co)algebra [8–10,12]. In this paperwe present a computer program that computes the explicit structural components of an A∞-(co)algebra deduced from a contraction, which is a special type of homotopy equivalence. Theinput is a contraction from a dg-(co)algebra to a simple dg-module and the output is a functionalobject that defines the operations {μi}i≥1 (resp. {�i}i≥1) and gives the A∞-structure on the simpledg-module [11]. In particular, this program is a set of programs enhancing the Kenzo system [5,18].Kenzo is a common Lisp program designed for computing in algebraic topology, in particular itallows the user to calculate homology and homotopy groups of complicated spaces. We use theprogram to examine some interesting applications and determine which of these operations arenon-vanishing; we give explicit formulae for computing these operations in some cases.

*Email: [email protected]

ISSN 0020-7160 print/ISSN 1029-0265 online© 2010 Taylor & FrancisDOI: 10.1080/00207160802247612http://www.informaworld.com

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936 A. Berciano-Alcaraz

Throughout this exposition we use the generic term ‘A∞-structure’ to refer to either anA∞-coalgebra or an A∞-algebra. For simplicity, examples and definitions will be given onlyfor A∞-coalgebras, with the analogous statements for A∞-algebras following by duality.

The paper is organized as follows. In Section 2, we review the preliminaries and establishour notation. Theoretical results underlying our computer program as well as some examplesdemonstrating the program are presented in Section 3. In Section 4, we conclude with a study ofthe complexity of the algorithms involved and some discussions and comments on future work.

2. Preliminaries and notation

Throughout this paper � denotes a commutative ring; in the examples, we set � = Z, the integers.We begin with a review of the basic definitions (for more detail see [13,22]). A differential gradeddg-module (DGM) is a �-module graded on the non-negative integers (M = ⊕

n≥0 Mn) andendowed with an endomorphism d of degree −1 such that dd = 0. An element x ∈ Mn hasdegree n, which will be expressed by |x| = n. In the case that M0 = �, M is called connectedand if, besides, M1 = 0, then it is called simply connected. Given a connected DGM, M , thereduced module M is the one with Mn = Mn for n > 1 and M0 = 0. The homology of M is thegraded module H∗(M) = {Hn(M) = Ker dn/Im dn+1}n≥0.

Given a DGM (M, dM), the suspension of M is the DGM (sM, dsM), where (sM)n = Mn−1

and dsM = −dM . Dually, the desuspension of M is the DGM (s−1M, ds−1M) given by (s−1M)n =Mn+1 with differential −dM . A morphism of degree i of graded modules f : M → N induces amorphism of suspensions sf : sM → sN defined by sf = (−1)if and dually of desuspensionss−1f : s−1M → s−1N . A dg-algebra (DGA) is a DGM (A, dA) endowed with an associativeproduct μA and a unit. Dually, a dg-coalgebra (DGC) is a DGM (A, dA) endowed with anassociative coproduct �A and a counit. A dg-Hopf algebra is a DGM (H, dH ) equipped with aproduct μH and a coproduct �H such that (H, dH , μH ) is a DGA, (H, dH , �H) is a DGC and

�HμH = (μH ⊗ μH) (1 ⊗ T ⊗ 1) (�H ⊗ �H) ,

where T is a morphism of dg-modules which interchanges the order on the tensor product.The tensor module of M is the DGM

T (M) =⊕n≥0

T n(M) =⊕n≥0

M⊗n,

where M⊗0 = �, and with tensor differential dt = d⊗M given by linear extension, i.e.

dt =n−1∑i=0

1⊗i ⊗ dM ⊗ 1⊗n−i−1.

A morphism f : M → N of DGMs induces a morphism T (f ) : T (M) → T (N) viaT (f )|M⊗n = f ⊗n. The tensor algebra of M, denoted by T a(M), is the DGM (T (M), dt ) togetherwith the product μ given by concatenation, i.e.

μ((a1 ⊗ · · · ⊗ an) ⊗ (an+1 ⊗ · · · ⊗ an+p)) = a1 ⊗ · · · ⊗ an+p.

The tensor coalgebra of M, denoted by T c(M), is the DGM (T (M), dt ) together with a coproduct� given by

�(a1 ⊗ · · · ⊗ an) =n∑

i=0

(a1 ⊗ · · · ⊗ ai) ⊗ (ai+1 ⊗ · · · ⊗ an).

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International Journal of Computer Mathematics 937

Given a connected DGA A, the reduced bar construction of A is the DGC B(A) = T s (A) withcofree coproduct

�B([a1| · · · |ar ]) =r−1∑i=1

[a1| · · · |ai] ⊗ [ai+1| · · · |ar ]

and differential dB = dt + ds , where dt is as before and

ds =r−1∑i=1

1⊗i−1 ⊗ μA ⊗ 1r−i−1

is the simplicial differential. Dually, given a simply connected DGC, C, the reduced cobarconstruction of C is the DGA �(A) = T s−1(C) with the concatenation product and differentiald� = dt + dc, where dt is as before and

dc =n∑

i=1

(−1)i−11⊗i−1 ⊗ � ⊗ 1⊗n−i .

The following connected commutative dg-Hopf algebras with null differential are important inour applications.

• The polynomial algebra P(v, 2n), n ≥ 1, generated by v of degree 2n, with productvivj = vi+j and coproduct �(vn) = ∑

i+j=n(i + j)!/i!j !vi ⊗ vj ;• The truncated polynomial algebra Qp(v, 2n) = P(v, 2n)/(vp);• The exterior algebra E(u, 2n + 1), n ≥ 0, generated by u of degree 2n + 1, with trivial product

u2 = 0 and trivial coproduct �(u) = u ⊗ 1 + 1 ⊗ u;• The divided power algebra �(w, 2n), n ≥ 1, generated by γ1(w) = w, with product

γiγj = (i + j)!/i!j !γi+j and coproduct �(γk(u)) = ∑i+j=k γi(u) ⊗ γj (u).

The notion of A∞-algebra, introduced by J.D. Stasheff [21] in the sixties, generalizes thenotion of algebra that is ‘associative up to homotopy’ [16,17]. A DGM (M, d) together witha non-associative morphism μ2 : M ⊗ M → M of degree zero that is compatible with d isassociative up to homotopy if there exists a morphism μ3 : M⊗3 → M of degree +1 such thatμ3d + dμ3 = μ2(μ2 ⊗ 1) − μ2(1 ⊗ μ2). An A∞-coalgebra (resp. A∞-algebra) is a graded mod-ule M together with a family of maps �i : M → M⊗i (resp. μi : M⊗i → M) of degree i − 2such that for all i ≥ 1

i∑n=1

i−n∑k=0

(−1)n+k+nk(1⊗i−n−k ⊗ �n ⊗ 1k

)�i−n+1 = 0.

(resp.

i∑n=1

i−n∑k=0

(−1)n+k+nkμi−n+1(1k ⊗ μn ⊗ 1⊗i−n−k

) = 0.

)

Next we review the main facts from homological perturbation theory used in this paper. Let(N, dN) and (M, dM) be DGMs.A contraction (or reduction) c : {N, M, f, g, φ} (see [6,20]) from(N, dN) to (M, dM) is a special type of homotopy equivalence given by morphisms f : N → M

and g : M → N of degree zero and a homotopy operator φ : N → N of degree +1 that satisfy

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the following conditions:

(c1)fg = 1M, (c2) φdN + dNφ + gf = 1N,

(c3)f φ = 0, (c4) φg = 0, (c5)φφ = 0.

Of course, the homology groups of N are canonically isomorphic to the homology groups ofM , but any additional algebraic structure on N (e.g. such as a DGA structure) does not transfer toM in general. In fact a contraction from a DGC to a DGM induces an A∞-(co)algebra structureon M [8,9] by means of a perturbation process.

A contraction c : {N, M, f, g, φ} between DGMs induces the following contractions ofsuspensions and tensor modules [7,8]:

• the suspension contraction of c, sc

sc : {sN, sM, sf, sg, sφ};• the tensor module contraction, T (c),

T (c) : {T (N), T (M), T (f ), T (g), T (φ)},where

T (φ)|T n(N) = φ[⊗n] =n−1∑i=0

1⊗i ⊗ φ ⊗ (g f )⊗n−i−1.

A morphism of graded modules f : N → N is called pointwise nilpotent whenever for allx ∈ N , x �= 0, there exists a positive integer n such that f n(x) = 0. A perturbation of a DGM N

consists of a morphism of graded modules δ : N → N of degree −1, such that (dN + δ)2 = 0.A perturbation datum of the contraction c : {N, M, f, g, φ} is a perturbation δ of the DGM N

satisfying that the composition φδ is pointwise nilpotent.A fundamental tool in homological perturbation theory is the basic perturbation lemma

(BPL) [20,4]), which is an algorithm whose input is a contraction c : {N, M, f, g, φ} and aperturbation datum δ of c and whose output is a new contraction cδ . The only requirement is thatthe composition φδ must be pointwise nilpotent so that the sums involved in the formulae arefinite for each x ∈ N :

Input: c : (N, dN)

φ

�� f��(M, dM)

g

�� + perturbation δ

Output: cδ : (N, dN + δ)

φδ

�� fδ

��(M, dM + dδ)

gδ

��

where fδ, gδ, φδ, dδ are given by the formulae

dδ = f δδcg, fδ = f (1 − δδ

cφ), gδ = δcg, φδ = δ

cφ

and δc = ∑

i≥0(−1)i(φδ)i .The procedure by which one obtains an A∞-coalgebra structure on a small DGM of a

contraction, known as tensor trick (see [8]), follows these steps.

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• As initial datum, let C be a DGC and let r : {C, M, f, g, φ} be a contraction.• Form the tensor module contraction on the desuspension

T s−1(r) : {T s−1(C), T s−1(M), T s−1(f ), T s−1(g), T s−1(φ)}.• Using the cosimplicial differential as a perturbation data of T s−1(r), apply the BPL and obtain

a new contraction

cb(r) : {�(C), �(M), cb(f ), cb(g), cb(φ)}where �(M) = T s−1(M) with the differential dT s−1 + dδ .

• Finally, the induced A∞-coalgebra structure on M can be extracted from the tilde cobardifferential dδ on �(M) and, for i ≥ 2, its explicit formula is given by (see [11]):

�i = (−1)[i/2]+i+1f ⊗i�[i]φ[⊗(i−1)] · · · φ[⊗2]�[2]g,

where

�[k] =k−2∑i=0

(−1)i1⊗i ⊗ � ⊗ 1⊗k−i−2.

Dually, if the initial data is a contraction r : {A, M, f, g, φ} with A as DGA, the A∞-algebrainduced on M is given by the formulae (see [11]):

μ1 = −dM

μn = (−1)n+1f μ(1) φ[⊗2]μ(2) · · · φ[⊗n−1]μ(n−1)g⊗n, n ≥ 2

where

μ(k) =k−1∑i=0

(−1)i+11⊗i ⊗ μA ⊗ 1⊗k−i−1.

Our program is organized as a module that enhances the Kenzo program [5], developedby F Sergeraert and some collaborators. Kenzo is a common lisp object system (CLOS) thatimplements numerous algebraic structures including DGAs, simplicial groups, morphisms,contractions and chain complexes, and uses them to compute various topological invariants suchas the homology groups of sophisticated spaces.

3. Algorithms to compute A∞-structures: ARAIA and CRAIC

The enhancement to Kenzo introduced here allows us to compute the A∞-(co)algebra structuresinduced on the small module of a contraction when the big DGM is a (co)algebra. This programcomponent consists of approximately 2000 lines of code and significantly enriches the set of Kenzoclasses (see [1,2,5]), and it is important to mention that as far as we know it is the first computerprogram available to compute A∞-structures given by contractions. The most important problemwe faced when creating this software was creating the appropriate data structures to handle thelarge quantity of data generated by the complicated formulae of the BPL. To make this sectionmore understandable to the reader, we have divided it into four subsections: in the first one, wewill present Kenzo system (with a small machine example); in Subsection 3.2 we will explaintheoretically the problems and the solutions adopted to create the software; Subsections 3.3 and3.4 are dedicated to two examples of computations.

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3.1 Kenzo system

Before describing our new programs, let us remark on the most important properties of Kenzoprogram: it is a program [5], developed by Professor Francis Sergeraert and some coworkers. Lispis a 16,000-line program devoted to symbolic computation in algebraic topology. It works with richand complex algebraic structures (chain complexes, differential graded algebras, simplicial sets,simplicial groups, morphisms between these objects, contractions, etc.) and has obtained someresults (e.g. homology groups of iterated loop spaces modified by a cell attachment, componentsof complex Postnikov towers, etc.), which had never been determined before. To clarify this lastpart let us see a simple example (extracted from the introduction of the manual of Kenzo [5]).In particular, we will consider now the Eilenberg–Mac Lane space K(Z, 1). This is an Abeliansimplicial group created in Kenzo by the function k-z. In this simplicial group, a simplex indimension n is mathematically represented by a sequence of integers, known as a bar object:

[a1 | a2 | · · · | an].In Kenzo, a non-degenerate simplex of K(Z, 1) in dimension n will be simply a list of n non-nullintegers, for instance, (2 3 4 5). In dimension 0, the only simplex is NIL (the base point).

> (setf kz1 (k-z 1))

[K38 Abelian-Simplicial-Group]

Because it is a simplicial set, it is possible to extract the faces of a simplex, let us suppose, e.g. a4-simplex of kz1 denoted by (2 3 4 5)

> (dotimes (i 5)(print (face kz1 i 4 ’(2 3 4 5))))

<AbSm - (3 4 5)><AbSm - (5 4 5)><AbSm - (2 7 5)><AbSm - (2 3 9)><AbSm - (2 3 4)>NIL

But this object is also a coalgebra and an algebra, and we may see the effect of the respectiveinduced coproduct (cprd) and product (aprd):

> (cprd kz1 4 ’(2 3 4 5))----------------------------------------------------{CMBN 4}<1 * <TnPr NIL (2 3 4 5)>><1 * <TnPr (2) (3 4 5)>><1 * <TnPr (2 3) (4 5)>><1 * <TnPr (2 3 4) (5)>><1 * <TnPr (2 3 4 5) NIL>>-----------------------------------------------------------

> (aprd kz1 6 (tnpr 2 ’(1 2) 4 ’(3 4 5 6)))---------------------------------------------------{CMBN 6}<1 * (1 2 3 4 5 6)><-1 * (1 3 2 4 5 6)><1 * (1 3 4 2 5 6)>

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<-1 * (1 3 4 5 2 6)><1 * (1 3 4 5 6 2)><1 * (3 1 2 4 5 6)><-1 * (3 1 4 2 5 6)><1 * (3 1 4 5 2 6)><-1 * (3 1 4 5 6 2)><1 * (3 4 1 2 5 6)><-1 * (3 4 1 5 2 6)><1 * (3 4 1 5 6 2)><1 * (3 4 5 1 2 6)><-1 * (3 4 5 1 6 2)><1 * (3 4 5 6 1 2)>----------------------------------------------------------

The printed results are the printed representation of combinations, i.e. integer linearcombinations of generators resulting from the application of the morphisms. The degree of thecombination is indicated by the information: CMBN n. In the same way, let us see which are then-homology groups with coefficients on Z of K(Z, 1) for n = {0, . . . , 4}:> (dotimes (i 5)

(homology kz1 i))

Homology in dimension 0 :Component Z---done---Homology in dimension 1 :Component Z---done---Homology in dimension 2 :---done---Homology in dimension 3 :---done---Homology in dimension 4 :---done---

So, the machine returns that the unique homology groups non-null are H0(K(Z, 1)) = Z andH1(K(Z, 1)) = Z.

3.2 New programs: symbolic representation

In our computational framework, we encode an A∞-coalgebra as a map of degree zero from M

to T a(M). When doing so, we actually ignore the degree of the maps. This allows us to write thecode as simply as possible. But apart from that, some difficult problems must be solved.

‘Infinite’ loops: The first problem is to translate mathematical categories into computationalclasses.

• Given coalgebras C and D, the tensor product C ⊗ D is a coalgebra with coproduct � : C ⊗D → (C ⊗ D)⊗2 given by (1 ⊗ T ⊗ 1)(�C ⊗ �D), where T interchanges the second andthird tensor factors.

• In our ‘computational world’ there is a class coalgebra, which must respect tensor products asindicated before. Thus, the program defines the morphism � by applying the rule. Realizing

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that (C ⊗ D)⊗2 must be a coalgebra, the program tries to define � recursively but in doing sofalls into an infinite loop.

Coherence:

• Mathematical coherence: An A∞-structure is much more complicated than a (co)algebra.Indeed, an A∞-algebra is an algebra in which associativity is satisfied up to homotopy.

• Computational coherence: We view the class of A∞-algebras as a superclass of the classof algebras, i.e. as a class of more general (weaker) objects. In particular, an A∞-algebra isimplemented in Kenzo as:

(DEFUN A-INFTY-ALGEBRA (&rest rest &key dfnt &allow-other-keys)(declare (type list dfnt))(already a-infty-algebra dfnt)(the a-infty-algebra

(apply #’make-instance ’a-infty-algebra rest)))

Here is a summary of our solutions to these problems.Inheritance: To avoid ‘infinite’ loops we used a lazy programming style. The slot-unbound

Lisp generic function allows us to implement a redundant slot dynamically only when requiredand thereby avoid infinite loops.

Extension of the class family: Since a DGA induces a chain complex, we adopt the followingnotation:

• CHCM=chain complex; A=DGA; C=DGC; HA=dg-Hopf algebra.• AA=A∞-algebra; AC=A∞-coalgebra.

Structure of classes of Kenzo: Thanks to the new point of view of computational ‘coherence’,the first step is to add the classes AA and AC, then the old (unenhanced) structure of Kenzo andthe new one can be expressed graphically as shown.

To obtain the A∞-structure induced by a contraction, we take as input a contraction, wherethe top chain complex (i.e. the DGM denoted by N in the definition of contraction) is a DGC. Acontraction (or reduction) in Kenzo 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

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idnm);; ORiGiN(orgn :type list :initarg :orgn :reader orgn)))

This class has seven slots:

• tcc, the object of type chain-complex representing the chain complex associated to thedg-module M;

• bcc, the object of type chain-complex representing the chain complex associated to thedg-module N ;

• f, the object of type morphism representing the morphism f ;• g, the object of type morphism representing the morphism g;• h, the object of type morphism representing the homotopy operator φ;• idmn, an integer, number plate for the object;• orgn, a comment list carefully chosen;

Applying the method CRAIC (Coalgebra Reduction A-Infinity Coalgebra; resp. ARAIA,Algebra Reduction A-Infinity Algebra) to a contraction, the output is the same contraction, butthe bottom chain complex (i.e. the small DGM denoted by M in the definition of contraction)now has an explicit A∞-structure induced. In particular, CRAIC (dually ARAIA) is implementedin Kenzo as a method, whose code is

(DEFMETHOD coalgebra-reduction-a-infty-coalgebra ((rdctreduction))

(the reduction(let ((c (tcc rdct))

(m (bcc rdct)))(declare (type chain-complex m)

(type coalgebra a))(unless (typep c ’coalgebra)

(error "In CRAIC, the TCC should be a coalgebra."))(if (typep m ’hopf-algebra)

(let ((imcprd (craic-mcprd rdct)))(declare (type morphism imcprd))(change-class m ’double-a-infty-clgb-h-algb)(setf (slot-value m ’imcprd) imcprd))

(if (typep m ’algebra)(let ((imcprd (craic-mcprd rdct)))

(declare (type morphism imcprd))(change-class m ’double-a-infty-clgb-algb)(setf (slot-value m ’imcprd) imcprd))

(if (typep m ’coalgebra)(let ((imcprd (craic-mcprd rdct)))

(declare (type morphism imcprd))(change-class m ’double-a-infty-clgb-clgb)(setf (slot-value m ’imcprd) imcprd))

(if (typep m ’a-infty-algebra)(let ((imcprd (craic-mcprd rdct)))

(declare (type morphism imcprd))(change-class m ’a-infty-clgb-a-infty-algb)(setf (slot-value m ’imcprd) imcprd))

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(if (typep m ’a-infty-coalgebra)(let ((imcprd (craic-mcprd rdct)))

(declare (type morphism imcprd))(change-class m ’a-infty-clgb-a-infty-clgb)(setf (slot-value m ’imcprd) imcprd))

(let ((mcprd (craic-mcprd rdct)))(declare (type morphism mcprd))(change-class m ’a-infty-coalgebra)(setf (slot-value m ’mcprd) mcprd)))))))

(reduction:f (f rdct):g (g rdct):h (h rdct):dfnt ‘(craic ,rdct))

rdct)))

where we can find the following auxiliary functions:

• (mtnpr-allp mtnpr): function that converts an mtnpr element to allp type;• (allp-mtnpr allp): function that converts an allp element to mtnpr type;• (degr-mtnpr-degr-allp degr mtnpr): given an element of tensor module, this function returns

the degree in the cobar construction;• (CMBN-ALLP-CMBN-MTNPR cmbn): cobar cmbn to mtnpr cmbn;• (extr-1cobar-cobar-cmbn cmbn): from a cobar cmbn returns a cmbn with only the elements of

cosimplicial degree 1;• (extr-n-cobar-cobar-cmbn n cmbn): from a cobar cmbn returns a cmbn with only the elements

of simplicial degree n;• (craic-mcprd-intr rdct);• (craic-mcprd rdct);• (coalgebra-reduction-a-infty-coalgebra (rdct reduction)). In short (craic).

We wish to compare the A∞-structure induced by a contraction with the trivial one wheneverthe bottom chain complex has non-trivial A∞-structure. Thus we must modify the classes and addsome new ones. We adopt the following additional notation.

• AAAA = an object with a double A∞-algebra, i.e. an A∞-algebra with another A∞-algebrainduced by a contraction; analogously AAAC (an A∞-algebra with an A∞-coalgebra inducedby a contraction), ACAC (an A∞-coalgebra with an A∞-coalgebra induced) and ACAA (anA∞-coalgebra with an A∞-algebra induced).

• DAA = an algebra plus an A∞-algebra, i.e., an algebra with another A∞-algebra induced by acontraction; analogously DCA (an algebra with an A∞-coalgebra induced), DAC (a coalgebrawith an A∞-algebra induced), DCC (a coalgebra with an A∞-coalgebra induced), DCHA (aHopf algebra with an A∞-coalgebra induced), DAHA (a Hopf algebra with an A∞-algebrainduced).

Theoretical steps: Given a contraction c : N → M from a dg-(co)algebra N , CRAIC (resp.ARAIA) is a CLOS method which, roughly speaking, follows this plan.

• Take as input a contraction c : N → M .• Check whether or not N is a DGC.• If so, create a new contraction T s−1c : T s−1(N) → T s−1(M).

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• Using the appropriate perturbation δ that in this case is the cosimplicial differential dc (see [21]for details), produce a new contraction �c : �N → (�M, dT + dδ) from the reduced cobarconstruction of N, i.e. the tensor algebra of N with a perturbed differential, to the tilde-cobarconstruction of M .

• Extract the new differential dδ induced on the small complex and transform it into a collectionof maps defining the induced A∞-coalgebra structure on M .

• Finally CRAIC returns the original contraction c : N → M having modified M from an objectin the class of DGMs to an object in the class of A∞-coalgebras.

More precisely, the object M has the same components as before plus a new one, imcprd(Induced Multi CoPRoDuct), which is a map from M to T M given by summing the �is withappropriate signs. The imcprd component is a functional object; if we wish to study the inducedA∞-coalgebra structure on M , we can apply this functional object to generators or some com-bination thereof and examine the results. This component is implemented by a Lisp statementformatted as follows:

(imcprd module (degree of element) element).The output is a combination (cmbn) of elements of M⊗n with n ≥ 2. For example, x = x1 ⊗

x2 ⊗ x3 ∈ M⊗3 is coded as<<Mtnpr [deg(x1) x1][deg(x2) x2][deg(x3) x3]>>

Let us consider two examples. The first one is trivial. Given a DGC, we compute the A∞-coalgebra structure induced by an automorphism. The second is more interesting and shows how,taking into account a contraction from the bar construction of a truncated polynomial algebrato a tensor product of an exterior algebra, E, with a divided power algebra, �, the coalgebrastructure on the bar construction of a truncated polynomial algebra is lost in the process inducingan A∞-coalgebra in the tensor product E ⊗ �.

3.3 Example 1: The trivial contraction P(u, 2) → P(u, 2)

In this first example, we will see that if we establish a contraction between DGCs and thiscontraction is compatible with the algebraic structure of the big DGM, the induced A∞-coalgebraon the small DGM via the contraction is nothing more than the original coproduct. To do it, letP(u, 2) be the polynomial algebra on one generator u of degree 2 with coefficients in Z andconsider the automorphism given by the identity on P(u, 2).

In general, a polynomial algebra P(u, n) is implemented in Kenzo as an object of the class H,with two inputs: a prime number zp to represent the ring ZP (with p = 0, the ring Z) and dmnswhich indicates the degree of generator of the polynomial algebra.

(DEFUN PLNM_ALGB (zp dmns)(declare (fixnum zp dmns))

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(unless (plusp dmns)(error "In POLYNOMIAL ALGEBRA, the dimension ˜D

should be positive." dmns))(the hopf-algebra

(let ((rslt (hopf-algebra:cmpr #’f-cmpr:basis (plnm_algb-basis dmns):bsgn 0:zp zp:dffr-intr #’zero-dffr-intr:dffr-strt :cmbn:aprd-intr #’plnm_algb-aprd-intr:aprd-strt :gnrt:cprd-intr (plnm_algb-cprd-intr zp):cprd-strt :gnrt:dfnt ‘(plnm_algb ,zp, dmns))))

(declare (type hopf-algebra rslt))(setf (slot-value (dffr rslt) ’dfnt)

‘(zero-mrph ,rslt ,rslt -1))rslt)))

In this case, it is possible to create P(u, 2) over Z; consider zp = 0 and dmns = 2:

> (setf p (plnm_algb 0 2))[K1 Hopf-Algebra]

Kenzo returns an object K1 of the class Hopf algebra. Consider the automorphism given by theidentity on P, which induces a trivial contraction c : {P(u, 2), P (u, 2), 1P , 1P , 0}, denoted by r:

> (setf r (trivial-rdct p))[K9 Reduction K1 => K1]

In this case, Kenzo has established a contraction from the object K1 to itself. Applying the methodCRAIC, we will be able to extract the A∞-coalgebra induced by this contraction.

> (craic r)[K9 Reduction K1 => K1]

The machine has returned apparently the same contraction, but let us examine the class of thesmall DGM.

> (setf m (bcc r))[K1 double-a-infty-clgb-h-algb]

The class has changed from Hopf algebra to hopf-alg+induced A∞-coalg. Thus the programreturns the initial coalgebra plus a new structure induced; so we can see the induced maps of theA∞-coalgebra over the generator of the polynomial algebra, i.e.

• �i(u)?:

> (imcprd m 2 1)----------------------{CMBN 2}<-1 * <<Mtnpr[0 0][2 1]>>><-1 * <<Mtnpr[2 1][0 0]>>>---------------------------

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In fact, the answer of the machine is u0 ⊗ u1 + u1 ⊗ u0.• �i(u

2)?:

> (imcprd m 4 2)----------------------{CMBN 4}

<-1 * <<Mtnpr[0 0][4 2]>>><-1 * <<Mtnpr[2 1][2 1]>>><-1 * <<Mtnpr[4 2][0 0]>>>

---------------------------

Now the answer of the machine is u0 ⊗ u2 + u1 ⊗ u1 + u2 ⊗ u0.• �i(u

3)?:

> (imcprd m 6 3)----------------------{CMBN 6}

<-1 * <<Mtnpr[0 0][6 3]>>><-1 * <<Mtnpr[2 1][4 2]>>><-1 * <<Mtnpr[4 2][2 1]>>><-1 * <<Mtnpr[6 3][0 0]>>>

-------------------------

Again the answer is u0 ⊗ u3 + u1 ⊗ u2 + u2 ⊗ u1 + u3 ⊗ u0.

Evidently, with the exception of �2, which is the original coproduct, all A∞-components of�i induced on P(u, 2n) vanish, so the machine returns the coproduct of P(u, 2) as the inducedA∞-coalgebra structure.

The computations were performed by a processor AMD 3000, with 512 mb of memory and40 Gb of hard disc. The time required to compute the maps was independent of the degree of thegenerator and less than 1 second.

3.4 Example 2: The contraction B(Q5(u, 2)) → E(v, 3) ⊗ �(w, 12)

Let us consider the reduced bar construction of a truncated polynomial algebra Q5(u, 2n) and thetensor product E ⊗ � of an exterior algebra and a divided power algebra. In this case, our threebasic objects are implemented in Kenzo as objects of the class and the explicit code is:

• Truncated polynomial algebra (TRPW_ALGB):

(DEFUN TRPW_ALGB (zp prm dmns)(declare (fixnum zp dmns))(unless (evenp dmns)

(error "In TRUNCATED POWER ALGEBRA, the dimension ˜Dshould be even." dmns))

(the hopf-algebra(let ((rslt (hopf-algebra

:cmpr #’f-cmpr:basis (trpw_algb-basis prm dmns):bsgn 0:zp zp:dffr-intr #’zero-dffr-intr:dffr-strt :cmbn:aprd-intr (trpw_algb-aprd-intr prm dmns)

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:aprd-strt :gnrt:cprd-intr (trpw_algb-cprd-intr zp prm):cprd-strt :gnrt:dfnt ‘(trpw_algb ,zp, prm, dmns))))



• Exterior algebra (EXTR_ALGB):

(DEFUN EXTR_ALGB (zp dmns)(declare (fixnum zp dmns))(unless (oddp dmns)

(error "In EXTERIOR ALGEBRA, the dimension ˜Dshould be odd." dmns))


:cmpr #’f-cmpr:basis (extr_algb-basis dmns):bsgn 0:zp zp:dffr-intr #’zero-dffr-intr:dffr-strt :cmbn:cprd-intr (extr_algb-cprd-intr zp):cprd-strt :gnrt:aprd-intr #’extr_algb-aprd-intr:aprd-strt :gnrt:dfnt ‘(extr_algb ,zp, dmns))))



• Divided power algebra (DVPW_ALGB):

(DEFUN DVPW_ALGB (zp dmns)(declare (type fixnum zp dmns))(unless (evenp dmns)

(error "In DIVIDED POWER ALGEBRA, the dimension ˜Dshould be even." dmns))


:cmpr #’f-cmpr:basis (dvpw_algb-basis dmns):bsgn 0:zp zp:dffr-intr #’zero-dffr-intr:dffr-strt :cmbn:aprd-intr (dvpw_algb-aprd-intr zp):aprd-strt :gnrt

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:cprd-intr (dvpw_algb-cprd-intr zp):cprd-strt :gnrt:dfnt ‘(dvpw_algb ,zp, dmns))))



To obtain the A∞-coalgebra induced on the tensor product E ⊗ �, let us consider the contrac-tion from the reduced bar construction of a truncated polynomial algebra Q5(u, 2n) to E ⊗ �.Furthermore, if we denote an element [ur1 | · · · |urm ] of B(Q5(u, 2n)) by its exponents [r1| · · · |rm],0 ≤ ri < 5, the morphisms f , g and φ are given by the following explicit formulae (see [6,14,15]):

f [r1|t1| . . . |rm|tm] ={

m∏k=1

δp,rk+tk

}γm(w),

f [r1|t1| . . . |rm|tm|l] = δ1,l

{m∏

k=1

δp,rk+tk

}v ⊗ γm(w),

where δi,j is the Kronecker delta, and

g(v) = [1], g(γk(w)) = [1|p − 1| . . . |1|p − 1].and

lφ1 = 0, φBQ[1] = 0,

φ[x] = −[1|x − 1] 1 < x < p,

φ[x|y] = −[1|x − 1|y],φ[x|y|z] = −[1|x − 1|y|z] − δp,x+y[1|p − 1|φ(z)].

Monomials in these algebras are encoded as shown, i.e. by sequences of exponentscorresponding to polynomials of the generator in successive factors of the ‘bar’tensor product. Thiscontraction is encoding on the file ‘small-cartan-reduction.cl’. Let us see how the machine works.

The first step is to define the truncated polynomial algebra with coefficients on Z, degree of thegenerator u = 2 and the prime p = 5 (denoted tr).

> (setf tr (trpw_algb 0 5 2))[K1 Hopf-Algebra]

The machine returns an object K1 of the class Hopf algebra. Then let us apply the contractiondefined theoretically before, to do thus, let us execute the command (rdct-bar tr), denoted by rrin short.

> (setf rr (rdct-bar tr))[K32 Reduction K10 => K27]

In this case, Kenzo establishes a contraction from the object K10, i.e. the bar construction ofthe truncated polynomial algebra to the object K27, i.e. E(v, 3) ⊗ �(w, 12). Now, applying themethod CRAIC, we will be able to extract the A∞-coalgebra induced on this tensor product.

> (craic rr)[K32 Reduction K10 => K27]

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Kenzo returns apparently the same contraction, but we can ask about the class of the smallDGM, i.e.:

> (setf m (bcc rr))[K27 double-a-infty-clgb-h-algb]

Now, the class E ⊗ � has changed from Hopf algebra to hopf-alg+induced A∞-coalg. So, nowwe can see which is the image of the maps of the induced A∞-coalgebra over the generator ofE ⊗ �:

• �i(u ⊗ γ0)?:

> (imcprd m 3 (tnpr 3 1 0 0))-------------------------------------{CMBN 1}

<-1 * <<Mtnpr[0 <TnPr 0 0>][3 <TnPr 1 0>]>>><-1 * <<Mtnpr[3 <TnPr 1 0>][0 <TnPr 0 0>]>>>

--------------------------------------------

That is, the induced map is exclusively the coproduct �2.• �i(u

0 ⊗ γ1)?:


<-1 * <<Mtnpr[0 <TnPr 0 0>][12 <TnPr 0 1>]>>><-1 * <<Mtnpr[12 <TnPr 0 1>][0 <TnPr 0 0>]>>><-1 * <<Mtnpr[3 <TnPr 1 0>][3 <TnPr 1 0>][3 <TnPr 1 0>]

[3 <TnPr 1 0>][3 <TnPr 1 0>]>>>------------------------------------------

In this case, the image is a combination of the coproduct �2 plus a map �5.• �i(u

1 ⊗ γ1)?:


<-1 * <<Mtnpr[0 <TnPr 0 0>][15 <TnPr 1 1>]>>><-1 * <<Mtnpr[3 <TnPr 1 0>][12 <TnPr 0 1>]>>><-1 * <<Mtnpr[15 <TnPr 1 1>][0 <TnPr 0 0>]>>>

-----------------------------------------

Again, the induced map is exclusively the coproduct �2.• Finally, let us see �i(u

0 ⊗ γ2)?:

> (imcprd m 24 (tnpr 0 0 24 2))----------------------------------------{CMBN 24}

<-1 * <<Mtnpr[0 <TnPr 0 0>][24 <TnPr 0 2>]>>><-1 * <<Mtnpr[12 <TnPr 0 1>][12 <TnPr 0 1>]>>><-1 * <<Mtnpr[24 <TnPr 0 2>][0 <TnPr 0 0>]>>><-1 * <<Mtnpr[3 <TnPr 1 0>][3 <TnPr 1 0>] [3 <TnPr 1 0>]

[3 <TnPr 1 0>][15 <TnPr 1 1>]>>><-1 * <<Mtnpr[3 <TnPr 1 0>][3 <TnPr 1 0>] [3 <TnPr 1 0>]



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[3 <TnPr 1 0>][3 <TnPr 1 0>]>>>-------------------------------------------------

So, we obtain two non-null maps, i.e. �2 and �5.

It is important to observe that the only non-vanishing structure maps in the inducedA∞-coalgebra structure on E(u, 3) ⊗ �(w, 12) are �2 and �5. Experimentally, we obtain theformula

�5(ui ⊗ γj (w)) =

∑k1+···+kp=j−1

ui+1γkl(w) ⊗ · · · ⊗ ui+1γkp

(w), with u ∈ {0, 1}.

Here is a timetable for computing �i as a function of the degree d of the generator:

Time used (seconds) d = 3 d < 48 d = 48 d = 60 d = 63

�′i s 0, 15′′ <1′′ 53′′ 20 h 28′30′′ Breaks

Finally, we expose the number of addends at different stages of the computation of �6(γ5(w))

in the case p = 5.

Number of summands i = 1 i = 2 i = 3 i = 4

(�[i]φ[⊗(i−1)] · · · φ[⊗2]�[2]g(〈γ5(w)〉) 135 945 4410 15,876

4. Computational cost involved in the process of obtaining A∞-(co)algebras

In this section, we are concerned about the theoretical study of the time and space investedin computing the maps of an A∞-structure induced by a contraction c : {A, M, f, g, φ}. Forsimplicity, we will focus on the case of A being an algebra (see [3] for more details).

Because the BPL appears on the process to obtain A∞-(co)algebras using contractions, theresulting algorithm implies high computational costs [19].

Recall the explicit formulae for the maps μn : M⊗n → M:

μn = (−1)n+1f μ(1) φ[⊗2] μ(2) · · · φ[⊗n−1] μ(n−1) g⊗n.

As for complexity in space, let us consider the number of addends generated in the sum shown.Taking into account that

φ[⊗k] =k−1∑i=0

1⊗i ⊗ φ ⊗ (g f )⊗k−i−1 and that μ(k) =k−1∑i=0

(−1)i+11⊗i ⊗ μA ⊗ 1⊗k−i−1,

the result of applying μn to an element x1 ⊗ x2 ⊗ · · · ⊗ xn has ((n − 1)!)2 addends.Concerning complexity in time, let us assume that each of the component morphisms of the

initial contraction, f, g, φ wastes a unit of time when applied (i.e. each of the morphism isconsidered a basic operation); we will also make this assumption for the composition g, f whichis applied in different terms of the morphisms φ[⊗k].

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Notice that taking an initial element x1 ⊗ x2 ⊗ · · · ⊗ xn, the one obtained by applying g⊗n isg(x1) ⊗ g(x2) ⊗ · · · ⊗ g(xn) and hence, g⊗n is O(n) in time.

On the other hand, the number of operations of each addend of the form 1⊗i ⊗ φ ⊗ (g f )⊗k−i−1

is k − i and that of each addend 1⊗i ⊗ μA ⊗ 1⊗k−i−1 is 1. That is, the number of basic operationscan be expressed by

n + 2(n − 1)!2 + (n − 1)!∑

ki∈{1,2,...,i}(k2 + 1 + k3 + 1 + · · · + kn−1 + 1),

where n comes from g⊗n, 2(n − 1)!2 from the two operations f, μ at the end of each addend andthe big sum corresponds to the operations on the composition

φ[⊗2] μ(2) · · · φ[⊗n−1] μ(n−1).

Notice that the sum is multiplied by (n − 1)! because of all the possibilities for taking an addend1⊗i ⊗ μA ⊗ 1⊗k−i−1 of each μ(k). Now, the previous sum can be expressed as

n + 2(n − 1)!2 + (n − 2)(n − 1)!2 + (n − 1)!n−1∑i=2

1 + · · · + i

i(n − 1)!,

and hence, the total number of operations is

n + n(n − 1)!2 + (n + 3)(n − 2)

4(n − 1)!2.

Therefore, the complexity of the algorithm becomes O(n!2) in time.

4.1 Conclusions and future work

This program gives useful hints when studying the delicate properties of A∞-structures obtainedfrom various processes. For example, one could use the program to study the tensor productof A∞-structures. A future and important extension of this work will be to compute the inducedstructure maps on the small DGM of a contraction when the big initial DGM is a dg-Hopf algebra.

Acknowledgements

This work was partially supported by a project of University of the Basque Country ‘EHU06/05’and by PAICYT researchproject FQM-296.

References

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[2] A. Berciano and F. Sergeraert, Software to compute A∞-(co)algebras: Araia ß Craic, http://www.ehu.es/aba/araia-craic.htm, 2005.

[3] A. Berciano, M.J. Jiménez, and P. Real, On the computation of A1-maps, LNCS 4770 (2007), pp. 43–57.[4] R. Brown, The twisted Eilenberg–Zilber theorem, Celebrazioni Archimedae del Secolo XX, Simposio di Topologia

(1967), pp. 34–37.[5] X. Dousson, J. Rubio, F. Sergeraert, and Y. Siret, The Kenzo program, http://www.fourier.ujfgrenoble.fr/sergeraert/

Kenzo/, 1999.[6] S. Eilenberg and S. Mac Lane, On the groups H(π , n), I, Ann. Math. 58 (1953), pp. 55–106.[7] V.K.A.M. Gugenheim and L.A. Lambe, Perturbation theory in differential homological algebra I, Illinois J. Math.

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[8] V.K.A.M. Gugenheim, L. Lambe, and J.D. Stasheff, Perturbation theory in differential homological algebra, II.Illinois J. Math. 35(3) (1991), pp. 357-373.

[9] J. Huebschmann and T. Kadeishvili, Small models for chain algebras, Math. Z. 207 (1991), pp. 245–280.[10] M.J. Jiménez, A∞-estructuras y Perturbación Homológica, PhD thesis, Universidad de Sevilla, 2003.[11] M.J. Jiménez and P. Real, Rectifications of A∞-algebras, Commun. Algebra 35(9) (2007), pp. 2731–2743.[12] T. Kadeishvili, On the homology theory of fibrations, Russian Math. Surv. 35(3) (1980), pp. 231–238.[13] S. Mac Lane, Homology, in Classics in Mathematics, Springer-Verlag, 1995.[14] A. Prouté, Algébres Diff érentielles Fortement Homotopiquement Associatives, PhD thesis, Université ParisVII, 1984.[15] P. Real, Homological perturbation theory and associativity, Homology, Homotopy Appl. 2(5) (2000), pp. 51–88.[16] S. Saneblidze and R. Umble, Diagonals on the Permutahedra, Multiplihedra and Associahedra, J. Homology,

Homotopy Appl. 6(1) (2004), pp. 363–411.[17] S. Saneblidze and R. Umble, The biderivative and A∞-bialgebras, J. Homology, Homotopy Appl. 7(2) (2005),

pp. 161–177.[18] F. Sergeraert, Kenzo system, http://www-fourier.ujf-grenoble.fr/sergerar/Kenzo/.[19] ———, The computability problem in algebraic topology, Adv. Math. 104(1) (1994), pp. 1–29.[20] W. Shih, Homologie des espaces fibrés, Inst. Hautes Etudes Sci. 13 (1962), pp. 93–176.[21] J.D. Stasheff, Homotopy associativity of H-spaces II, Trans. A.M.S. 108 (1963), pp. 293–312.[22] C.A.Weibel, An introduction to homological algebra, in Cambridge Studies in Advanced Mathematics 38, Cambridge

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A computational approach of A ∞ -(co)algebras