MANUAL OF ARAIA AND CRAICMANUAL OF ARAIA AND CRAIC Technical and theoretical methods to compute A ∞-(co)algebras Ainhoa Berciano Alcaraz Departamento de Matematica Aplicada, Estad´

MANUAL OF ARAIA AND CRAIC

Technical and theoretical methods to computeA∞-(co)algebras

Ainhoa Berciano Alcaraz

Departamento de Matematica Aplicada, Estadıstica e Investigacion Operativa.Facultad de Ciencia y Tecnologıa-Zientzia eta Teknologiaren Fakultatea.

Universidad del Paıs Vasco-Euskal Herriko Unibertsitatea.

2

Contents

Introduction 5

1 Araia and Craic 11.0.1 Theoretical method . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 System of classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Bar-Cobar 112.0.4 Specific Functions . . . . . . . . . . . . . . . . . . . . . . . . . 11

Bibliography 13

3

4 Contents

Introduction

In this paper we would like to explain in detail how does ARAIA-CRAIC work! In fact,it is a module added to kenzo program (http://), a symbolic software to compute someimportant invariants in Algebraic Topology Framework. In particular, it is specialized into compute homology groups of iterated loop spaces.

So, our purpose in this lecture is to present the first symbolic software to compute A∞-(co)algebras given by contractions (special type of homotopy equivalences).

Let us say that the notion of A∞-algebra appears in the literature on the sixties as a gen-eralization of “associative up to homotopy”, introduced by J. D. Stasheff [Sta63].

Given M a differential graded module (dg-module), a morphism µ2 : M ⊗ M → Mof degree zero compatible with the differential of the dg-module, not associative, is saidassociative up to homotopy if it verifies that there exists µ3 : M⊗3 → M of degree +1such that µ3(d⊗ 1⊗ 1 + 1⊗ d⊗ 1 + 1⊗ 1⊗ d) = µ2(µ2 ⊗ 1)− µ2(1⊗ µ2).

Let (X, ∗) be a topological space with a base point ∗ and let ΩX denote the space ofbased loops in X: a point of ΩX is a continuous map f : S1 → X taking the base pointof the circle to the base point ∗.

Let us take as multiplication the composi-tion map

µ2 : ΩX × ΩX → ΩX

(f1, f2) −→ f1 ∗ f2

µ2 is not associative because of

5

6 Introduction

(f1∗f2)∗f3

6=

f1∗(f2∗f3)

But, it is associative up to homotopy, because there exists a Homotopy operator betweenthem

µ3 : [0, 1]× (ΩX)3 → ΩX.

In fact, the algebraic definition of an A∞-algebra is the following:

Definition 0.1. An A∞-algebra is a graded module A, with a family of graded mapsµi : A⊗i → A, of degree i− 2, such that for all i ≥ 1:

i∑n=1

i−n∑k=0

(−1)n+k+nkµi−n+1(1⊗k ⊗ µn ⊗ 1⊗i−n−k) = 0.

• µ1µ1 = 0⇒ µ1 is a differential.

• µ1µ2 = µ2(µ1 ⊗ 1 + 1⊗ µ1)⇒ µ2 compatible with dif.

• µ3(µ1 ⊗ 12 + 12 ⊗ µ1 + 1⊗ µ1 ⊗ 1) + µ1µ3 = µ2(µ2 ⊗ 1− 1⊗ µ2).

Example 0.1. Every dg-algebra is, in particular, an A∞-algebra with µ1 = d, µ2 = µ andµi = 0 for all i ≥ 3.

Example 0.2. The chain complex of the loop space of X , C∗(ΩX) has an A∞-algebrastructure.

Definition 0.2. An A∞-coalgebra is a graded module C, with a family of graded maps∆i : C → C⊗i, of degree i− 2, such that for all i ≥ 1:

i∑n=1

i−n∑k=0

(−1)n+k+nk(1⊗i−n−k ⊗∆n ⊗ 1⊗k)∆i−n+1 = 0.

• ∆1∆1 = 0⇒ ∆1 is a differential.

7

• ∆2∆1 = (∆1 ⊗ 1 + 1⊗∆1)∆2 ⇒ ∆2 compatible with dif.

Example 0.3. Every dg-coalgebra is, in particular, an A∞-coalgebra with ∆1 = d, ∆2 =∆ and ∆i = 0 for all i ≥ 3.

Definition 0.3. A contraction c : N, M, f, g, φ is a 5-tuple, such that

• (N, dN), (M, dM) dg-modules.

• f : N∗ →M∗ and g : M∗ → N∗ morphisms of degree zero;

• φ : N∗ → N∗+1 is a homotopy operator;

• fg = 1M , φdN + dNφ + gf = 1N ;

• fφ = 0 , φg = 0 , φφ = 0.

Corollary 0.1. H∗(N) ' H∗(M)

Theorem 0.2 (GLS74).

r : N →M and N algebra︸︷︷︸⇓

M inherits an A∞ − algebra structure

Theorem 0.3 (JR02).

If M has an A∞ − algebra structure⇓

∃r : N →M contraction, where N algebra

Corollary 0.4. Mathematical Consequence For us, it is the same to speak about A∞-(co)algebras or to speak about contractions!

Corollary 0.5. Computational Consequence There exist real algorithms to compute allthe maps involved in an A∞-(co)algebra structure by a computer!

CRAIC theoretical method:

• Take as input a contraction r : N →M .

• Check if N is a dg-coalgebra.

8 Introduction

• Create a new contraction Ts−1r : Ts−1(N)→ Ts−1(M).

• Produce a new contraction Br : BN → (BM, dM + d∞).

• Extract the new differential induced on the small complex d∞.

• Return the original contraction r : N →M and the maps induced on M .

9

Chapter 1

ARAIA and CRAIC

In this chapter we show some specific functions and methods to compute the A∞-(co)algebrastructure induced on the small dg-module of a reduction if the big one is a (co)algebra.This program component is about 2000 lines long and it was in particular necessary tosignificantly enrich the set of Kenzo classes (see [BS05], [DRSS99]). In particular, wegive some simple examples of computation and comments about the complexity of thealgorithms.

1.0.1 Theoretical method

Given a reduction c : N → M , if N is a dg-(co)algebra, CRAIC, Coalgebra ReductionA-Infinity Coalgebra (resp. ARAIA, Algebra Reduction A-Infinity Algebra) is a CLOSmethod which roughly follows this plan:

1. Take as input a reduction c : N →M .

2. Check if N is a dg-coalgebra.

3. Create a new reduction Ts−1c : Ts−1(N)→ Ts−1(M).

4. Produce a new reduction from the Bar construction of N (the tensor coalgebra ofN with a new differential) to a bar tilde construction of M : Bc : BN → (BM, dT +d∞),using an appropriate perturbation δ, (see [Sta63] for the details).

5. Extract the new differential induced on the small complex d∞ to transform it in acollection of maps of the A∞-coalgebra induced on M .

1

2 Chapter 1. Araia and Craic

6. Finally CRAIC returns the original reduction c : N →M , with the modification ofthe class of M from a dg-module to an object of the class of A∞-coalgebras.

More explicitly, the object M has the same components as before plus a new one imcprd(Induced Multi CoPRoDuct), that is a map from M to TM , defined as a direct sum of∆ii with some appropriate signs. The imcprd component is a functional object and ifwe want to study the A∞-coalgebra structure induced on M, we can make this functionalobject work on some generators or some combinations of them and examine the results.Making this component work is obtained by a Lisp statement formatted as follows:

(imcprd module (degree of element) element).

And the answer is a combination (cmbn) of elements of M⊗n with n ≥ 2, for instance, ifx ∈M⊗3, x = x1⊗ x2⊗ x3, it is coded as

<<Mtnpr [deg(x1) x1][deg(x2) x2][deg(x3) x3]>>

1.1 System of classes

An small remark is necessary to be done. The structure of classes of the original Kenzosystem has to be modified to this new one:

1.2. New Functions 3

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1.2 New FunctionsIn “araia.cl” we can find the functions:

• (mtnpr-abar mtnpr) =function that converts an mtnpr element to abar type.

• (abar-mtnpr abar) =function that converts an abar element to mtnpr type.

• (degr-mtnpr-degr-abar degr mtnpr) =given an element of tensor module, this func-tion returns the degree in the bar construction.

• (CMBN-ABAR-CMBN-MTNPR cmbn) =Abar cmbn to mtnpr cmbn

• (extr-1abar-abar-cmbn cmbn) =from a bar cmbn returns a cmbn with only the ele-ments of simplicial degree 1.


• (extr-n-abar-abar-cmbn n cmbn) =from a bar cmbn returns a cmbn with only theelements of simplicial degree n.

• (araia-mprd-intr rdct) =internal method:given an algebra reduction, returns the a-infty algebra in the small model.

• (araia-mprd rdct)

• (algebra-reduction-a-infty-algebra (rdct reduction))

In the other hand, in “craic.cl” we can find the dual 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 func-tion 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 theelements of cosimplicial degree 1.

• (extr-n-cobar-cobar-cmbn n cmbn) =from a cobar cmbn returns a cmbn with onlythe elements of simplicial degree n.

• (craic-mcprd-intr rdct)

• (craic-mcprd rdct)

• (coalgebra-reduction-a-infty-coalgebra (rdct reduction)). In a short way, (craic)

1.3 ExamplesLet us consider three examples, the first one is a trivial example, where we consider adg-coalgebra and we would like to see the A∞-coalgebra structure induced by an iso-morphism from it to itself. The second one is a little more complicated, and it showsthe transference from the bar construction of a truncated power algebra (that is a dg-coalgebra) to the tensor product of an exterior algebra, E, with a divided power algebra,Γ. The third example is a generalization of the second one: in this case we consider thetensor product of E ⊗ Γ with itself, and we show which can be the formulas in the newA∞-coalgebra.

1.3. Examples 5

1.3.1 Example 1: The trivial reduction P (u, 2)→ P (u, 2)

Let P (u, 2) be the polynomial algebra with one generator u of degree 2 with coefficientsin Z and consider an isomorphism given by the identity in P . Such an isomorphism in-duces a trivial reduction c : P (u, 2), P (u, 2), 1P , 1P , 0, and the A∞-components ∆i

induced on P (u, 2n) are null except ∆2 which is the original one, that is, the result is thesame genuine coalgebra. To create this algebra, we can run:

(setf p (plnm_algb 0 2))← PoLyNoMial ALGeBra in Z, deg(u)=2.[K1 Hopf-Algebra] ← object of class Hopf-Algebra.> (setf r (trivial-rdct p)) ← Construction of the trivial reduction.[K9 Reduction K1 => K1]> (craic r) ← Applying CRAIC.[K9 Reduction K1 => K1]> (setf m (bcc r)) ← Ask for the small dg-module.[K1 double-a-infty-clgb-h-algb]← The class changes from hopf-alg to hopf-alg+induced A∞-coalg.

> (imcprd m 2 1) ← ¿A∞-coalg. over u?----------------------CMBN 2<-1 * <<Mtnpr[0 0][2 1]>>> ← (0, u0)⊗ (2, u1) ∈M⊗2

<-1 * <<Mtnpr[2 1][0 0]>>> ← (2, u1)⊗ (0, u0) ∈M⊗2

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

> (imcprd m 4 2) ← ¿A∞-coalg. over u2?----------------------CMBN 4<-1 * <<Mtnpr[0 0][4 2]>>> ← (0, u0)⊗ (4, u2) ∈M⊗2

<-1 * <<Mtnpr[2 1][2 1]>>> ← (2, u1)⊗ (2, u1) ∈M⊗2

<-1 * <<Mtnpr[4 2][0 0]>>> ← (4, u2)⊗ (0, u0) ∈M⊗2

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

> (imcprd m 6 3) ← ¿A∞-coalg. over u3?----------------------CMBN 6<-1 * <<Mtnpr[0 0][6 3]>>> ← (0, u0)⊗ (6, u3) ∈M⊗2

<-1 * <<Mtnpr[2 1][4 2]>>> ← (2, u1)⊗ (4, u2) ∈M⊗2

<-1 * <<Mtnpr[4 2][2 1]>>> ← (4, u2)⊗ (2, u1) ∈M⊗2

<-1 * <<Mtnpr[6 3][0 0]>>> ← (6, u3)⊗ (0, u0) ∈M⊗2

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


In fact, we can see that the machine returns the coproduct of P (u, 2), as the A∞-coalgebrainduced.

1.3.2 Example 2: The reduction B(Q5(u, 2))→ E(v, 3)⊗ Γ(w, 12)

Let us consider as second example the contraction from the bar construction of a trun-cated power algebra to the tensor product of an exterior algebra to a divided power alge-bra, 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 have explicit formulas (see [?] or [?]).

f [r1|t1| . . . |rm|tm] = ∏m

k=1 δp,rk+tkγm(w),

f [r1|t1| . . . |rm|tm|l] = δ1,l∏m

k=1 δp,rk+tkv ⊗ γm(w),

where δi,j is the Kronecker symbol.

g(v) = [1],

g(γk(w)) = [1|p− 1| k times. . . |1|p− 1].

The elements of these algebras are coded as before, a sequence of natural numbers codingthe “bar” tensor product of the corresponding powers.

> (setf tr (trpw_algb 0 5 2)) ← TRuncated PoWer-ALGeBra in Z, p =5, deg(u)=2.

[K1 Hopf-Algebra] ← object of class Hopf-Algebra.> (setf rr (rdct-bar tr)) ← Construction of the reduction.[K32 Reduction K10 => K27]> (craic rr) ← Applying CRAIC.[K32 Reduction K10 => K27]> (setf m (bcc rr)) ← Ask for the small dg-module.[K27 double-a-infty-clgb-h-algb]← The class changes from hopf-alg to hopf-alg+induced A∞-coalg.

> (imcprd m 3 (tnpr 3 1 0 0)) ← ¿A∞-coalg. over u⊗ γ0?-------------------------------------CMBN 1<-1 * <<Mtnpr[0 <TnPr 0 0>][3 <TnPr 1 0>]>>> ← An element of M⊗2

<-1 * <<Mtnpr[3 <TnPr 1 0>][0 <TnPr 0 0>]>>> ← An element of M⊗2

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

1.3. Examples 7

> (imcprd m 12 (tnpr 0 0 12 1)) ← ¿A∞-coalg. over u0 ⊗γ1?-------------------------------------CMBN 15<-1 * <<Mtnpr[0 <TnPr 0 0>][12 <TnPr 0 1>]>>> ← An element ofM⊗2

<-1 * <<Mtnpr[12 <TnPr 0 1>][0 <TnPr 0 0>]>>> ← An element ofM⊗2

<-1 * <<Mtnpr[3 <TnPr 1 0>][3 <TnPr 1 0>] ← An element of M⊗5

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

> (imcprd m 15 (tnpr 3 1 12 1)) ← ¿A∞-coalg. over u1 ⊗γ1?-------------------------------------CMBN 15<-1 * <<Mtnpr[0 <TnPr 0 0>][15 <TnPr 1 1>]>>> ← An element ofM⊗2



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

> (imcprd m 24 (tnpr 0 0 24 2)) ← ¿A∞-coalg. over u0 ⊗γ2?----------------------------------------CMBN 24<-1 * <<Mtnpr[0 <TnPr 0 0>][24 <TnPr 0 2>]>>> ← An element ofM⊗2



<-1 * <<Mtnpr[3 <TnPr 1 0>][3 <TnPr 1 0>] ← An element of M⊗5

[3 <TnPr 1 0>][3 <TnPr 1 0>][15 <TnPr 1 1>]>>><-1 * <<Mtnpr[3 <TnPr 1 0>][3 <TnPr 1 0>] ← An element of M⊗5





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

An important observation is the following: the only maps non-null in the A∞-coalgebrastructure induced in E(u, 3)⊗Γ(w, 12) are ∆2 and ∆5, and experimentally we obtain theformula of

∆5(ui ⊗ γj(w)) =

∑k1+···+kp=j−1

ui+1γkl(w)⊗ · · · ⊗ ui+1γkp(w).

1.3.3 Example 3: Algorithms to compute A∞-structures on the ten-sor product

An important application of this program is considered now. It makes possible the exper-imental study of the properties of several A∞-structures and to obtain the explicit corre-sponding closed formulas unknown so far, at least in small degrees.

Let us consider for example the tensor product E(u, 3) ⊗ Γ(γ, 8) with itself. We use thenatural notion of tensor product of reductions.

> (setf tr2 (trpw_algb 0 3 2)) ← Q3(u, 2).

[K1 Hopf-Algebra] ← object of class Hopf-Algebra.> (setf r2 (rdct-bar tr2)) ← Construction of the reduction.[K32 Reduction K10 => K27]> (setf rr2 (tnsr-prdc r2 r2)) ← reduction⊗reduction.[K42 Reduction K12 => K33]> (craic rr2) ← Applying CRAIC.[K42 Reduction K12 => K33]> (setf m (bcc rr2))[K33 double-a-infty-clgb-h-algb]

> (imcprd m 3 (tnpr 0 (tnpr 0 0 0 0) 3 (tnpr 3 1 0 0)))--------------------------------------------------CMBN 3<-1 * <<Mtnpr[0 <TnPr <TnPr 0 0> <TnPr 0 0>>]

[3 <TnPr <TnPr 0 0> <TnPr 1 0>>]>>> ← An element of M⊗2

<-1 * <<Mtnpr[3 <TnPr <TnPr 0 0> <TnPr 1 0>>][0 <TnPr <TnPr 0 0> <TnPr 0 0>>]>>> ← An element of M⊗2

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

1.3. Examples 9

> (imcprd m 8 (tnpr 0 (tnpr 0 0 0 0) 8 (tnpr 0 0 8 1)))----------------------------------------CMBN 8<-1 * <<Mtnpr[0 <TnPr <TnPr 0 0> <TnPr 0 0>>]



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




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





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

> (imcprd m 32 (tnpr 16 (tnpr 0 0 16 2) 16 (tnpr 0 0 16 2)))--------------------------------------CMBN 32<-1 * <<Mtnpr[0 <TnPr <TnPr 0 0> <TnPr 0 0>>]



+ terms of M⊗2

<1 * <<Mtnpr[6 <TnPr <TnPr 1 0> <TnPr 1 0>>][6 <TnPr <TnPr 1 0> <TnPr 1 0>>][11 <TnPr <TnPr 1 0> <TnPr 0 1>>][11 <TnPr <TnPr 0 1> <TnPr 1 0>>]>>> ← An element of M⊗4

<1 * <<Mtnpr[6 <TnPr <TnPr 1 0> <TnPr 1 0>>][11 <TnPr <TnPr 0 1> <TnPr 1 0>>][6 <TnPr <TnPr 1 0> <TnPr 1 0>>]



+ terms of M⊗4

<1 * <<Mtnpr[11 <TnPr <TnPr 0 1> <TnPr 1 0>>][6 <TnPr <TnPr 1 0> <TnPr 1 0>>][6 <TnPr <TnPr 1 0> <TnPr 1 0>>][11 <TnPr <TnPr 1 0> <TnPr 0 1>>]>>> ← An element of M⊗4

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

We obtain (experimentally) that the maps involved in the A∞-coalgebra in the tensorproduct (E(u, 3)⊗ Γ(γ, 8))⊗2 are ∆2, ∆3 and ∆4.

Chapter 2

Hopf algebras and Bar-Cobars

2.0.4 Specific FunctionsOur generators are arrays of dim 3, n rows, m columns and pairs (gnr1, degr1) in eachposition (i,j).

• (BRCBR ßrest list) =BaRCoBaR element.

• (DEFCONSTANT +NULL-BRCBR+ (make-brcbr :list +empty-list+))

• (bar-cobar-cmpr cmpr) = We consider the degrees of the elements, row by row, ifdegr(aij) > degr(bij)⇒ brcbr(A) > brcbr(B).

• (sign2 array i j) =me da la suma de los grados de los elementos anteriores a laposicion ij de mi array!!

• (BAR-COBAR-TN-DFFR halgb) =Bar cobar tensor differential

• (bar-cobar-dffr-intr dffr zp cmpr) =bar cobar tensor internal differential.

• (dffr-int-i-j array i j dffr) =differential over the element (i,j) of the matrix.

• (VRTC-BAR-COBAR hopf-algebra) =vertical bar cobar.

• (BAR-COBAR-FV-DFFR halgb) =The differential given by the product of H.

• (bar-cobar-intr-fv-dffr aprd zp cmpr) =internal differential associated to the productof H.

• (bar-cobar-intr-f-v-i array i aprd) =differential over the line i?

• (BAR-COBAR-FH-DFFR halgb) =the differential given by the coproduct of H.

11

12 Chapter 2. Bar-Cobar

• (bar-cobar-intr-fh-dffr cprd zp cmpr) =the internal differential associated to the co-product of H.

• (bar-cobar-intr-f-h-i array i cprd) =differential over the column i.

• (dsc halgb) =sum of differentials.

• (dsc-intr vrtc-dffr hrzn-dffr zp cmpr)= internal method.

• (CPDF-INTR mrph2 mrph1) = internal method of composition of differentials.

• (BCVH halgb) =Composition of hrzn-mrph and latter the vertical one.

• (BCHV halgb) =Composition of vrtc-mrph and latter the horizontal one.

• (BAR-COBAR hopf-algebra) =total bar-cobar complex.

• (BAR-COBAR-INTR-DFFR tnsr-dffr vrtc-dffr hrzn-dffr zp cmpr) =the total differ-ential on the bar cobar complex.

Bibliography

[Ber06] A. Berciano-Alcaraz, Symbolic Description of Hopf Algebras and Perturbation,Proceedings EACA, pp. 37-40, 2006.

[Ber06b] A. Berciano-Alcaraz, Calculo simbolico y Tecnicas de Control de A∞-estructuras. Tesis Doctoral Universidad de Sevilla, 2006.

[Ber06c] A. Berciano-Alcaraz, Effective Homology of Hopf Algebras. ”Global Inte-grability of Field Theories”. Karlsruhe University Press, pp. 27-38, 2006.

[BJR06] A. Berciano, M.J. Jimenez and P. Real, Reducing computational cost of thebasic perturbation lemma, Lecture Notes in Computer Science, LNCS 4194, pp.33-48, 2006.

[BR04] A. Berciano and P. Real, The A∞-coalgebra structure of the Zp-homology ofEilenberg-Mac Lane spaces, Proceedings EACA, 29-33, 2004.

[BS05] A. Berciano-Alcaraz and F. Sergeraert: Software to compute A∞-(co)algebras:Araia ß Craic. http://www.ehu.es/aba/araia-craic.htm

[Bro57] E.H. Brown, Finite computability of Postnikov complexes, Annals of Math. 65(2) (1957), 1-20.

[Bro59] E.H. Brown, Twisted tensor products I, Annals of Math. 65 (1959), 223-246.

[Bro67] R. Brown, The twisted Eilenberg-Zilber theorem, Celebrazioni Archimedae delSecolo XX, Simposio di Topologia (1967), 34-37.

[DRSS99] X. Dousson, J. Rubio, F. Sergeraert e Y. Siret: The Kenzo program. InstitutFourier, Grenoble (1999). http://www-fourier.ujf-grenoble.fr/ sergeraert/Kenzo/

[May67] J.P. May, Simplicial Objects in Algebraic Topology, D. Van Nostrand Com-pany (1967).

[RS02] J. Rubio y F. Sergeraert, Constructive algebraic topology, Bull. Sci. Math, 126(2002), 389-412.

13

14 Bibliography

[Ser87] F. Sergeraert, Homologie effective I, II, C. R. Acad. Sc. Paris 304 (1987), 279-281y 319-321.

[Ser94] F. Sergeraert, The computability problem in algebraic topology. Adv. Math. 104(1) (1994), 1-29.

[Shi62] W. Shih, Homologie des espaces fibres, Inst. Hautes Etudes Sci., 13 (1962), 93-176.

[Sta63] J.D. Stasheff, Homotopy associativity of H-spaces I, II, Transactions of A.M.S.108 (1963), 275-292, 293-312.

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