Margot Vulliez

Matthew A Gleason

Aurélien Souto-Lebel

Yann Quinsat

Claire Lartigue

Steven Kordel

Adam C Lemoine

Christopher A Brown
email: brown@wpi.edu.

Steven P Kordell

Multi-scale Curvature Analysis and Correlations with the Fatigue Limit on Steel Surfaces after Milling

Keywords: roughness, fatigue, machining

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The objective of this work is to investigate the strength of correlations between the curvatures of the topographies of ball-end milled steel specimens and the fatigue limits in four-point bending. Curvatures are calculated from profiles extracted from measured topographies. The curvatures change with position and with the scale of observation and provide an indication of the integrity of the surface topography.

A quantitative understanding of the relation between the topography and the fatigue limit can facilitate better product and process design for parts where fatigue resistance is important. The discovery of strong correlations between topographic characterization parameters and the fatigue limit could depend on identifying the appropriate characterization parameter and the appropriate scale, which would be similar to correlations that have been found with adhesion, friction, fracture surfaces, and mass transfer [START_REF] Brown | Fundamental scales of adhesion and area-scale fractal analysis[END_REF][START_REF] Berglund | Milled Die Steel Surface Roughness Correlation with Steel Sheet Friction[END_REF][START_REF] Berglund | On Discovering Relevant Scales in Surface Roughness Measurement-An Evaluation of a Band-Pass Method[END_REF][START_REF] Cantor | Scale-based correlations of relative areas with fracture of chocolate[END_REF][START_REF] Moreno | Effect of food surface roughness on oil uptake by deep-fat fried products[END_REF].

State-of-the-art

Although it is generally accepted that topography influences the fatigue limit, quantitative descriptions with experimental confirmation appear to be rare. Geometric stress concentration (K t ) is based on curvature (k=1/ρ) and the depth of a notch (t) [START_REF] Peterson | Stress concentration factors[END_REF]:

K t = 1 + 2 (t/ρ) ½ (1)
Models have been proposed that use conventional roughness characterizations in the calculation of stress concentrations. Arola and Ramulu [START_REF] Arola | An examination of the effects from surface texture on the strength of fiber-reinforced plastics[END_REF] studied rough surface topographies of machined fiber reinforced composites. They replaced notch depth with the conventional roughness characterization parameters: R a average roughness, R t maximum peak to valley depth, R z average of the greatest peak to valley depth in five sections. This produces an expression for tensile loading:

K t = 1 + 2 (R a /ρ)(R t /R z ) (2)
For machined metals, however, the surface roughness parameters appear not to be sufficient, at least on an individual basis, for estimating fatigue strength [START_REF] Novovic | The effect of machined topography and integrity on fatigue life[END_REF]. Cracks that lead to fatigue failure of mechanical parts have been found to originate from micro-scratches on the surface, which are not described by conventional roughness parameters [START_REF] Guillemot | Prediction of the endurance limit taking account of the microgeometry after finishing milling[END_REF]. Cracks are not restricted to originating on topographic features and could originate on microstructural elements. Useful expressions for the stress concentrations could depend on the material and the preparation of the surface. The dependence of the stress concentration on the curvatures calculated from measured topographies seems like a logical extension of the fundamental development of stress concentrations for notches. There are established procedures for strength analysis in shafts and axles that are based on the static strength with stress concentration factors (e.g., DIN 743 [START_REF]DIN 743-3 Shafts and axles, calculation of load capacity -Part 3:Strength of materialsstandard[END_REF]).

The roughness of topographies that results from machining of metals has regular and irregular, or chaotic, components [START_REF] Quinsat | Surface topography in ball end milling process: Description of a 3D surface roughness parameter[END_REF][START_REF] Quinsat | Characterization of 3D surface topography in 5-axis milling[END_REF]. Machined surfaces also have regular components at larger scales, which are usually associated with the feed. The irregular components have been modeled as stochastic, successfully reproducing Abbot curves [START_REF] Denkena | Kinematic and Stochastic Surface Topography of Machined TiAl6V4-Parts by means of Ball Nose End Milling[END_REF]. These irregular topographies contain features with a range of curvatures that vary with scale and position. These features defy any simple curvature descriptions as smooth notches.

Recently, methods of calculating curvatures of roughness profiles on a multi-scale basis have been proposed [START_REF] Gleason | Characterizing Radii on the Tips of Biopsy Needles, .Structured and Freeform Surfaces[END_REF][START_REF] Gleason | Profile Curvatures by Heron's Formula as a Function of Scale and Position on an Edge Rounded by Mass Finishing[END_REF]. The current work applies these recent developments of the multi-scale calculation of curvatures on profiles to the characterization of milled surfaces, and then uses it for correlation with the fatigue limit.

A strong correlation is being sought here between some representation of the curvature of features on the surface and the fatigue limit in four point bending. The fatigue specimens are all machined with a ball-nose endmill, the same feed per tooth but with different tool inclinations, and different feed directions. Because the influence of residual stress and micro-geometry can be coupled, in four specimens the residual stresses were removed, leaving only the influence of the micro-geometry. The curvature is calculated as a function of scale and position. It is not immediately obvious how to use this myriad of calculated curvatures to find a strong correlation with the fatigue limit. Several statistical representations of curvature are investigated. A method is proposed for finding the appropriate scale for calculating the curvature (patent pending).

Methods

Bainitic 50CrMo4 steel (Re=480 Mpa; Rm=1080 Mpa) was machined by ball-end milling on a Mikron UCP710 five-axis vertical machining centre to create six different types of surfaces on approximately 80 specimens for fatigue texting. These specimens have been studied extensively previously [9,11,12 ,16, 17]. The tool inclination relative to the surface is defined by the lead angle β, as shown in Fig. 1. Two values of β were studied (β = 45° and β = -3°). The specimen dimensions are 100x20x4mm. The machining conditions are listed in Table 1. Four of the machined specimens were subjected to a stress-relieving heat treatment at 470⁰C for one hour, with a cooling rate of 90⁰C/hr. As indicated below in Table 2 during the machining the milling tool was fed both transversely and longitudinally with respect to the eventual loading direction, i.e., tensile stress axis, in four-point bending fatigue tests [START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF]. Each of the two inclinations was repeated in both directions for a total of four specimens. These four specimens were subjected to the stress relieving heat treatment. Two additional specimens were machined in the transverse direction and were not subjected to the heat treatment.

The nominal scallop height, h c , can be calculated to be 5μm from the step over and the tool nose radius by theoretical approximation (h c =a e ²/8r). This expression is based on smooth surface generation by a smooth tool edge with no ploughing.

The machined surfaces were measured with a 3D Stil Microstation. This is a horizontal profile-scanning instrument with a confocal chromatic height sensor with a 100μm vertical range. All measurements were cropped to 700x700μm. A modal outlier filter [START_REF] Legoic | Outlier filtering: a new method for improving the quality of surface measurements[END_REF] was used to eliminate spike artefacts from the measurements. This modal filtering is characterized by two parameters: the risk (α=10 -5 here) and the number of modes for the calculation (250 here). Thresholding and Gaussian filtering were attempted but not used. According to area-scale analysis [START_REF]Geometrical product specifications (GPS) --Surface texture: Areal --Part 2: Terms, definitions and surface texture parameters[END_REF][START_REF]Surface Texture: Roughness, Waviness and Lay[END_REF], these filters did not work as well as the modal filter for removing spike-like measurement artifacts. The relative areas were higher for these methods. In the final analysis, only the modal filtering was used.

Curvature calculations were done on a total of 1120 profiles extracted parallel to the axis of maximum tensile stress. Because the sampling intervals in the profiling direction were different for two of the measurements, 0.625μm versus 0.5μm, the common interval of 0.625μm was chosen and the profiles closest to that sampling interval were extracted.

A geometric method, based on Heron's formula [14. 15], is used to calculate the curvatures based on three heights equally separated laterally. The lateral spacing of the two exterior heights is used to represent the scale. For the calculation of the curvature from three measured heights, the algorithm steps along a profile of height sample by height sample, calculating the curvature as a function of position at one scale, and then the calculation repeats for each scale.

A strict protocol was followed for the fatigue tests, which were done in four-point bending [START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF]. The two extreme loading points (or, more precisely, loading lines) were 50mm apart, and the central, interior loading points were 20mm apart.

Results

The fatigue results are shown in Table 2, along with the machining conditions and an indication of whether or not they were stress relieved. The specimens with polished surfaces have a fatigue limit of 755MPa [START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF].

Examples of renderings of measured surfaces are shown in Figures 2 a&b, from specimens milled with inclination of -3⁰ and 45⁰ with the stress relieved. The fine-scale roughness, which could be described as variations from an otherwise smooth surface that is defined only by the tool shape and motion, is clearly evident, appearing as irregular or chaotic topographic elements. The roughness values for the height parameters are shown in Table 3. The S parameters are means and standard deviations of the areal measurements. It is noted that the identical machining conditions do not produce surfaces with identical roughness parameters. The peak-to-valley roughness values (St) differ clearly from the theoretical calculation of the cusp height (h c =5μm) in all cases. At the larger scales, the values of the curvatures diminish and tend towards zero, consistent with the flat nature of the specimens at larger scales. Figures 4a&b go up to a scale of 600μm, where it is difficult to discern any change in curvature with position in contrast to the finer scales. At the finest scales changes in curvature with position are obvious. Therefore, additional plots are shown at the larger scales with the curvature axis re-scaled in order to make the changes in curvature with position along the profile more obvious. Several criteria, combinations of the curvature and roughness parameters, were regressed linearly with the fatigue limit over a range of scales. The most successful criteria, shown in Table 4, come from analyses of the distribution of the curvatures. The mean and standard deviations of the curvatures are calculated on each profile at each scale. The maximum values from these calculations from all of the 1120 profiles might match the most dangerous feature on the surface for initiating a fatigue crack. Regression analyses with the fatigue limit versus these maximum values are performed over all the scales in the measurements. The results are shown in Table 4 with the regression coefficients and scales. 0.96 610 Fig. 6 shows a plot of the regression coefficient R² as a function of scale for the most successful criterion, maximum mean of the curvatures plus two standard deviations (max(μ(k) + 2σ(k))). The maximum R² (0.96) is reached at a scale of 610μm. Fig. 7 shows the plot of max(μ(k) + 2σ(k)) versus the fatigue limit at 610μm. This remarkably strong correlation is found only at a scale of 610μm. Fig. 6. The coefficient of linear regression curvature (mean of the curvatures, plus two standard deviations) regressed with the fatigue limit, plotted versus scale. Fig. 7. Mean of the curvatures, plus two standard deviations versus the fatigue limit at a scale of 610μm. The two circled results were from specimens that were not heat treated.

Discussion

The curvature calculated from measured profiles provides an indication of the stress concentration that would influence fatigue strength. Curvature, or radius, of notch roots has been used in the traditional, theoretical developments of the calculation of stress concentrations.

Fatigue cracks can initiate on things other than topographic features. However, in this study, it has been decided to focus on the machining and resulting topography. However, the strong correlation of the fatigue limit with the curvature confirms our choice, regardless of the heat treatment and the direction of machining relative to the applied tensile stress.

This work suggests that a systematic approach to finding the relations between roughness and fatigue limit could be based on determining the scale at which the mean curvature correlates most strongly with the fatigue limit.

The scale at which the strongest correlation is found is larger than the feed per tooth. It would be interesting to see if the maximum R² would also be found at this same scale if the machining were to be performed with finer feeds per tooth.

The negative curvatures (convex up) should not be expected to influence the fatigue limit. At the scale of the strongest correlation it is noted that there is little negative curvature. In any event the consideration of the mean plus two standard deviations of the curvatures at each scale is robust with respect to the inclusion of negative curvatures. This approach guarantees that the features that are the most likely to provide dangerous stress concentrations will be considered.

The stress-relieving heat treatment seems to have little influence on the correlation. As indicated in Fig. 5 the two specimens without stress relief appear to be on the same curve as those with stress relief. In situations where residual stresses play a larger role, the influence of the residual stress on the fatigue limit might be considered by a super position of the residual stress on the applied stresses. Then the same sort of calculation of curvature could be used to determine the influence of the microgeometry.

Conclusions

 The maximum mean curvature plus two standard deviations correlates strongly with the limit in fatigue at a scale of 610μm of the ball-end milled steel parts in this study.  The strength of the correlations with the multi-scale curvature parameters and the fatigue limit varies with the scale of calculation.  A method has been developed and demonstrated for determining for finding the appropriate scale for multi-scale parameters for predicting fatigue behaviour when applied to certain material and manufacturing process.

Fig. 1 .

 1 Fig. 1. Schematic of ball-end milling with an inclined tool indicating only a regular component of the machined surface topography

Fig. 2a .

 2a Fig. 2a. Rendering of a measurement of the surface milled with an inclination of -3º and then stress relieved and subjected to transverse loading as indicated by the line in the upper left image.

Fig. 2b .

 2b Fig. 2b. Rendering of a measurement of the surface milled with an inclination of 45º and then stress relieved and subjected to longitudinal loading, as indicated by the line in the upper left image.

Figures

  Figures 3a&b show and curvatures versus position at scales of 150 and 500μm. The positive curvatures are concave up and tend to be at the bottoms of valleys, more likely for fatigue crack initiation.

Fig. 3a .

 3a Fig. 3a. Profile, extracted as indicated in Fig. 2a, with corresponding curvatures versus position at scales of 150 and 500μm.Figures 4&5 show the curvatures versus position and scale. Curvature is plotted on the vertical axis is and is also represented by colour. The horizontal axes denote the positions along the extracted profile and the scale at which the curvature is calculated. As the scale increases, there are fewer positions at which the curvature can be calculated, and the plot along the profile axis becomes narrower.

Fig.

  Fig. 3b. Profile, extracted as indicated in Fig. 2b, with corresponding curvatures versus position at scales of 150 and 500μm.

Fig. 4a .

 4a Fig. 4a. Profile curvatures (vertical axis, colour) as a function of scale and position from the surface shown in Figures 2a and3a milled at -3º.

Fig. 4b .

 4b Fig. 4b. Profile curvatures (vertical axis, colour) as a function of scale and position from the surface shown in Figures 2b and3b milled at 45º.

Figures

  Figures 5a&b show the curvatures versus position at the largest scales.

Fig. 5a .

 5a Fig. 5a. Profile curvatures (vertical axis, colour) as a function of scale and position from the surface shown in Figures 2a and3a milled at -3º.

Fig. 5b .

 5b Fig. 5b. Profile curvatures (vertical axis, colour) as a function of scale and position from the surface shown in Figures 2b and3b milled at 45º.

Table 1 .

 1 Machining conditions

	__________________________________________
	Spindle speed	ω	9550 rpm
	Feed per tooth	f	400μm per tooth
	Depth of cut	p	500μm
	Step over	ae	450μm
	Tool nose radius	r	5mm
	Inclination angles	βf	45⁰, -3⁰_________

Table 2 .

 2 Fatigue limits, machining conditions, and stress relief

	_____________________________
	stress Angles	Feed	Fatigue limit
	relieved βf deg	direction	MPa
	no	-3	transverse	670
	yes	-3	transverse	633
	yes	45	transverse	675
	no	45	transverse	680
	yes	-3	longitudinal	735
	__ yes	45	longitudinal	715_____

Table 3 .

 3 Conventional roughness characterizations of means of heights. All values are in micrometres.

	Specimen* Sa	Sq	St	Sv
	N-3T	1.39 1.79	11.2	3.81
	Y-3T	1.45 1.73	17.2	5.71
	Y45T	1.59 1.92	11.8	6.35
	N45T	1.46 1.76	10.3	5.29
	Y-3L	1.72 2.05	13.8	6.66
	Y45L_____ 2.28 2.71	20.48	7.99
	*Stress relief Y/N, angle, direction

Table 4 .

 4 Regression coefficients for several candidate criteria

	Method	max R² scale (μm)
	max(k)	0.76	620
	μ (10 max(k))	0.8	618
	max(μ(k) + σ(k))	0.65	620
	max(μ(k) + 2σ(k))

Acknowledgements

The authors gratefully acknowledge the steel manufacturer ASCOMETAL, for providing the steel and performing fatigue tests; DigitalSurf for generously providing MountainsMap software for calculating conventional roughness characterization parameters and rendering images. Sfrax was used for area-scale calculations and was provided by Surfract, which is owned by co-author Brown. A provisional patent has been filed in the United States on this method for determining the fatigue limit from the surface topography.


Alexandre Quesney
email: alexandre.quesney@univ-nantes.fr

Homotopy BV-algebra structure on the double cobar construction

Keywords: Cobar construction, Homotopy G-algebra, BV-algebra, Hopf algebra. 2000 MSC: 55P48, 55U10, 16T05

We show that the double cobar construction, Ω 2 C * (X), of a simplicial set X is a homotopy BValgebra if X is a double suspension, or if X is 2-reduced and the coefficient ring contains the field of rational numbers Q. Indeed, the Connes-Moscovici operator defines the desired homotopy BV-algebra structure on Ω 2 C * (X) when the antipode S : ΩC * (X) → ΩC * (X) is involutive. We proceed by defining a family of obstructions

When X is a suspension, the only obstruction remaining is O 2 := E 1,1 -τE 1,1 where E 1,1 is the dual of the ⌣ 1 -product. When X is a double suspension the obstructions vanish.

Introduction

Adams' cobar construction provides a model of the loop space of a 1-connected topological space [START_REF] Adams | On the cobar construction[END_REF]. The cobar construction is a functor from differential graded coalgebras to differential graded algebras; for iteration, a coproduct is thus needed. For a 1-reduced simplicial set X, Baues defined [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF] a DG-bialgebra structure on its first cobar construction ΩC * (X). The resulting double cobar construction is an algebraic model for the double loop space. We show that the double cobar construction, Ω 2 C * (X), of a simplicial set X is a homotopy BValgebra (a homotopy G-algebra in the sense of Gerstenhaber-Voronov [START_REF] Gerstenhaber | Homotopy G-algebras and moduli space operad[END_REF] together with a degree one operator) if X is a double suspension, or if X is 2-reduced and the coefficient ring contains the field of rational numbers Q.

Baues' coproduct on ΩC * (X) is equivalent to a homotopy G-coalgebra structure {E k,1 } k≥1 on the DG-coalgebra C * (X), that is a family of operations

E k,1 : C * (X) → C * (X) ⊗k ⊗ C * (X), k ≥ 1,
satisfying some relations. This corresponds also to the coalgebra structure of C * (X) over the second stage filtration operad F 2 χ of the surjection operad χ given in [START_REF] Berger | Combinatorial operad actions on cochains[END_REF][START_REF] Mcclure | Multivariable cochain operations and little n-cubes[END_REF], see Section 4.1. Since a bialgebra structure determines the antipode (whenever it exists), the antipode S : ΩC * (X) → ΩC * (X) on the cobar construction is then determined by the homotopy G-coalgebra structure on C * (X).

As a model for the chain complex of double loop spaces, the double cobar construction is expected to be a Batalin-Vilkovisky algebra up to homotopy. Indeed, it is well-known that the circle action on the double loop space Ω 2 X by rotating the equator defines a BV operator; thereby the homology H * (Ω 2 X) is a Batalin-Vilkovisky algebra [START_REF] Getzler | Batalin-Vilkovisky algebras and two-dimensional topological field theories[END_REF].

The cobar construction ΩH of an involutive Hopf algebra turns out to be the underlying complex in the Hopf-cyclic Hochschild cohomology of H, [START_REF] Connes | Cyclic cohomology and Hopf algebra symmetry[END_REF]. The cyclic operator requires an involutive antipode on the underlying bialgebra H. Assuming that the antipode is involutive, Menichi proved [START_REF] Menichi | Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras[END_REF], that for a unital (ungraded) Hopf algebra H, the Connes-Moscovici operator induces a Batalin-Vilkovisky algebra on the homology of the cobar construction H * (ΩH).

By defining a family of operations

O n : C * (X) → C * (X) ⊗n , n ≥ 2,
we provide a criterion for the involutivity of S in terms of the operations

E k,1 : C * (X) → C * (X) ⊗k ⊗ C * (X).
By Kadeishvili's work [START_REF] Kadeishvili | On the cobar construction of a bialgebra[END_REF] the double cobar construction is a homotopy G-algebra, it is endowed with a family of operations

E 1,k : Ω 2 C * (X) ⊗ Ω 2 C * (X) ⊗k → Ω 2 C * (X), k ≥ 1,
satisfying some relations. In particular, the following bracket

{a; b} = E 1,1 (a ⊗ b) -(-1) (|a|+1)(|b|+1) E 1,1 (b ⊗ a), a, b ∈ Ω 2 C * (X),
together with the DG-product of Ω 2 C * (X), induces a Gerstenhaber algebra structure on the homology H * (Ω 2 C * (X)). The vanishing of the operations

O n : C * (X) → C * (X) ⊗n , n ≥ 2,
gives a sufficient condition for the extension of the homotopy G-algebra structure on Ω 2 C * (X)

given by Kadeishvili to the homotopy BV-algebra structure whose BV-operator is the Connes-Moscovici operator. We establish the following for a general homotopy G-coalgebra, Proposition 3.3. Let (C, d, ∇ C , E k,1 ) be a homotopy G-coalgebra.

1. The cobar construction ΩC is an involutive DG-Hopf algebra if and only if all the obstructions O n : C → C ⊗n defined in (3.5) for n ≥ 2 are zero. 2. Let C be 2-reduced i.e. C 0 = R and C 1 = C 2 = 0. If all the obstructions O n : C → C ⊗n are zero, then the double cobar construction Ω 2 C is a homotopy BV-algebra given by the Connes-Moscovici operator.

We apply this criterion to the homotopy G-coalgebra C * (X) of a 1-reduced simplicial set X. We show that, when ΣX is a simplicial suspension, the family of operations

O n : C * (ΣX) → C * (ΣX) ⊗n reduces to O 2 : C * (ΣX) → C * (ΣX) ⊗ C * (ΣX).
This operation O 2 is the deviation from the cocommutativity of the operation

E 1,1 : C * (ΣX) → C * (ΣX) ⊗ C * (ΣX).
In fact, E 1,1 is the only non-trivial operation of the family {E k,1 } k≥1 defining the homotopy Galgebra structure on C * (ΣX), see Propositions 4.4 and 4.5. However, in the case of a double simplicial suspension Σ 2 X, the operation E 1,1 is also trivial, and all the obstructions

O n : C * (Σ 2 X) → C * (Σ 2 X) ⊗n , n ≥ 2,
are zero. As a consequence, the cobar construction ΩC * (Σ 2 X) is the free tensor DG-algebra with the shuffle coproduct. Thus we have, Theorem 4.6. Let Σ 2 X be a double suspension. Then:

• the homotopy G-coalgebra structure on C * (Σ 2 X) corresponding to Baues' coproduct

∇ 0 : ΩC * (Σ 2 X) → ΩC * (Σ 2 X) ⊗ ΩC * (Σ 2 X),
has trivial higher operations i.e. E 1,k = 0 for k ≥ 1;

• the double cobar construction Ω 2 C * (Σ 2 X) is a homotopy BV-algebra with the Connes-Moscovici operator as BV-operator.

On the other hand, if the ground ring contains the field of rational numbers Q, we deform (see Section 4.4 for the precise statement) (ΩC * (X), ∇ 0 , S) into a cocommutative DG-Hopf algebra (ΩC * (X), ∇ ′ 0 , S ′ ). Therefore, (ΩC * (X), ∇ ′ 0 , S ′ ) has an involutive antipode and the homotopy BValgebra structure we consider on the deformed double cobar construction Ω(ΩC * (X), ∇ ′ 0 , S ′ ) follows. Thus, we have Theorem 4.8. Let X be a 2-reduced simplicial set. Then the double cobar construction Ω(ΩC * (X), ∇ ′ 0 , S ′ ) over R ⊃ Q coefficients is a homotopy BV-algebra with BV-operator the Connes-Moscovici operator.

The paper is organized as follows.

In the first section we review background materials on the bar-cobar constructions and Hopf algebras. The second section is devoted to the structures of homotopy G-algebras [START_REF] Kadeishvili | On the cobar construction of a bialgebra[END_REF] and homotopy BValgebras [START_REF] Menichi | Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras[END_REF] on the cobar construction. These first two sections fix notations and sign conventions.

In the third section we define the family of operations O n : C → C ⊗n , n ≥ 2, on a homotopy G-coalgebra (C, E k,1 ).

In the section 4 we give applications : In the subsection 4.1 we set the convention for the homotopy G-coalgebra structure on C * (X). We compare the homotopy G-coalgebra structure on C * (X) coming from Baues' coproduct with the action of the surjection operad given in [START_REF] Berger | Combinatorial operad actions on cochains[END_REF][START_REF] Mcclure | Multivariable cochain operations and little n-cubes[END_REF]. Subsections 4.2 and 4.3 give applications to simplicial suspensions. In the case of single suspension ΣX the family of operations O n : C * (ΣX) → C * (ΣX) ⊗n , n ≥ 2 reduces to O 2 . We show that, for a double suspension, this last obstruction O 2 vanishes. The last subsection 4.4 is devoted to the rational case. We prove that the double cobar construction of a 2-reduced simplicial set is a homotopy BV-algebra. In Appendix we recall and specify some facts about the Hirsch and the homotopy G-algebras. In particular, we make explicit the signs related to our sign convention.

Notations and preliminaries

Conventions and notations

Let R be a commutative ring. A graded R-module M is a family of R-modules {M n } where indices n run through the integers. The degree of a ∈ M n is denoted by |a|, so here |a| = n. The r-suspension s r is defined by (s r M) n = M n-r . Algebras (respectively coalgebras) are understood as associative algebras (respectively coassociative coalgebras). A unital R-algebra (A, µ, η) is called augmented if there is an algebra morphism ǫ : A → R. We denote by A the augmentation ideal Ker(ǫ). For a coalgebra (C, ∇) the n-iterated coproduct is denoted by

∇ (n) : C → C ⊗n+1
for n ≥ 1. We use the Sweedler notation,

∇(c) = c 1 ⊗ c 2 and ∇ (n) (c) = c 1 ⊗ c 2 ⊗ • • • ⊗ c n+1 ,
where we have omitted the sum. A counital R-coalgebra (C, ǫ) is called coaugmented if there is a coalgebra morphism η : R → C. We denote by C = Ker(ǫ) the reduced coalgebra with the reduced coproduct

∇(c) = ∇(c) -c ⊗ 1 -1 ⊗ c. A (co)algebra A is called connected if it is both (co)augmented and A n = 0 for n ≤ -1 and A 0 ∼ = R. A (co)algebra, A is called n-reduced if it is both connected and A k = 0 for 1 ≤ k ≤ n. A coaugmented coalgebra C is called conilpotent if the following filtration F 0 C := R F r C := R ⊕ {c ∈ C|∇ n (c) = 0, n ≥ r} for r ≥ 1, is exhaustive, that is C = r F r C.

The bar and cobar constructions

We refer to [START_REF] Loday | Algebraic Operads. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete[END_REF]Chapter 2] for the background materials related to the bar and cobar constructions.

The cobar construction is a functor

Ω : DGC 1 → DGA 0 (C, d C , ∇ C ) → ΩC = (T(s -1 C), d Ω )
from the category of 1-connected DG-coalgebras to the category of connected DG-algebras. Here, T(s -1 C) is the free tensor algebra on the module s -1 C and d Ω is the unique derivation such that

d Ω (s -1 c) = -s -1 d C (c) + (s -1 ⊗ s -1 )∇ C (c) for all c ∈ C.
The bar construction is a functor

B : DGA 0 → DGC c (A, d A , µ A ) → B A = (T c (s A), d B )
from the category of connected DG-algebras to the category of conilpotent DG-coalgebras. Here, T c (s A) is the cofree tensor coalgebra on the module s A and d B is the unique coderivation with components

T c (s A) s A ⊕ (s A) ⊗2 s A. -s d A s -1 + s µ A (s -1 ⊗ s -1 )
We recall the bar-cobar adjunction, Theorem 1.1. [14, Theorem 2.2.9] For every augmented DG-algebra Λ and every conilpotent DG-coalgebra C there exist natural bijections

Hom DG-Alg (ΩC, Λ) ∼ = Tw(C; Λ) ∼ = Hom DG-Coalg (C, B Λ).
The set Tw(C; Λ) of twisting cochains from C to Λ is the set of degree 1 linear maps f : C → Λ verifying the twisting condition:

∂ f := d f + f d = -µ Λ ( f ⊗ f )∇ C .

Hopf algebras

Definition 1.2. For a DG-bialgebra (H, d, µ, η, ∇, ǫ) an antipode S : H → H is a chain map which is the inverse of the identity in the convolution algebra Hom(H, H); the convolution product being f ⌣ g = µ( f ⊗ g)∇. Explicitly, S satisfies S(a 1 )a 2 = ηǫ(a) = a 1 S(a 2 ) for all a ∈ H. (c) a 2 S(a 1 ) = ηǫ(a). vi. If H is commutative or cocommutative, then S 2 = id.

Homotopy structures on the cobar construction

Homotopy G-algebra on the cobar construction

We present the homotopy G-algebra structure on the cobar construction of a 1-reduced DGbialgebra given in [START_REF] Kadeishvili | On the cobar construction of a bialgebra[END_REF]. The homotopy G-algebras are also known as the Gerstenhaber-Voronov algebras defined in [START_REF] Gerstenhaber | Homotopy G-algebras and moduli space operad[END_REF]. We define the Hirsch algebras and the homotopy G-algebras via the bar construction.

Definition 2.1.

A Hirsch algebra Λ is the data of a connected DG-algebra (Λ, d, µ Λ ) together with a map µ : B Λ ⊗ B Λ → B Λ making B Λ into an associative unital DG-bialgebra.

For a connected DG-algebra Λ a DG-product µ : B Λ ⊗ B Λ → B Λ corresponds to a twisting cochain E ∈ Tw(B Λ ⊗ B Λ, Λ), cf. Theorem 1.1. After a correct use of desuspensions (see sign convention below), the latter is a family of operations {E i,j } i,j≥1 , E i,j : Λ ⊗i ⊗ Λ ⊗j → Λ satisfying some relations, see [START_REF] Kadeishvili | On the cobar construction of a bialgebra[END_REF]. We denote by µ E : B Λ ⊗ B Λ → B Λ such a DG-product.

A Hirsch algebra is a particular B ∞ -algebra whose underlying A ∞ -algebra structure is a DGA structure. Recall that the bar construction B Λ of a DG-algebra Λ is the cofree tensor coalgebra T c (sΛ) together with the derivation induced by the differential and the product of Λ.

Λ T c (sΛ) DGA (d Λ , µ Λ ) DGC d induced by d Λ and µ Λ A ∞ {µ k } k≥0 DGC d induced by µ k for k ≥ 0 Hirsch (d Λ , µ, {E i,j } i,j≥1 ) DG-bialgebra d induced by d Λ and µ ; DG-product µ E B ∞ ({µ k } k≥0 , {E i,j } i,j≥1 ) DG-bialgebra d induced by µ k for k ≥ 0 ; DG-product µ E Definition 2.2. A homotopy G-algebra (Λ, d Λ , µ Λ , {E 1,j } j≥1 ) is a Hirsch algebra (Λ, d Λ , µ Λ , {E i,j } i,j≥1
) such that E i,j = 0 for i ≥ 2.

Sign convention

Let E ∈ Tw(B Λ ⊗ B Λ, Λ) be a twisting cochain. Its (i, j)-component is E i,j : (s Λ) ⊗i ⊗ (s Λ) ⊗j → Λ. We denote by E i,j :

Λ ⊗i ⊗ Λ ⊗j → Λ the component E i,j (a 1 ⊗ ... ⊗ a i ⊗ a i+1 ⊗ ... ⊗ a i+j ) := (-1) ∑ i+j-1 s=1 |a s |(i+j-s) E i,j (s ⊗i ⊗ s ⊗j )(a 1 ⊗ ... ⊗ a i ⊗ a i+1 ⊗ ... ⊗ a i+j ) = E i,j (s a 1 ⊗ ... ⊗ s a i ⊗ s a i+1 ⊗ ... ⊗ s a i+j ). (2.1)
The degree of E i,j is i + j -1. We denote

E i,j ((a 1 ⊗ ... ⊗ a i ) ⊗ (b 1 ⊗ ... ⊗ b j )) by E i,j (a 1 , ..., a i ; b 1 , ...b j ).
Let B be a 1-reduced DG-bialgebra. For a ∈ B and b

:= s -1 b 1 ⊗ ... ⊗ s -1 b s ∈ ΩB, we set a ⋄ b = s -1 (a 1 b 1 ) ⊗ ... ⊗ s -1 (a s b s ).
Proposition 2.3. [START_REF] Kadeishvili | On the cobar construction of a bialgebra[END_REF] Let B be a 1-reduced DG-bialgebra. Then the cobar construction (ΩB, d Ω ) together with the operations E 1,k , k ≥ 1 defined by

E 1,k (s -1 a 1 ⊗ ... ⊗ s -1 a n ; b 1 , ..., b k ) = ∑ 1≤i 1 ≤i 2 ≤...≤i k ≤n ± s -1 a 1 ⊗ ... ⊗ s -1 a i 1 -1 ⊗ a i 1 ⋄ b 1 ⊗ ... ⊗ s -1 a i k -1 ⊗ a i k ⋄ b k ⊗ s -1 a i k +1 ⊗ ... ⊗ s -1 a n (2.2)
when n ≥ k and zero when n < k is a homotopy G-algebra.

In particular the operation E 1,1 : ΩB ⊗ ΩB → ΩB is given by

E 1,1 (s -1 a 1 ⊗ ... ⊗ s -1 a m ; s -1 b 1 ⊗ ... ⊗ s -1 b n ) := m ∑ l=1 (-1) γ s -1 a 1 ⊗ ...(a 1 l • b 1 ) ⊗ ... ⊗ s -1 (a n l • b n ) ⊗ ... ⊗ s -1 a m (2.3) with γ = n ∑ u=1 |b u | m ∑ s=l+u |a s | + m -l -u + 1 + κ n (a l ) + n ∑ u=2 (|a u l | -1) u-1 ∑ s=1 |b s | -u + 1 + n ∑ s=1 (|a s l | -1)(2n -2s + 1) + n-1 ∑ s=1 (|a s l | + |b s |)(n -s),
where

κ n (a) := ∑ 1≤2s+1≤n |a 2s+1 | if n is even; ∑ 1≤2s≤n |a 2s | if n is odd.
We postpone the construction of this operation to Appendix.

Homotopy BV-algebra on the cobar construction

We define a homotopy BV-algebra as a homotopy G-algebra (in the sense of Gerstenhaber-Voronov [START_REF] Gerstenhaber | Homotopy G-algebras and moduli space operad[END_REF]) together with a BV-operator. The main example, that we will discuss in details, is the cobar construction of a 1-reduced involutive DG-Hopf algebra H. The BV-operator on ΩH is the Connes-Moscovici operator defined in [START_REF] Connes | Cyclic cohomology and Hopf algebra symmetry[END_REF].

Definition 2.4.

A homotopy BV-algebra Λ is a homotopy G-algebra together with a degree one DG-operator ∆ : Λ → Λ subject to the relations:

∆ 2 = 0; {a; b} = (-1) |a| ∆(a • b) -∆(a) • b -(-1) |a| a • ∆(b) + d Λ H(a; b) + H(d Λ a; b) + (-1) |a| H(a; d Λ b) for all a, b ∈ Λ, where {-; -} is the Gerstenhaber bracket {a; b} = E 1,1 (a; b) -(-1) (|a|+1)(|b|+1) E 1,1 (b; a) and H : Λ ⊗ Λ → Λ is a degree 2 linear map.
A straightforward calculation shows that:

∆({a; b}) = {∆(a); b} + (-1) |a|+1 {a; ∆(b)} -∂H(a; b), where H(a; b) := ∆H(a; b) + H(∆(a); b) + (-1) |a| H(a; ∆(b)) ∀a, b ∈ Λ.
Therefore ∆ is a derivation for the bracket {-; -} if ∆ is a derivation for H, or more generally if ∆ is a derivation for H up to a homotopy.

In the example we consider hereafter, the homotopy H is itself antisymmetric, more precisely, we have H(a; b)

:= H 1 (a; b) -(-1) (|a|+1)(|b|+1) H 1 (b; a)
. It turns out that the operator ∆ is not a derivation for H in general. However we do not know yet if there exists a homotopy H ′ such that H = ∂H ′ . A. Connes and H. Moscovici defined in [START_REF] Connes | Cyclic cohomology and Hopf algebra symmetry[END_REF] a boundary map on the cobar construction of an involutive Hopf algebra H. This operator, hereafter called the Connes-Moscovici operator, induces a BV-operator on the homology H * (ΩH). More precisely, let (H, d, µ, η, ∇, ǫ, S) be an involutive DG-Hopf algebra, that is with an involutive antipode S. With our sign convention, the Connes-Moscovici operator ∆ : ΩH → ΩH is zero on both R = (s -1 H) ⊗0 and (s -1 H) ⊗1 , and is given on the n-th component (s -1 H) ⊗n , n ≥ 2 by:

∆(s -1 a 1 ⊗ s -1 a 2 ⊗ ... ⊗ s -1 a n ) = n-1 ∑ i=0 (-1) i π n τ i n (s -1 a 1 ⊗ s -1 a 2 ⊗ ... ⊗ s -1 a n ), (2.4)
where τ n is the cyclic permutation

τ n (s -1 a 1 ⊗ s -1 a 2 ⊗ ... ⊗ s -1 a n ) := (-1) (|a 1 |-1)(∑ n i=2 |a i |-1) s -1 a 2 ⊗ s -1 a 3 ⊗ ... ⊗ s -1 a n ⊗ s -1 a 1 ,
and

π n := (s -1 µ) ⊗n-1 τ n-1,n-1 (S ⊗n-1 ⊗ 1 ⊗n-1 )((τ∇) (n-2) ⊗ 1 n-1 ) s ⊗n , τ n,n (a 1 ⊗ a 2 ⊗ ... ⊗ a n ⊗ b 1 ⊗ ... ⊗ b n ) := (-1) ∑ n-1 i=1 |b i |(∑ n j=i+1 |a j |) a 1 ⊗ b 1 ⊗ a 2 ⊗ b 2 ⊗ ... ⊗ a n ⊗ b n . More explicitly, ∆(s -1 a 1 ⊗ s -1 a 2 ⊗ ... ⊗ s -1 a n ) = n ∑ k=1 ± s -1 S(a n-1 k )a k+1 ⊗ s -1 S(a n-2 k )a k+2 ⊗ ... ⊗ s -1 S(a 1 k )a k-1 . (2.5)
The involutivity of the antipode of H makes ∆ into a square zero chain map. L. Menichi proved [17, Proposition 1.9] that for a unital (ungraded) Hopf algebra H, the Connes-Moscovici operator induces a Batalin-Vilkovisky algebra on the homology of the cobar construction H * (ΩH). In our context, Proposition 2.5. [START_REF] Menichi | Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras[END_REF] Let H be a 1-reduced involutive DG-Hopf algebra. Then the cobar construction (ΩH, d Ω ) is a homotopy BV-algebra whose the BV-operator is defined in (2.4).

Proof. Let us define Menichi's homotopy H(a; b)

:= H 1 (a; b) -(-1) (|a|+1)(|b|+1) H 1 (b; a). H 1 (s -1 a 1 ⊗ ... ⊗ s -1 a m ; s -1 b 1 ⊗ ... ⊗ s -1 b n ) := ∑ 1≤j≤p≤m-1 (-1) ξ j +n+1 π s m+n-1 τ s,n+m-1-j m+n-1 ρ (p-j+1) n+m (s -1 a 1 ⊗ ... ⊗ s -1 a m ⊗ s -1 b 1 ⊗ ... ⊗ s -1 b n ) (2.6)
with

ξ j =          m ∑ s=1 |a s | for j = 1; m ∑ s=m-j+1 |a s | for j > 1.
Where,

π s m = π m (s -1 ) ⊗m τ s,i m := (τ s m ) i τ m (a 1 ⊗ a 2 ⊗ ... ⊗ a m ) := (-1) |a 1 |(∑ n i=2 |a i |) a 2 ⊗ a 3 ⊗ ... ⊗ a m ⊗ a 1 ρ (i) m+1 := (1 ⊗i-1 ⊗ µ ⊗ 1 ⊗m-i )(1 ⊗i-1 ⊗ τ m-i+1,1 )s ⊗m+1 τ k,1 (a 1 ⊗ ... ⊗ a k ⊗ b) := (-1) |b|(∑ k i=2 |a i |) a 1 ⊗ b ⊗ a 2 ⊗ ... ⊗ a k .
By Proposition [17, Proposition 1.9] we have

{a; b} = (-1) |a| ∆(a • b) -∆(a) • b -(-1) |a| a • ∆(b) + d 1 H(a; b) + H(d 1 a; b) + (-1) |a| H(a; d 1 b)
on the cobar construction (ΩH,

d 0 + d 1 )
, where d 1 is the quadratic part of the differential d Ω . The operators involved in the above equation commute with d 0 . Therefore we can replace d 1 by d Ω in the previous equation.

Involutivity of the antipode of ΩC in terms of the homotopy G-coalgebra C

A Hirsch coalgebra C is the formal dual of a Hirsch algebra, that is it corresponds to a DG-coproduct ∇ : ΩC → ΩC ⊗ ΩC making ΩC into a DG-bialgebra. Baues' coproduct [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF] ∇ 0 : ΩC * (X) → ΩC * (X) defined on the cobar construction of a simplicial set X corresponds to a homotopy G-coalgebra structure E k,1 on C * (X), that is a particular case of Hirsch coalgebra structure. With this example in mind we consider a homotopy G-coalgebra (C, E k,1 ). Then there exists a unique antipode S : ΩC → ΩC on its cobar construction. The purpose of this section is to give a criterion for the involutivity of the antipode S in terms of the operations E k,1 . This takes the form of a family of operations

O n : s -1 C → (s -1 C) ⊗n , n ≥ 2.
For convenience, we set

C := s -1 C. Thus O n : C → C ⊗n , n ≥ 2. Definition 3.1. A Hirsch coalgebra C is the data of a 1-reduced DG-coalgebra (C, d, ∇ C
) together with a map ∇ : ΩC → ΩC ⊗ ΩC making ΩC into a coassociative counital DG-bialgebra. The corresponding operations on C are denoted by E i,j : C → C ⊗i ⊗ C ⊗j . The degree of E i,j is 0. A homotopy G-coalgebra is a Hirsch coalgebra whose operations E i,j = 0 for j ≥ 2.

Let the cobar construction (ΩC, d, µ Ω , ∇) be a DG-bialgebra. We can define the antipode S : ΩC → ΩC by S(η(1)) = η(1) and for σ ∈ C n with n ≥ 1 by

S(σ) := -σ -µ Ω (S ⊗ 1)∇(σ) (3.1) which makes sense since ∇(σ) ⊂ i+j=n 0<i,j<n (ΩC) i ⊗ (ΩC) j . Indeed, µ Ω (S ⊗ 1)∇(η(1)) = ηǫη(1) = η(1) gives immediately that S(η(1)) = η(1). Moreover, since µ Ω (S ⊗ 1)∇(σ) = 0 for all σ ∈ C n with n ≥ 1, we have µ Ω (S ⊗ 1)∇(σ) = µ Ω (S ⊗ 1)∇(σ) + µ Ω (S(σ) ⊗ η(1) + S(η(1)) ⊗ σ) = µ Ω (S ⊗ 1)∇(σ) + S(σ) + σ = 0.
Remark 3.2. Equivalently, we can define S, see [14, section 1.3.10 p.15], as the geometric serie (Id) ⌣-1 = ∑ n≥0 (ηǫ -Id) ⌣n with (ηǫ -Id) ⌣0 = ηǫ and where ⌣ denote the convolution product from Definition 1.2. Note that this sum is finite when it is evaluated on an element. This presentation is combinatorial and gives for

[σ] ∈ (s -1 C) n , S([σ]) = ηǫ([σ]) + (ηǫ -Id)([σ]) + µ Ω ∇([σ]) -µ (2) Ω ∇ (2) ([σ]) + ... + (-1) n-1 µ (n) Ω ∇ (n) ([σ]). (3.2)
By the iii of Proposition 1.4 the antipode is an algebra antimorphism that is an algebra morphism from (ΩC, µ Ω ) to ΩC (12) := (ΩC, µ Ω (12) := µ Ω τ). Moreover, it is also a DG-map, therefore it corresponds to a twisting cochain F ∈ Tw(C, ΩC [START_REF] Kassel | Quantum groups[END_REF] ). An antipode is determined by the underlying bialgebra structure. Here the latter is equivalent to a homotopy G-algebra structure E k,1 on C. We make explicit the twisting cochain F in terms of E k,1 . We recall the notation C := s -1 C. We write

F i : C → C ⊗i for the i-th component of F. The relation 1 µ Ω (1 ⊗ S)∇ = ηǫ gives F 1 = -Id C ; (3.3) F n = ∑ 1≤s≤n-1 n 1 +...+n s =n-1 n i ≥1 (-1) s+1 (1 ⊗n 1 +...+n s-1 ⊗ E n s ,1 )...(1 ⊗n 1 ⊗ E n 2 ,1 )E n 1 ,1 , n ≥ 2. (3.4)
Let the operation E n i ,1 : C → C ⊗n i ⊗ C be represented by the tree in Figure 1. Then the summands of the equation (3.4) are represented by the tree in Figure 2.

1 • • • n i Figure 1: The operation E n i ,1 . 1 • • • n 1 1 • • • n 2 1 • • • n s Figure 2: A summand of F n
The three first terms are

F 1 = -Id C , F 2 = E 1,1 and F 3 = -(1 ⊗ E 1,1 )E 1,1 + E 2,1 .
Formulated in these terms, the n-th component of the difference S 2 -Id is zero for n = 1 and

O n := n ∑ s=1 ∑ n 1 +...+n s =n µ (s) Ω (12) (F n 1 ⊗ ... ⊗ F n s )F s (3.5) for n ≥ 2. The terms (F n 1 ⊗ ... ⊗ F n s )F s have codomain C ⊗n 1 ⊗ C ⊗n 2 ⊗ • • • ⊗ C ⊗n s . The s-iterated product µ (s) Ω (12)
permutes these s blocks as the permutation

1 2 • • • s -1 s s s -1 • • • 2 1     in S s ,
where S s denotes the symmetric group on s objects. The two first terms we obtain are

O 2 = F 2 -τF 2 = E 1,1 -τE 1,1
and

O 3 = (1 + τ 3 )F 3 + τ 2,1 (F 2 ⊗ 1)F 2 + τ 1,2 (1 ⊗ F 2 )F 2 = (τ 1,2 -1 -τ 3 )(1 ⊗ E 1,1 )E 1,1 + τ 2,1 (E 1,1 ⊗ 1)E 1,1 + (1 + τ 3 )E 2,1 ,
1 We can also consider µ Ω (S ⊗ 1)∇ = ηǫ. It gives equivalent F n but with a more complicated description because of the apparition of permutations. For example

F 3 = -(E 1,1 ⊗ 1)E 1,1 -(τ ⊗ 1)E 2,1 .
where the permutations are τ In fact we can make more precise the second point of the previous proposition. Let M be a DGmodule and let M ≤n be the sub-module of elements m ∈ M of degree |m| ≤ n. The homotopy G-coalgebra structure on a connected coalgebra C preserves the filtration C ≤n . Indeed the degree of the operations E k,1 : C → C ⊗k ⊗ C is 0. We have,

1,2 (a ⊗ b ⊗ c) = ±b ⊗ c ⊗ a, τ 2,1 (a ⊗ b ⊗ c) = ±c ⊗ a ⊗ b and τ 3 (a ⊗ b ⊗ c) = ±c ⊗ b ⊗ a;
Proposition 3.4. Let (C, d, ∇ C , E k,1 ) be an i-reduced homotopy G-coalgebra, i ≥ 2. If there exists an integer n such that O k = 0 for ki ≤ n -1, then Ω 2 (C ≤n ) is a homotopy BV-algebra. Proof. The degree of O k : C → C ⊗k is 0. Then on s -1 (C ≤n ) = C ≤n-1 the only eventually non-zero operations O k : C ≤n → ( C ≤n ) ⊗k are those with ki ≤ n -1.
Remark 3.5. The two previous propositions work for a Hirsch coalgebra C instead of a homotopy G-coalgebra : the operations from (3.5) have the same definition in terms of F n ; only the F n 's defining the antipode differ. However, the involved techniques above are quite similar for Hirsch coalgebras.

Remark 3.6. For a general Hirsch coalgebra (C, E i,j ) the condition of cocommutativity of the coproduct on ΩC is E i,j = τE j,i for all (i, j). Accordingly, with Proposition 1.4 we see (already in component three) that condition for the antipode to be involutive is weaker than condition for coproduct to be cocommutative. When the Hirsch coalgebra is a homotopy G-coalgebra, the cocommutativity of the coproduct means that the homotopy G-coalgebra is quasi trivial: E 1,1 = τE 1,1 and E k,1 = 0 for k ≥ 2.

Applications

On a homotopy G-coalgebra structure of C * (X)

The chain complex C * (X) of a topological space or a simplicial set X has a rich algebraic structure, it is an E ∞ -coalgebra. This was treated in many papers including J. McClure and J. Smith [START_REF] Mcclure | Multivariable cochain operations and little n-cubes[END_REF], C. Berger and B. Fresse [START_REF] Berger | Combinatorial operad actions on cochains[END_REF]. For example, the surjection operad χ introduced in [START_REF] Mcclure | Multivariable cochain operations and little n-cubes[END_REF] acts on the normalized chain complex C * (X) of a simplicial set X making it into a coalgebra over χ, see [START_REF] Berger | Combinatorial operad actions on cochains[END_REF][START_REF] Mcclure | Multivariable cochain operations and little n-cubes[END_REF]. It has a filtration of suboperads

F 1 χ ⊂ F 2 χ ⊂ • • • ⊂ F n-1 χ ⊂ F n χ ⊂ • • • ⊂ χ.
This structure leads to a coproduct ∇ 0 : ΩC * (X) → ΩC * (X) ⊗ ΩC * (X) first defined by Baues in [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF]. The third stage filtration F 3 χ gives a homotopy cocommutativity ∇ 1 : ΩC * (X) → ΩC * (X) ⊗ ΩC * (X) to Baues' coproduct, see [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF]. In turn the operation ∇ 1 : ΩC * (X) → ΩC * (X) ⊗ ΩC * (X) is cocommutative up to a homotopy ∇ 2 : ΩC * (X) → ΩC * (X) ⊗ ΩC * (X) and so on. The resulting structure is known as a structure of DG-bialgebra with Steenrod coproduct ∇ i . This was achieved by Kadeishvili in [START_REF] Kadeishvili | Cochain operations defining Steenrod ⌣ i -products in the bar construction[END_REF] where the corresponding operations in χ are given.

Baues' coproduct [3, p.334], [4, (2.9) equation ( 3)] on the cobar construction ΩC * (X) corresponds to a homotopy G-coalgebra on C * (X). By a direct comparison, we see that this homotopy G-coalgebra structure coincides with the one given in [START_REF] Berger | Combinatorial operad actions on cochains[END_REF][START_REF] Mcclure | Multivariable cochain operations and little n-cubes[END_REF]. To be more precise, let

E k,1 0 : C * (X) → C * (X) ⊗k ⊗ C * (X)
be the operations defined by Baues' coproduct

∇ 0 : ΩC * (X) → ΩC * (X) ⊗ ΩC * (X). Let E 1,k : C * (X) → C * (X) ⊗ C * (X) ⊗k
the operations defined by

E 1,k (s -1 σ) = ∑ 0≤j 1 <j 2 <...<j 2k ≤n ± s -1 σ(0, ..., j 1 , j 2 , ..., j 3 , j 4 , • • • , j 2k-1 , j 2k , ..., n) ⊗ s -1 σ(j 1 , ..., j 2 ) ⊗ s -1 σ(j 3 , ..., j 4 ) ⊗ • • • ⊗ s -1 σ(j 2k-1 , ...j 2k ), for s -1 σ ∈ C n (X). These are the operations denoted by AW(1, 2, 1, 3, ..., k -1, 1, k) in [5, section 2.2] where AW(u) : C * (X) → C * (X) ⊗n is defined for a surjection u ∈ χ(n) d . The operad F 2 χ is generated by the surjections (1, 2) and (1, 2, 1, 3, ..., 1, k, 1, k + 1, 1) for k ≥ 1. Then the operations E 1,k := AW(1, 2, 1, 3, ..., k, 1, k + 1, 1),
define a homotopy G-coalgebra structure on C * (X). We have Proposition 4.1. Let X be a 1-reduced simplicial set. Then

E k,1 0 = ±τ 1,k E 1,k ,
where τ 1,k orders the factors as the following permutation

1 2 • • • k k + 1 2 3 • • • k + 1 1     .
Proof. First we recall Baues' coproduct ∇ 0 . We adopt the same conventions as in [START_REF] Baues | The double bar and cobar constructions[END_REF]. For

σ i ∈ X, the tensor [σ 1 |σ 2 |...|σ n ] is the tensor s -1 σ i 1 ⊗ s -1 σ i 2 ⊗ ... ⊗ s -1 σ i r where the indices i j are such that σ i j ∈ X n i j with n i j ≥ 2. For a subset b = {b 0 < b 1 < ... < b r } ⊂ {0, 1, .
.., n} we denote by i b the unique order-preserving injective function

i b : {0, 1, ..., r} → {0, 1, ..., n} such that Im(i b ) = b. Let σ ∈ X n and 0 ≤ b 0 < b 1 ≤ n. We denote by σ(b 0 , ..., b 1 ) the element i * b σ ∈ X b 1 -b 0 where b = {b 0 , b 0 + 1, ..., b 1 -1, b 1 }. Let b ⊂ {1, ..., n -1}, we denote by σ(0, b, n) the element i * b ′ σ where b ′ = {0} ∪ b ∪ {n}. Baues' coproduct ∇ 0 [3, p.334] is defined on σ ∈ X n , n ≥ 2 by: ∇ 0 : ΩC * (X) → ΩC * (X) ⊗ ΩC * (X) s -1 σ = [σ] → ∑ b={b 1 <b 2 <...<b r } b⊂{1,...,n-1} (-1) ζ [σ(0, ..., b 1 )|σ(b 1 , ..., b 2 )|...|σ(b r , ..., n)] ⊗ [σ(0, b, n)] (4.1)
where

ζ = r|σ(0, .., b 1 )| + r ∑ i=2 (r + 1 -i)|σ(b i-1 , ..., b i )| -r(r + 1)/2,
it is extended as an algebra morphism on the cobar construction. We show that the coproduct ∇ 0 defines the same homotopy G-coalgebra structure on C * (X) when X is 1-reduced. Let σ be an n-simplex, then Baues' coproduct is a sum over the subset b = {b 1 < b 2 < ... < b r } ⊂ {1, ..., n -1}. For such a b, we set b 0 := 0, b r+1 := n and we defined β l ∈ b ∪ ∅ as follows. For l = 1,

β 1 := min 1≤i≤r+1 {b i | b i -b i-1 ≥ 2},
and let η 1 be the index such that b η 1 = β 1 ; for l ≥ 2,

β l := min η l-1 +1≤i≤r+1 {b i | b i -b i-1 ≥ 2}
where η l is the index such that b η l = β l . Moreover, we set α l := b η l -1 . Let k be the integer such that 1 ≤ l ≤ k. Explicitly, k is given by:

k = #{1 ≤ i ≤ r + 1 | b i -b i-1 ≥ 2}.
Thus the coproduct ∇ 0 is

∇ 0 ([σ]) = n ∑ k=0 ∑ 0≤α 1 ≤β 1 ≤α 2 ≤•••≤α k ≤β k ≤n β i -α i ≥2, 1≤i≤k ± s -1 σ(α 1 , ..., β 1 ) ⊗ s -1 σ(α 2 , ..., β 2 ) ⊗ • • • • • • ⊗ s -1 σ(α k , ..., β k ) ⊗ s -1 σ(0, ..., α 1 , β 1 , ..., α 2 , • • • , α k , β k , ..., n).
Thus we have,

E k,1 (s -1 σ) = ∑ ± s -1 σ(α 1 , ..., β 1 ) ⊗ s -1 σ(α 2 , ..., β 2 ) ⊗ • • • ⊗ s -1 σ(α k , ..., β k )⊗ s -1 σ(0, ..., α 1 , β 1 , ..., α 2 , • • • , α k , β k , ..., n).
The result is obtained by setting j 2l-1 = α l and j 2l = β l . Since X is 1-reduced, the elements σ(j l , ..., j l+1 ) such that j l+1j l = 1 are elements in X 1 = * = s 0 ( * ) and then are degenerate.

In the two next subsections 4.2 and 4.3 we adopt the above notations for the homotopy Gcoalgebra structure E 1,k on C * (X).

Obstruction to the involutivity of the antipode of ΩC * (ΣX)

Let ΣX be a simplicial suspension of a simplicial set X. We show that the family of obstructions O n : C * (ΣX) → C * (ΣX) ⊗n defined in (3.5) can be reduced to O 2 : C * (ΣX) → C * (ΣX) ⊗2 ; the latter being governed by the (lack of) cocommutativity of the operation E 1,1 . Definition 4.2. [15, Definition 27.6 p.124] Let X be a simplicial set such that X 0 = * and with face and degeneracy operators s j : X n → X n+1 and d j : X n → X n-1 . The simplicial suspension ΣX is defined as follow. The component (ΣX) 0 is just an element a 0 and (ΣX) n = {(i, x) ∈ N ≥1 × X n-i }/ (i, s n 0 ( * )) = s n+i 0 (a 0 ) . We set a n = s n 0 (a 0 ). The face and degeneracy operators are generated by:

• d 0 (1, x) = a n for all x ∈ X n ; • d 1 (1, x) = a 0 for all x ∈ X 0 ; • d i+1 (1, x) = (1, d i (x)) for all x ∈ X n , n > 0; • s 0 (i, x) = (i + 1, x); • s i+1 (1, x) = (1, s i (x)),
with the other face and degeneracy operators determined by the requirement that ΣX is a simplicial set. Proposition 4.3. [START_REF] Hess | A chain coalgebra model for the James map[END_REF] The differential of (ΩC * (ΣX), d 0 + d 1 ) is reduced to its linear part d 0 .

Proof. The only non degenerate elements in ΣX are (1, x) and we have that d 0 (1, x) = a n is degenerate. Therefore the Alexander-Whitney coproduct on C * (ΣX) is primitive (we recall that C * are the normalized chains). Then, the reduced coalgebra C * (ΣX) is a trivial coalgebra; so the quadratic part d 1 of the differential d Ω = d 0 + d 1 on the cobar construction is trivial.

A natural coproduct to define is the shuffle coproduct (primitive on cogenerators) which gives a cocommutative DG-Hopf structure to ΩC * (ΣX); applying Proposition 2.5 we obtain on Ω 2 C * (ΣX) a homotopy BV-algebra structure. However, this coproduct does not correspond to Baues' coproduct ∇ 0 . Indeed, the homotopy G-algebra structure on C * (ΣX) is not completely trivial. Because of the cocommutativity of the Alexander-Whitney coproduct on C * (ΣX), the operation E 1,1 must be a chain map but not necessary the zero map. We obtain, Proposition 4.4. The homotopy G-coalgebra structure E k,1 on C * (ΣX) given by Baues' coproduct is

E 1,1 (s -1 σ) = ∑ 1<l<n ± s -1 σ(0, ..., l) ⊗ s -1 σ(0, l, l + 1, ..., n); E k,1 = 0 for k ≥ 2.
Proof. For a non-degenerate σ ∈ (ΣX) n , we have d 0 σ = σ(1, ..., n) is degenerate. Consequently, the operation

E 1,1 (s -1 σ) = ∑ k<l ± s -1 σ(k, ..., l) ⊗ s -1 σ(0, ..., k, l, l + 1, ..., n) reduces to E 1,1 (s -1 σ) = ∑ 1<l<n ± s -1 σ(0, ..., l) ⊗ s -1 σ(0, l, l + 1, ..., n).
The higher operations E k,1 for k ≥ 2 given by

E 1,k (s -1 σ) = ∑ 0≤j 1 <j 2 <...<j 2k ≤n ± s -1 σ(0, ..., j 1 , j 2 , ..., j 3 , j 4 , • • • , j 2k-1 , j 2k , ..., n) ⊗ s -1 σ(j 1 , ..., j 2 ) ⊗ s -1 σ(j 3 , ..., j 4 ) ⊗ • • • ⊗ s -1 σ(j 2k-1 , ...j 2k )
are zero since the terms σ(j 3 , ..., j 4 ) are degenerate.

The triviality of higher operations E k,1 for k ≥ 2 does not imply the vanishing of higher obstructions O n since the E 1,1 operation appears in all the O n 's. However, we can reduce the family of obstructions O n to only O 2 in this case.

Proposition 4.5. Let (C, d, ∇ C , E k,1 ) be a homotopy G-coalgebra with E k,1 = 0 for k ≥ 2. If O 2 = E 1,1 -τE 1,1 is zero, then so is O n for n ≥ 2.
Proof. By making explicit the coassociativity of the coproduct ∇ : ΩC → ΩC we obtain, in particular, the equation

(1 ⊗ E 1,1 )E 1,1 -(E 1,1 ⊗ 1)E 1,1 = (τ ⊗ 1)E 2,1 + (1 ⊗ 1)E 2,1 .
Therefore the triviality of the higher operation E 2,1 implies that E 1,1 is coassociative. Together with the vanishing of O 2 we obtain a coassociative and cocommutative operation E 1,1 . We use this fact to vanish the O n 's : by (3.4), the terms F i , i ≤ n involved in O n are

F n =(-1) n (1 ⊗n-2 ⊗ E 1,1 ) • • • (1 ⊗ E 1,1 )E 1,1 .
Now because of the coassociativity of E 1,1 we can write each term µ

(s) Ω (12) (F n 1 ⊗ ... ⊗ F n s )F s of the sum O n = n ∑ s=1 ∑ n 1 +...+n s =n µ (s) Ω (12) (F n 1 ⊗ ... ⊗ F n s )F s
as ±σ • F n where σ is a permutation (depending of the term we consider). Now using the cocommutativity and again the coassociativity of E 1,1 we can remove all transpositions to obtain ±F n . A direct counting shows that positive and negative terms are equal in number : the sign of (F n 1 ⊗ ... ⊗ F n s )F s is (-1) n 1 +...+n s +s = (-1) n+s and the number of partitions of n into s integers ≥ 1 is ( n-1 s-1 ). Therefore we can consider O 2 as the only obstruction to the involutivity of the antipode on the cobar construction.

Homotopy BV-algebra structure on

Ω 2 C * (Σ 2 X)
Here we prove that the homotopy G-coalgebra structure on C * (Σ 2 X) is trivial and then that ΩC * (Σ 2 X) is an involutive DG-Hopf algebra with the shuffle coproduct. Theorem 4.6. Let Σ 2 X be a double suspension. Then :

• the homotopy G-coalgebra structure on C * (Σ 2 X) corresponding to Baues' coproduct

∇ 0 : ΩC * (Σ 2 X) → ΩC * (Σ 2 X) ⊗ ΩC * (Σ 2 X),
has trivial higher operations i.e. E 1,k = 0 for k ≥ 1;

• the double cobar construction Ω 2 C * (Σ 2 X) is a homotopy BV-algebra with the BV-operator from (2.4).

Proof. By Proposition 4.4 the operations E k,1 for k ≥ 2 are trivial. Therefore to prove the first statement it remains to prove that E 1,1 is trivial. This follows that

d 1 (1, (1, x)) = (1, d 0 (1, x)) = (1, a n ) = s n+1 (1, a 0 ) is degenerate. Indeed, the operation E 1,1 (s -1 σ) = ∑ 1<l<n ± s -1 σ(0, ..., l) ⊗ s -1 σ(0, l, l + 1, ..., n) is trivial since l > 1.
For the second statement since the homotopy G-coalgebra structure is trivial, all the obstructions O n are zero and so, by Proposition 3.3, we obtain the announced homotopy BV-algebra structure on Ω 2 C * (Σ 2 X).

We make explicit the homotopy BV-algebra structure on Ω 2 C * (Σ 2 X). Let us first observe that Baues' coproduct ∇ 0 is a shuffle coproduct. Indeed, the homotopy G-coalgebra structure on C * (Σ 2 X) being trivial, for any element a ∈ C * (Σ 2 X) we have

∇ 0 (s -1 a) = (E 1,0 + E 0,1 )(s -1 a) = s -1 a ⊗ 1 + 1 ⊗ s -1 a ∈ ΩC * (Σ 2 X) ⊗ ΩC * (Σ 2 X).
Therefore its extension ∇ 0 : ΩC * (Σ 2 X) → ΩC * (Σ 2 X) ⊗ ΩC * (Σ 2 X) as algebra morphism is the shuffle coproduct, see [START_REF] Kassel | Quantum groups[END_REF]Theorem III.2.4]. We write ∇ 0 as

∇ 0 ([a 1 |...|a n ]) = ∑ I 0 ∪I 1 ±[a I 0 ] ⊗ [a I 1 ]
where the sum is taken over all partitions I 0 ⊔ I 1 of I = {1, ..., n} with

I 0 = {i 1 < i 2 < ... < i k } and I 1 = {j 1 < j 2 < ... < j n-k }.
We denote a I 0 to be a i 1 ⊗ a i 2 ⊗ ... ⊗ a i k . Also we denote a I -1 0 to be a i k ⊗ ... ⊗ a i 2 ⊗ a i 1 and similarly for a I 1 . Now we make explicit the BV-operator on a 2 and 3-tensor. For the sake of simplicity we do not keep track of signs; also to avoid confusion we write a = [a 1 |a 2 |...|a k ] for the basic elements of ΩC * (Σ 2 X) and [a 1 , a 2 , ..., a n ] for the basic elements of

Ω 2 C * (Σ 2 X). Thus [a 1 , a 2 , ..., a n ] is [[a 1,1 |a 1,2 |...|a 1,k 1 ], [a 2,1 |a 2,2 |...|a 2,k 2 ], ..., [a n,1 |a n,2 |...|a n,k n ]].
With these conventions, the BV-operator ∆ is given by

∆([a, b]) = ∆([[a 1 |...|a m ], [b 1 |...|b n ]]) = [[a m |...|a 1 |b 1 |...|b n ]] + [[b n |...|b 1 |a 1 |...|a m ]]. ∆([[a 1 |...|a m ], [b 1 |...|b n ], [c 1 |...|c r ]]) = [[a I -1 1 |b 1 |...|b n ], [a I -1 0 |c 1 |...|c r ]] + [[b I -1 1 |c 1 |...|c r ], [b I -1 0 |a 1 |...|a m ]] + [[c I -1 1 |a 1 |...|a m ], [c I -1 0 |b 1 |...|b n ]].
The homotopy H from Proposition 2.5 is given by

H([a, b]) = 0.
To know explicitly each term of

∆([a, b, c]) -[a, b, ∆([c])] -[∆([a, b]), c] -{[a, b]; [c]} = ∂H([a, b]; [c])
we need to know the following terms:

H([a, b]; [c]) = H([[a 1 |...|a m ], [b 1 |...|b n ]]; [[c 1 |...|c r ]]) = ±[[b n |...|b 1 |a 1 |...|a m |c 1 |...|c r ]], H([a, b, c]; [d]) = H([[a 1 |...|a m ], [b 1 |...|b n ], [c 1 |...|c r ]]; [[d 1 |...|d s ]]) = ±[[c I -1 1 |a 1 |...|a m |d 1 |...|d s ], [c I -1 0 |b 1 |...|b n ]] ± [[c I -1 1 |a 1 |...|a m ], [c I -1 0 |b 1 |...|b n |d 1 |...|d s ]] ± [[b I -1 1 |c 1 |...|c r ], [b I -1 0 |c 1 |...|c r |d 1 |...|d s ]],
and

H([a, b]; [c, d]) = ±[[c I -1 1 |a 1 |...|a m |d 1 |...|d s ], [c I -1 0 |b 1 |...|b n ]].
Remark 4.7. From a topological point of view, let us consider X to be a connected (countable) CWcomplex with one vertex. The James/Milgram's models J i (X) are H-spaces homotopically equivalent to Ω i Σ i X, see [18, theorem 5.2]. Moreover, for i ≥ 1 the cellular chain complex C * (J i (ΣX)) is a cocommutative, primitively generated DG-Hopf algebra, and C * (J i+1 (X)) is isomorphic to ΩC * (J i (ΣX)), [18, theorem 6.1 and 6.2]. Then using Proposition 2.5 we get a homotopy BValgebra structure on it which is similar to the one obtained in the simplicial context. Also, we have a DGA-quasi-isomorphism

ΩC * (J 1 (ΣX)) C * (J 2 (X)) C * (Ω 2 Σ 2 X). C * (j 2 )
4.4. Homotopy BV-algebra structure on Ω 2 C * (X) over R ⊃ Q

In [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF], Baues gives an explicit construction of the cobar construction of a 1-reduced simplicial set X over the integer coefficient ring as a cocommutative up to homotopy DG-bialgebra. By using methods of [START_REF] Anick | Hopf algebras up to homotopy[END_REF], he shows that over a ring R containing Q as a subring, this DG-bialgebra can be deformed into a strictly cocommutative DG-bialgebra [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF]Theorem 4.7]. The latter is isomorphic as DG-Hopf algebras to the universal enveloping algebra of the Lie algebra L(s -1 C * X) generated by the desuspension of the reduced coalgebra of normalized chain complex, combine [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF]Theorem 4.8] and [START_REF] Kassel | Quantum groups[END_REF]Proposition V.2.4]. Hence, over such a ring R, the cobar construction ΩC * (X) is (can be deformed into) an involutive DG-Hopf algebra. By applying Proposition 2.5 to the resulting involutive cobar construction (ΩC * (X), ∇ ′ 0 , S ′ ), we obtain Theorem 4.8. Let X be a 2-reduced simplicial set. Then the double cobar construction Ω(ΩC * (X), ∇ ′ 0 , S ′ ) over R ⊃ Q coefficients is a homotopy BV-algebra with BV-operator the Connes-Moscovici operator.

In the sequel we detail how the antipode is deformed. We extend the deformation (given in [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF]) of the DG-bialgebra structure of the cobar construction to a deformation of the DG-Hopf structure. The resulting cobar construction (ΩC * (X), ∇ ′ 0 , S ′ ) comes with (anti)derivation homotopies (see below) connecting the obtained DG-Hopf structure (∇ ′ 0 , S ′ ) with the initial one (∇ 0 , S).

First of all we extend some definitions from [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF].

Let DGA 0 be the category of connected DG-algebras (associative). For a free DG-module V, we denote by L(V) the DG-Lie algebra which is the free graded Lie algebra on the free module V. We set V ≤n (resp. V <n ) the sub DG-module of elements v ∈ V such that |v| ≤ n (resp. |v| < n). Similarly, V ≥n (resp. V >n ) denotes the subset of elements v ∈ V such that |v| ≥ n (resp. |v| > n).

d(V ≤n+1 ) ⊂ L(V ≤n ) ⊂ T(V) = A (4.7)
V ≤n ⊂ ker(∇). (4.8)

Appendix

Here we recall and develop some facts about the Hirsch and the homotopy G-algebras. A Hirsch (Λ,

d Λ , •, {E 1,k } k≥1 ) corresponds to a product µ E : B Λ ⊗ B Λ → B Λ such that (B Λ, d B Λ , µ E ) is a unital DG-bialgebra.
We write down the relations among the E i,j , i, j ≥ 1 coming from both the associativity of µ E and the Leibniz relation

d B Λ µ E = µ E (d B Λ ⊗ 1 + 1 ⊗ d B Λ ).
We detail the construction of the operation E 1,1 defined in (2.3).

Unit condition

For all s a = s a 1 ⊗ ... ⊗ s a i ∈ BΛ we have:

µ E (1 Λ ⊗ s a) = µ E (s a ⊗ 1 Λ ) = s a (4.10)
The product being determined by its projection on Λ we have:

prµ E (1 Λ ⊗ s a) = E 0,i (1 Λ ⊗ s a) = pr(s a 1 ⊗ ... ⊗ s a i ) = a 1 if i = 1; 0 if i = 1,
and also the symmetric relation. Thus,

E 0,i = E i,0 = 0 for all i = 1 and E 0,1 = E 1,0 = Id Λ . (4.11)

Associativity condition

With the sign convention (2.1), the associativity of µ E gives on Λ ⊗i ⊗ Λ ⊗j ⊗ Λ ⊗k :

E 1,k (E i,j (
+ E i,1 (a 1 , ..., a i ; E j,k (b 1 , ..., b j ; c 1 , ...c k )) (4.12)
where

α 1 = n ∑ u=1 i u+1 ∑ s=i u +1 |a s | + i u+1 -i u j u ∑ s=1 |b s | + j u α 2 = m ∑ u=1 j u+1 ∑ s=j u +1 |b s | + j u+1 -j u k u ∑ s=1 |c s | + k u with i 0 = 0; j 0 = 0; k 0 = 0; i n+1 = i; j n+1 = j m+1 = j; k m+1 = k.
For i = j = k = 1, that is on Λ ⊗ Λ ⊗ Λ, the relation (4.12) gives:

E 1,1 (E 1,1 (a; b); c) = E 1,1 (a; E 1,1 (b; c)) + E 1,2 (a; b, c) + (-1) (|b|-1)(|c|-1) E 1,2 (a; c, b) -E 2,1 (a, b; c) -(-1) (|a|-1)(|b|-1) E 2,1 (b, a; c). (4.13) Leibniz relation On Λ ⊗i ⊗ Λ ⊗j , the projection of d B µ E = µ E (d B ⊗ 1 + 1 ⊗ d B ) gives: d Λ E i,j (a 1 , ..., a i ; b 1 , ..., b j ) + ∑ 0≤i 1 ≤i 0≤j 1 ≤j (-1) β 2 E i 1 ,j 1 (a 1 , ..., a i 1 ; b 1 , ..., b j 1 ) • E i-i 1 ,j-j 1 (a i 1 +1 , ..., a i ; b j 1 +1 , ..., b j ) = = i ∑ l=1 (-1) β 3 E i,j (a 1 , ..., d Λ (a l ), ..., a i ; b 1 , ..., b j ) + i-1 ∑ l=1 (-1) β 4 E i-1,j (a 1 , ..., a l • a l+1 , ..., a i ; b 1 , ..., b j ) + j ∑ l=1 (-1) β 5 E i,j (a 1 , ..., a i ; b 1 , ..., d Λ (b l ), ..., b j ) + j-1 ∑ l=1 (-1) β 6 E i,j-1 (a 1 , ..., a i ; b 1 , ..., b l • b l+1 , ..., b j ) (4 .14)
where

β 2 = i 1 ∑ s=1 |a s | + j 1 ∑ s=1 |b s | + i 1 + j 1 + i ∑ s=i 1 +1 |a s | + i -i 1 j 1 ∑ s=1 |b s | + j 1 β 3 =η 0 (s a) β 4 =η 1 (s a) β 5 = i ∑ s=1 |a s | + i + η 0 (s b) β 6 = i ∑ s=1 |a s | + i + η 1 (s b). For s a := s a 1 ⊗ • • • ⊗ s a i , the following signs η 0 (s a) = l-1 ∑ s=1 |a s | + l η 1 (s a) = l ∑ s=1 |a s | + l,
are the signs of the differential of the bar construction. And similarly for s

b = s b 1 ⊗ • • • ⊗ s b j .
Remark 4.15. When the twisting cochain E : B Λ ⊗ B Λ → Λ has E i,j = 0 except E 1,0 and E 0,1 , then Λ is a commutative DG-algebra.

We recall (cf. Definition 2.2) that when the twisting cochain E : B Λ ⊗ B Λ → Λ satisfies E i,j = 0 for i ≥ 2, then (Λ, d, µ Λ , {E 1,j }) is called a homotopy G-algebra. Remark 4.16. [13, Proposition 3.2] The condition E i,j = 0 for i ≥ 2 is equivalent to the following condition: for each integer r, I r := ⊕ n≥r (s -1 Λ) ⊗n is a right ideal for the product µ

E : B Λ ⊗ B Λ → B Λ.
For a homotopy G-algebra Λ, the equation (4.14) gives the three following equalities.

On Λ ⊗1 ⊗ Λ ⊗1 :

d Λ E 1,1 (a; b) -E 1,1 (d Λ a; b) + (-1) |a| E 1,1 (a; d Λ b) = (-1) |a| a • b -(-1) |a||b| b • a . (4.15) On Λ ⊗2 ⊗ Λ ⊗1 : E 1,1 (a 1 • a 2 ; b) = a 1 • E 1,1 (a 2 ; b) + (-1) |a 2 |(|b|-1) E 1,1 (a 1 ; b) • a 2 . (4.16) On Λ ⊗1 ⊗ Λ ⊗2 : d Λ E 1,2 (a; b 1 , b 2 ) + E 1,2 (d Λ a; b 1 , b 2 ) + (-1) |a|+1 E 1,2 (a; d Λ b 1 , b 2 ) + (-1) |a|+|b 1 | E 1,2 (a; b 1 , d Λ b 2 ) = (-1) |a|+|b 1 |+1 E 1,1 (a; b 1 )b 2 + (-1) (|a|-1)|b 1 | b 1 E 1,1 (a; b 2 ) -E 1,1 (a; b 1 b 2 ) . (4.17)
The sign in (2.3)

Now we give the construction of the operation E 1,1 in (2.3). Recall that B is a DG-bialgebra and that E 1,1 :

Λ ⊗ Λ → Λ with Λ := ΩB. First, we set E 1,1 (s -1 a; s -1 b) := s -1 (a • b) for all a, b ∈ B.
Next, using the equation (4.16), we extend this to:

E 1,1 (s -1 a 1 ⊗ ... ⊗ s -1 a m ; s -1 b) := m ∑ l=1 (-1) γ 1 s -1 a 1 ⊗ ... ⊗ s -1 (a l • b) ⊗ s -1 a l+1 ⊗ ... ⊗ s -1 a m , for all homogeneous elements a 1 , a 2 , • • • , a m , b ∈ B, where γ 1 = |b| m ∑ s=l+1 |a s | -m + l .
On the other hand, using a slight abuse of notation, we set

E 1,1 (s -1 a; s -1 b 1 ⊗ s -1 b 2 ) := (-1) |a 2 ||b 1 |+|a 2 | s -1 (a 1 • b 1 ) ⊗ s -1 (a 2 • b 2 ).
The abuse of notation comes from the fact that the terms (-1)

|a||b 1 |+|a| s -1 (b 1 ) ⊗ s -1 (a • b 2 ) and s -1 (a • b 1 ) ⊗ s -1 (b 2
) belong to the above terms i.e. the coproduct of B evaluated on the element a is not reduced. Using the equation (4.14) we extend E 1,1 (using the same abuse of notation) to: = f 1 ({a; b}).

E 1,1 (s -1 a; s -1 b 1 ⊗ ... ⊗ s -1 b n ) := (-1) γ 2 +|a|+1 s -1 (a 1 • b 1 ) ⊗ ... ⊗ s -1 (a n • b n ), with γ 2 = γ 2 (a) = κ n (a) + n ∑ u=2 (|a u | -1) u-1 ∑ s=1 |b s | -u + 1 + n ∑ s=1 (|a s | -1)(2n -2s + 1) + n-1 ∑ s=1 (|a s | + |b s |)(n -s), where κ n (a) := ∑ 1≤2s+1≤n |a 2s+1 | if n is even; ∑ 1≤2s≤n |a 2s | if n is odd.

Definition 1 . 3 .Proposition 1 . 4 .

 1314 A DG-Hopf algebra (H, d, µ, η, ∇, ǫ, S) is a DG-bialgebra H endowed with an antipode S. If moreover, the antipode is involutive (i.e. S 2 = id) we call H an involutive DG-Hopf algebra.An antipode satisfies the following properties. [19, Proposition 4.0.1] i. S(a 2 ) ⊗ S(a 1 ) = (-1) |a 1 ||a 2 | S(a) 1 ⊗ S(a) 2 (coalgebra antimorphism). ii. S(η(1)) = η(1) (unital morphism). iii. S(ab) = (-1) |a||b| S(b)S(a) (algebra antimorphism). iv. ǫ(S(a)) = ǫ(a) (counital morphism). v. The following equations are equivalent: (a) S 2 = S • S = id; (b) S(a 2 )a 1 = ηǫ(a);

Proposition 3 . 3 . 1 .

 331 the signs being given by the Koszul sign rule. We conclude, Let (C, d, ∇ C , E k,1 ) be a homotopy G-coalgebra. The cobar construction ΩC is an involutive DG-Hopf algebra if and only if all the obstructions O n : C → C ⊗n defined in (3.5) for n ≥ 2 are zero. 2. Let C be 2-reduced. If all the obstructions O n : C → C ⊗n are zero, then the double cobar construction Ω 2 C is a homotopy BV-algebra whose the BV-operator is the Connes-Moscovici operator defined in (2.4).

Definition 4 . 9 .. 5 ) 4 . 10 .Definition 4 . 11 .

 495410411 Let f , g : A → B be two maps between two DG-modules A, B. A derivation homotopy between f and g is a mapF : A → B satisfying dF + Fd = fg (4.2)F(ab) = F(a)g(b) + (-1) |F||a| f (a)F(b) ∀a, b ∈ A.(4.3)An antiderivation homotopy between f and g is a mapΓ : A → B satisfying dΓ + Γd = fg (4.4) Γ(ab) = (-1) |a||b| (Γ(b)g(a) + (-1) |Γ||a| f (b)Γ(a)) ∀a, b ∈ A. (4Definition A homotopy DG-bialgebra (A, ∇, G 1 , G 2) is an object A in DGA 0 together with a coproduct ∇ in DGA 0 making (A, ∇) into a coalgebra in DGA 0 cocommutative up to G 1 , coassociative up to G 2 , both being derivation homotopies. Let (A, ∇, G 1 , G 2 ) be a homotopy DG-bialgebra such that A = TV, V being a DG-module with V 0 = 0. The counit is the augmentation of TV. We call (A, ∇, G 1 , G 2 , V) n-good if the following conditions (4.6), (4.7),(4.8) hold.∇ = τ∇ and (∇ ⊗ 1)∇ = (1 ⊗ ∇)∇ on V ≤n(4.6)

Proposition 4 . 19 .

 419 Let (Λ, d Λ , •, E 1,k ) be a homotopy G-algebra. Then the degree 1 bracket {a; b} = E 1,1 (a; b) -(-1) (|a|-1)(|b|-1) E 1,1 (b; a)defines a Gerstenhaber algebra structure on the homology H(Λ, d Λ ).

Proof.

  The equality(4.15) shows the commutativity of the product of Λ up to homotopy. Indeed, it suffices to set E # 1,1 (a; b) := (-1) |a| E 1,1 (a; b) for each homogeneous element to obtain the desired homotopy. The symmetry condition (4.20) is satisfied by construction. The Jacobi relation (4.22) comes from (4.13). Indeed, let us first observe that applying (4.13) we have:E 1,1 (a; {b; c}) = E 1,1 (a; E 1,1 (b; c)) -(-1) (|b|-1)(|c|-1) E 1,1 (a; E 1,1 (c; b)) = E 1,1 (E 1,1 (a; b); c) -(-1) (|b|-1)(|c|-1) E 1,1 (E 1,1 (a; c); b) + R where R = -E 1,2 (a; b, c) -(-1) (|b|-1)(|c|-1) E 1,2 (a; c, b) -(-1) (|b|-1)(|c|-1) -E 1,2 (a; c, b) -(-1) (|b|-1)(|c|-1) E 1,2 (a; b, c) = 0. From this {a; {b; c}} =E 1,1 (a; {b; c}) -(-1) (|a|-1)(|b|+|c|) E 1,1 ({b; c}; a) =E 1,1 (a; {b; c}) -(-1) (|a|-1)(|b|+|c|) E 1,1 (E 1,1 (b; c); a) -(-1) (|b|-1)(|c|-1) E 1,1 (E 1,1 (c; b); a) =E 1,1 (E 1,1 (a; b); c) -(-1) (|b|-1)(|c|-1) E 1,1 (E 1,1 (a; c); b) -(-1) (|a|-1)(|b|+|c|) E 1,1 (b; E 1,1 (c; a)) -(-1) (|b|-1)(|c|-1) E 1,1 (c; E 1,1 (b; a)) + L,whereL = -(-1) (|a|-1)(|b|+|c|) E 1,2 (b; c, a) + (-1) (|c|-1)(|a|-1) E 1,2 (b; a, c) -(-1) (|b|-1)(|c|-1) E 1,2 (c; b, a) + (-1) (|b|-1)(|a|-1) E 1,2 (c; a, b) . By definition -{{a; b}; c} = -E 1,1 (E 1,1 (a; b); c) + (-1) (|a|-1)(|b|-1) E 1,1 (E 1,1 (b; a); c) + (-1) (|c|-1)(|a|+|b|) E 1,1 (c; E 1,1 (a; b)) -(-1) (|a|-1)(|b|-1) E 1,1 (c;E 1,1 (b; a)) . Thus we obtain, {a; {b; c}} -{{a; b}; c} = -(-1) (|b|-1)(|c|-1) E 1,1 (E 1,1 (a; c); b) -(-1) (|a|-1)(|b|+|c|) E 1,1 (b; E 1,1 (c; a)) + (-1) (|a|-1)(|b|-1) E 1,1 (E 1,1 (b; a); c) + (-1) (|c|-1)(|a|+|b|) E 1,1 (c; E 1,1 (a; b)) + L.

Finally, applying once

  again the equality (4.13) to both the 3-th and 4-th term, we obtain {a; {b; c}} -{{a; b}; c} = -(-1) (|b|-1)(|c|-1) E 1,1 (E 1,1 (a; c); b) -(-1) (|a|-1)(|b|+|c|) E 1,1 (b; E 1,1 (c; a)) + (-1) (|a|-1)(|b|-1) E 1,1 (b; E 1,1 (a; c)) + (-1) (|c|-1)(|a|+|b|) E 1,1 (E 1,1 (c; a); b) + L + L ′ = -(-1) (|b|-1)(|c|-1) E 1,1 ({a; c}; b) + (-1) (|a|-1)(|b|-1) E 1,1 (b; {a; c}) + L + L ′ =(-1) (|a|-1)(|b|-1) {b; {a; c}} + L + L ′ ,whereL ′ = (-1) (|a|-1)(|b|-1) E 1,2 (b; a, c) + (-1) (|a|-1)(|c|-1) E 1,2 (b; c, a) + (-1) (|c|-1)(|a|+|b|) -E 1,2 (c; a, b) -(-1) (|a|-1)(|b|-1) E 1,2 (c; b, a) .The equality L + L ′ = 0 is easily verified.The Poisson relation (4.21) follows from the equations (4.16) and(4.

17

 17

  ). Indeed, take (-1)|b 1 | E 1,2 (a; b 1 , b 2 )instead of E 1,2 (a; b 1 , b 2 ) in (4.17), we obtain: {a; bc} =E 1,1 (a; bc) -(-1) (|a|-1)(|b|+|c|-1) E 1,1 (bc; a)∼E 1,1 (a; b)c + (-1) (|a|-1)|b| bE 1,1 (a; c) -(-1) (|a|-1)(|b|+|c|-1) bE 1,1 (c; a) + (-1) |c|(|a|-1) E 1,1 (b; a)c ∼ E 1,1 (a; b) -(-1) (|a|-1)(|b|+|c|-1)+|c|(|a|-1) E 1,1 (b; a) c + b (-1) (|a|-1)|b| E 1,1 (a; c) -(-1) (|a|-1)(|b|+|c|-1) E 1,1 (c; a) ∼{a; b}c + (-1) (|a|-1)|b| b{a; c}, where ∼ is the equivalence relation: a ∼ b iff a is homotopic to b.The commutativity between the bracket and the differential follows from (4.15). Indeed,d Λ {a; b} =d Λ E 1,1 (a; b) -(-1) (|a|-1)(|b|-1) d Λ E 1,1 (b; a) = -E 1,1 (d Λ a; b) -(-1) |a| E 1,1 (a; d Λ b) + (-1) |a| a • b -(-1) |a||b| b • a -(-1) (|a|-1)(|b|-1) E 1,1 (d Λ b; a) -(-1) |b| E 1,1 (b; d Λ a) + (-1) |b| b • a -(-1) |a||b| a • b = -E 1,1 (d Λ a; b) -(-1) |a| E 1,1 (a; d Λ b) -(-1) (|a|-1)(|b|-1) E 1,1 (d Λ b; a) -(-1) |b| E 1,1 (b; d Λ a) + (-1) |a| a • b -(-1) |a||b| b • a -(-1) (|a|-1)(|b|-1)+|b| b • a -(-1) |a||b| a • b = -E 1,1 (d Λ a; b) -(-1) |a| E 1,1 (a; d Λ b) -(-1) (|a|-1)(|b|-1) E 1,1 (d Λ b; a) -(-1) |b| E 1,1 (b; d Λ a) = -{d Λ a; b} -(-1) |a| {a; d Λ b}.Proposition 4.20. An ∞-morphism of homotopy G-algebras induces in homology a morphism of Gerstenhaber algebras.Proof. Let us take the following convention f i := f i (s ⊗i ). Then the equation (4.18) reads for k = l = 1:E 1,1 ( f 1 (a); f 1 (b))f 1 E 1,1 (a; b) = (-1) |a|-1 f 2 (a; b) + (-1) |a|(|b|-1) f 2 (b; a). (4.23)We obtain{ f 1 (a); f 1 (b)} = E 1,1 ( f 1 (a); f 1 (b)) -(-1) |a|-1)(|b|-1) E 1,1 ( f 1 (b); f 1 (a)) = f 1 E 1,1(a; b) + (-1) |a|-1 f 2 (a; b) -(-1) |a|(|b|-1) f 2 (b; a) -(-1) |a|-1)(|b|-1) f 1 E 1,1 (b; a) + (-1) |b|-1 f 2 (b; a) -(-1) |b|(|a|-1) f 2 (a; b)

  a 1 , ..., a i ; b 1 , ...b j ); c 1 , ..., c k )+ E n+1,k (E i 1 ,j 1 (a 1 , ..., a i 1 ; b 1 , ...b j 1 ), ... ..., E i-i n ,j-j n (a i n +1 , ..., a i ; b j n +1 , ...b j ); c 1 , ..., c k ) E i,m+1 (a 1 , ..., a i ;E j 1 ,k 1 (b 1 , ..., b j 1 ; c 1 , ...c k 1 ), ... ..., E j-j m ,k-k m (b j m +1 , ..., b j ; c k m +1 , ...c k ))

		i+j
	∑ n=1 (-1) α 1 = ∑ 0≤i 1 ≤...≤i n ≤i 0≤j 1 ≤...≤j n ≤j j+k ∑ ∑ (-1) α 2
	m=1	0≤j 1 ≤...≤j m ≤j 0≤k 1 ≤...≤k m ≤k

Acknowledgement. I would like to thank Jean-Claude Thomas for his very helpful remarks, in particular for suggesting the study of suspensions. I am grateful to Muriel Livernet for the useful conversation at IHP and her many comments about the content of this paper.

If moreover there exists a chain map S : A → A such that the condition: µ(1 ⊗ S)∇ = ηǫ = µ(S ⊗ 1)∇ on V ≤n (4.9) is satisfied, then A is called n-good + .

Lemma 4.12. Let A := (A = TV, ∇, G 1 , 0, S, V) be a homotopy DG-bialgebra with antipode S. We suppose it is n-good + for a map S : A → A. Then there exists a homotopy DG-bialgebra (A, ∇ n+1 , G n+1 1 , G n+1 2 , S n+1 , V n+1 ) which both extends A and is (n + 1)-good + . Moreover, there is a derivation homotopy F n+1 : ∇ ≃ ∇ n+1 and an antiderivation homotopy Γ n+1 : S ≃ S n+1 with F n+1 (a) = 0 and Γ n+1 (a) = 0 for |a| < n, a ∈ A.

Proof. The part (n + 1)-good as bialgebra is already done in [START_REF] Baues | The cobar construction as a Hopf algebra[END_REF]Theorem 4.5] where ∇ n+1 , G n+1 1 ,G n+1 2 , F n+1 and V n+1 are defined. Therefore we only need to construct S n+1 and Γ n+1 . We recall that (

The homotopy F n+1 between ∇ and ∇ n+1 is such that

we obtain the desired homotopy. Indeed,

Next we extend S n+1 on T((V n+1 ) ≤n+1 ) as an algebra antimorphism and we extend Γ n+1 as an antiderivation homotopy. We have µ(S n+1 ⊗ 1)∇ n+1 = ηǫ on (V n+1 ) ≤n+1 since ∇ n+1 is coassociative on (V n+1 ) ≤n+1 . Lemma 4.13. Let (A = TV, ∇, G 1 , S) be a DG-Hopf algebra cocommutative up to the homotopy G 1 and with S as antipode. Suppose that (A, ∇, G 1 , S, V) is 1-good + . Then there is a coproduct ∇ ′ and an antipode S ′ such that (A, ∇ ′ , S ′ ) is a cocommutative DG-Hopf algebra. Moreover there is a derivation homotopy F : ∇ ≃ ∇ ′ and an antiderivation homotopy Γ : S ≃ S ′ .

Proof. An iteration of Lemma 4.12 yields, for each n ≥ 1, an n-good

The derivation homotopy F is defined as ∑ n≥1 F n and the antiderivation homotopy Γ is defined as ∑ n≥1 Γ n . Proposition 4.14. Let Q ⊂ R and let X be a 1-reduced simplicial set. Then there is both a coproduct ∇ ′ and an antipode S ′ on the cobar construction ΩC * X such that (ΩC * (X), ∇ ′ , S ′ ) is a cocommutative DG-Hopf algebra. Moreover there is a derivation homotopy F : ∇ 0 ≃ ∇ ′ and an antiderivation homotopy Γ : S 0 ≃ S ′ , where ∇ 0 and S 0 are respectively Baues' coproduct [4, (3)] and the associated antipode.

Proof. We denote by G 1 the homotopy to the cocommutativity of the coproduct ∇ 0 . It is defined in [4, (4)]. We apply Lemma 4.13 to (ΩC * (X), ∇ 0 , G 1 , 0, S, s -1 C * X) which is 1-good + . Finally, we find the operation in (2.3):

where

Definition 4.17. An ∞-morphism between two homotopy G-algebras, say Λ and Λ ′ , is a morphism of unital DG-algebras between the associated bar constructions:

Such a morphism is a collection of maps

for all k ≥ 1, l ≥ 1, and

for all n ≥ 1, where, ∂ is the differential of Hom(Λ ⊗n , Λ ′ ).

Now we show that the homology of a homotopy G-algebra is a Gerstenhaber algebra. To fix the convention: