A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops

Let $M$ be a compact oriented $d$-dimensional smooth manifold and $X$ a topological space. Chas and Sullivan \cite{Chas-Sullivan:stringtop} have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM):=H_{*+d}(LM)$. Getzler \cite{Getzler:BVAlg} has defined a structure of Batalin-Vilkovisky algebra on the homology of the pointed double loop space of $X$, $H_*(\Omega^2 X)$. Let $G$ be a topological monoid with a homotopy inverse. Suppose that $G$ acts on $M$. We define a structure of Batalin-Vilkovisky algebra on $H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)$ extending the Batalin-Vilkovisky algebra of Getzler on $H_*(\Omega^2BG)$. We prove that the morphism of graded algebras $$H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)\to\mathbb{H}_*(LM)$$ defined by Felix and Thomas \cite{Felix-Thomas:monsefls}, is in fact a morphism of Batalin-Vilkovisky algebras. In particular, if $G=M$ is a connected compact Lie group, we compute the Batalin-Vilkovisky algebra $\mathbb{H}_*(LG;\mathbb{Q})$.


Introduction
We work over an arbitrary principal ideal domain k.Algebraic topology gives us two sources of Batalin-Vilkovisky algebras (Definition 6): Chas Sullivan string topology [1] and iterated loop spaces.More precisely, let X be be a pointed topological space, extending the work of Cohen [3], Getzler [8] has shown that the homology H * (Ω 2 X) of the double pointed loopspace on X is a Batalin-Vilkovisky algebra.Let M be a closed oriented manifold of dimension d.Denote by LM := map(S 1 , M) the free loop space on M. By Chas and Sullivan [1], the (degree-shifted) homology H * (LM) := H * +d (LM) of LM has also a Batalin-Vilkovisky algebra.In this paper, we related this two a priori very different Batalin-Vilkovisky algebras.
Initially, our work started with the following Theorem of Felix and Thomas.
Theorem 1. [5, Theorem 1] Let G be a topological monoid acting on a smooth compact oriented manifold M. Consider the map Θ G,M : ΩG × M → LM which sends the couple (w, m) to the free loop t → w(t).m.

The map induced in homology
H * (Θ G,M ) : is a morphism of commutative graded algebras.
In [5], Félix and Thomas stated this theorem with G = aut 1 (M), the monoid of self equivalences homotopic to the identity.But their theorem extends to any topological monoid G since any action on M factorizes through Hom(M, M) and aut 1 (M) is the path component of the identity in Hom(M, M).Note also that the assumptions of [5] "k is a field " and "M is simply connected" are not necessary.
If G = M is a Lie group, Θ G,G : ΩG ×G → LG is a homeomorphism.Therefore we recover the following well-known isomorphism (See for example [13, Proof of Theorem 10]).
Corollary 2. Let G be a path-connected compact Lie group.The algebra H * (LG) is isomorphic to the tensor product of algebras H * (ΩG) ⊗ H * (G).
The following is our main theorem.Remark 4. We assume that the homology with coefficients in k, H * (ΩG), is torsion free (this hypothesis is of course satisfied if k is a field), since we want H * (ΩG) to have a diagonal.
This Batalin-Vilkovisky algebra H * (ΩG) ⊗ H * (M) contains H * (ΩG) as sub Batalin-Vilkovisky algebra (Corollary 23).Denote by BG the classifying space of G.We show that the sub Batalin-Vilkovisky algebra H * (ΩG) is isomorphic to the Batalin-Vilkovisky algebra H * (Ω 2 BG) introduced by Getzler (Proposition 14).So finally ,we have obtained a morphism of Batalin-Vilkovisky (Theorem 24) from the homology of the double pointed loop space on BG to the free loop space homology on M.
Assuming that H * (Ωaut 1 (M)) is torsion free, Theorem 15 can also be applied to the monoid of self equivalences aut 1 (M) as we shall now explain.Since a smooth manifold has a CW-structure, M is a finite CW-complex.So by a theorem of Milnor [15], Hom(M, M) has the homotopy type of a CW-complex K. Therefore aut 1 M has the homotopy type of a path component of K. Recall that a path-connected homotopy associative H-space which has the homotopy of a CW-complex has naturally a homotopy inverse [20, Chapter X. Theorem 2.2], i.e. is naturally a H-group [17, p. 35].Therefore aut 1 M has a H-group structure.
Remark 5.In [5], Felix and Thomas posed the following problem: Is always surjective?The answer is no.Take M := S 2n , n ≥ 1. Rationally aut 1 S 2n has the same homotopy type as S 4n−1 .Therefore H * (Ωaut 1 S 2n ; Q)⊗H * (S 2n ; Q) is concentrated in even degree.It is easy to see that H * (LS 2n ; Q) is not trivial in odd degree.So H * (Θ aut 1 (S 2n ),S 2n ) cannot be surjective.
We give now the plan of the paper: Section 2: We give the definition and some examples of Batalin-Vilkovisky algebras.
Section 3: We compare the S 1 -action on Ω 2 BG to a S 1 -action defined on ΩG.Therefore the obvious isomorphism of algebras H * (ΩG) → H * (Ω 2 BG) is in fact an isomorphim of Batalin-Vilkovisky algebras.
Section 6: Using a second result of Felix and Thomas in [5], we show that π ≥1 (Ωaut Section 7: This section is devoted to calculations of the Batalin-Vilkovisky algebra H * (ΩG) ⊗ H * (M) over the rationals.In particular, for any path-connected compact Lie group G, we compute H * (LG : Q).
Section 8: This section is an appendix on split fibrations.
several papers (including this one).I would like especially to thank the referee for his serious job that led me to greatly inprove this paper.
2. Batalin-Vilkovisky algebras Definition 6.A Batalin-Vilkovisky algebra is a commutative graded algebra A equipped with an action of the exterior algebra such that The bracket { , } of degree +1 defined by Koszul [12, p. 3] (See also [8,Proposition 1.2] or [16]) has shown that { , } is a Lie bracket and therefore that a Batalin-Vilkovisky algebra is a Gerstenhaber algebra.
Example 8. (Tensor product of Batalin-Vilkovisky algebras) Let A and A ′ be two Batalin-Vilkovisky algebras.Denote by B A and B A ′ their respective operators.Consider the tensor product of algebras A ⊗ A ′ .Consider the operator B A⊗A ′ on A ⊗ A ′ given by Let X and X ′ be two pointed topological spaces.Let M and M ′ be two compact oriented smooth manifolds.It is easy to check that the Kunneth morphisms are morphisms of Batalin-Vilkovisky algebras.

The circle action on the pointed loops of a group
In this section, we define an action up to homotopy of the circle S 1 on the pointed loops ΩG of a H-group G.When G is a monoid, we show (Propositions 13 and 14) that the algebra H * (ΩG) equipped with the operator induced by this action is a Batalin-Vilkovisky algebra isomorphic to the Batalin-Vilkovisky algebra H * (Ω 2 BG) introduced by Getzler [8].Therefore, in this paper, instead of working with H * (Ω 2 BG), we always consider H * (ΩG).
Consider the free loop fibration ΩX j ֒→ LX ev ։ X.The evaluation map ev : LX ։ X admits a trivial section s : X ֒→ LX.
Suppose now that X is a H-group G.The following is easy to check if X is a group.For the H-group case, see appendix 8. Consider the map r : LG → ΩG unique up to homotopy such that the composite LG is homotopic to the map that sends the loop l ∈ LG to the loop t ∈ I → l(t)l(0) −1 .The map Θ G,G : ΩG × G → LG that maps (w, g) to the free loop t → w(t)g is a homotopy equivalence.Its homotopy inverse is the map LG → ΩG × G, l → (r(l), l(0)).In particular, r : LG ։ ΩG is a retract of j and the composite r • s is homotopically trivial.Theorem 9. (Compare with [13, Proposition 28]) Let X be a topological space.The retract r : LΩX ։ Ω 2 X is a morphism of S 1 -spaces up to homotopy (i.e. in the homotopy category of spaces).
Theorem 9 follows from Propositions 10i), 12 and 13.In [13, Proposition 28]), the same theorem is proved but the retract was defined in a different way.

Proposition 10.
i) Let G be a H-group.Then ΩG is equipped with an action of S 1 up to homotopy.
ii) Let G be a H-group acting up to homotopy on a topological space X.Then ΩG × X is equipped with an action of S 1 up to homotopy such that Θ G,X : ΩG × X → LX is a morphism of S 1 -spaces up to homotopy.
Proof.i) We define the action of S 1 on ΩG as the composition of ii) We define the action of S 1 on ΩG×X by s.(w, x) := (s.w, w(s).x),s ∈ S 1 , w ∈ Ω and x ∈ X.Here s.w denote the action of s ∈ S 1 on w ∈ ΩG given in i).It is easy to see that Θ G,X is a morphism of S 1 -spaces up to homotopy.Proposition 11.Let G 1 and G 2 be two H-groups.Let f : (G 1 , e) → (G 2 , e) be an homomorphism of H-spaces (in the sense of [17, p. 35]).Then Ωf : Ω(H 1 , e) → Ω(H 2 , e) is a morphism of S 1 -spaces up to homotopy.
Proof.A homomorphism of H-spaces between H-groups is necessarily a homomorphism of H-groups.The S 1 -structure on ΩG is clearly functorial with respect to homomorphisms of H-groups.Therefore the S 1 -structure on ΩG depends functorially only on the multiplication of G.
Proposition 12.The retract r : LG ։ ΩG is a morphism of S 1 -spaces up to homotopy.
Proof.Denote by p ΩG : ΩG × G → ΩG the projection on the first factor.By definition of r, the diagram y y t t t t t t t t t ΩG commutes up to homotopy.By Proposition 10 ii), Θ G,G is a morphism of S 1 -spaces up to homotopy.By definition of the action of the circle S 1 on ΩG × G, the projection p ΩG is also a morphism of S 1 -spaces up to homotopy.Proposition 13.Let (X, * ) be a pointed space.The action of S 1 up to homotopy on ΩG when G = ΩX given by Proposition 10 is homotopic to the action of S 1 on map ((E 2 , S 1 ), (X, * )) given by rotating the disk E 2 .
Proof.The map I × S 1 → E 2 sending (t, x) ∈ R × C to the barycenter of x with weight t and of 1 with weight 1 − t, gives the canonical homeomorphism Since j is a monomorphism in the homotopy category, to see that θ commutes with the S 1 -action up to homotopy, it suffices to see that the two maps j • action • (S 1 × θ) and j • θ • action are homotopic.
The adjoint of j • θ • action is the map The maps of pairs of spaces 2 .are homotopic: to construct the homotopy, fill the triangle of vertices e is , e is x and 1. Therefore j • action • (S 1 × θ) and j • θ • action are homotopic.
Proposition 14.Let G be a topological monoid which is also a Hgroup.Then the algebra H * (ΩG) equipped with the H * (S 1 )-module structure given by Proposition 10i) is a Batalin-Vilkovisky algebra isomorphic to the Batalin-Vilkovisky algebra H * (Ω 2 BG) given by [8].
Proof.Consider the classifying space of G, BG.There exists a homomorphism of H-spaces h : G ≃ → ΩBG which is a weak homotopy equivalence since π 0 (G) is a group.By Propositions 11 and 13, Ωh : ΩG ≃ → Ω 2 BG is a morphism of S 1 -spaces up to homotopy.Therefore, H * (Ωh) is both an isomorphism of graded algebras and of H * (S 1 )modules.Since H * (Ω 2 BG) is a Batalin-Vilkovisky algebra [8], H * (ΩG) is also a Batalin-Vilkovisky algebra.

The Batalin-Vilkovisky algebra
This section is the heart of the paper.We show (Theorem 15) that the tensor product of algebras H * (ΩG) ⊗ H * (M) equipped with an operator B ΩG×M is a Batalin-Vilkovisky algebra related to the Batalin-Vilkovisky algebra H * (LM) of Chas and Sullivan.Extending a result of Hepworth (Corollary 20), we give an explicit formula for this operator B ΩG×M .We deduce then the morphism of Batalin-Vilkovisky algebra from H * (ΩG) to H * (LM) (Corollary 23).
Let us start by giving a short proof of the Felix and Thomas theorem.
Proof of Theorem 1.Since e.m = m for m ∈ M, the map Θ G,M : ΩG×M → LM is a morphism of fiberwise monoids from the projection map p M : ΩG × M ։ M to the evaluation map ev : LM ։ M.
Therefore by [9, part 2) of Theorem 8.2], the composite is a morphism of graded algebras.
is an isomorphism of algebras.Consider the action of S 1 on ΩG up to homotopy given by Proposition 10 i) and the action of S 1 on LM given by rotation of the loops.Consider the induced diagonal action of S 1 on ΩG × LM.Explicitly, if G is a group, the diagonal action of s ∈ S 1 on (w, l) ∈ ΩG × LM is simply given by the pointed loop t → w(t + s)w(s) −1 and the free loop t → l(t + s).
Consider the twisted action of S 1 on ΩG × LM defined by s.(w, l) = (s.w,t → w(s).l(t+ s)) where s.w is the action of s ∈ S 1 on w ∈ ΩG given by Proposition 10 i).With respect to the twisted action on the source and the diagonal action on the target, Φ G,M is a morphism of S 1 -spaces up to homotopy.
The algebra H * (ΩG) ⊗ H * (LM) equipped with the H * (S 1 )-module structure given by the diagonal action is the tensor product of the Batalin-Vilkovisky algebra H * (ΩG) given by Proposition 14 and of the Batalin-Vilkovisky algebra H * (LM) given by Chas and Sullivan [1].Therefore by Example 8, it is a Batalin-Vilkovisky algebra.
Since the isomorphism H * (Φ G,m ) is both a morphism of algebras and a morphism of H * (S 1 )-modules, H * (ΩG) ⊗ H * (LM) equipped with the H * (S 1 )-module structure given by the twisted action is also a Batalin-Vilkovisky algebra.
Consider the trivial section s : M ֒→ LM mapping x ∈ M to the free loop constant on x.It is well-known that H * (s) : H * (M) → H * (LM) is a morphism of algebras.The map ΩG×s : ΩG×M → ΩG×LM is S 1equivariant with respect to the S 1 -action on ΩG ×M given by Proposition 10 ii) and to the twisted S 1 -action on ΩG × LM.Therefore, since The composite ) is the composite of the following morphisms of Batalin-Vilkovisky algebras: According to [10, Lemma 7], this homology suspension coincides with the usual one studied in [20,Chapter VIII].Since this paper was written almost completely before the preprint of Hepworth appeared, we will never use in this paper this fact that regretfully, we did not notice.However, we felt that it was necessary to use his terminology and we rewrote our paper accordingly.
In [10,Theorem 5], Hepworth computed the Batalin-Vilkovisky algebra on the modulo 2 free loop space homology on the special orthogonal group, H * (LSO(n); F 2 ).When n ≥ 4, Lemma 7 of [10] is required in order to achieve this interesting computation.
Proposition 18.Let G be a H-group.Let X be a topological space.Let act X : G × X → X be an action up to homotopy of G on X. Suppose that H * (ΩG) is torsion free.Denote by B ΩG (respectively B ΩG×X ) the operator given by the action of H * (S 1 ) on H * (ΩG) (respectively H * (ΩG × X)) given by Proposition 10.Then for any a ∈ H * (ΩG), x ∈ H * (X), Here ∆a = a (1) ⊗ a (2) is the diagonal of a ∈ H * (ΩG).
Proof.By Proposition 10 ii), the action of H * (S 1 ) on H * (ΩG) ⊗ H * (X) is the composite where ∆ H * (S Let ε H * (ΩG) : H * (ΩG) ։ k be the augmentation of the Hopf algebra On the other hand, by Definition 17 Lemma 19.Let G be a path-connected Lie group.Then H * (ΩG) is kfree and concentrated in even degree.So for any a ∈ H * (ΩG), B ΩG a = 0.
Proof.Let Ω 0 G be the path-component of the constant loop ê in ΩG.Since (ΩG, ê) is a H-group, the composite of the inclusion map and of the multiplication is an isomorphism (of algebras since H * (ΩG) is commutative).Let G be the universal cover of G. Then we have an isomorphism of algebras From Remark 16, Proposition 18 and Lemma 19, since here (−1) |a (1) | is equal to 1, we immediately obtain the following Corollary due to Hepworth.

Corollary 20. [10, Theorem 1] Let G be a path-connected compact Lie group. Then as Batalin-Vilkovisky algebra,
Corollary 21.Let G be a H-group acting up to homotopy on a topological space X.Assume that H * (ΩG) is torsion free.Then H * (X) is a trivial sub H * (S 1 )-module of the H * (S 1 )-module H * (ΩG) ⊗ H * (X) given by Proposition 10 ii).

Note that the composite
→ LX is homotopic to s : X ֒→ LX the trivial section mapping x ∈ X to x the free loop constant on x.Through s, X is a trivial sub S 1 -space of LX.Proposition 18.By definition, in Proposition 10 i), the action of any t ∈ S 1 on the constant loop ê ∈ Ω(G, e) is ê, since r : LG → ΩG is a pointed map.Therefore by definition, in Proposition 10 ii), the action of t ∈ S 1 on (ê, x) ∈ ΩG × X is t.(ê, x) = (t.ê,ê(s).x)= (ê, e.x).Therefore X ∼ = {ê} × X is up to homotopy a trivial sub S 1 -space of ΩG × X.And in homology, if H * (X) is considered as a trivial H * (S 1 )-module, the morphism

First proof of Corollary 21 without using
Second proof of Corollary 21 using Proposition 18. Again, we observe that the constant loop ê ∈ ΩG on the neutral element is a fixed point under the S 1 -action of ΩG.Therefore B ΩG (1) = 0.
Let ε H * (S 1 ) : H * (S 1 ) ։ k be the augmentation.Since the restriction of σ to S 1 × {ê} is the composite By Proposition 18, is a morphism of H * (S 1 )-modules.
Second proof using shriek maps.The trivial fibration proj : ΩG×M → ΩG is S 1 -equivariant.Therefore by [2, Section 2.3 Borel Construction], the integration along the fiber of proj, proj !: Theorem 24.Assume the hypothesis of Theorem 15.Then the composite is a morphism of Batalin-Vilkovisky algebras.
As pointed in the introduction of this paper, this theorem can be deduce using Theorem 15.But we prefer to give an independent proof.
Proof without using Theorem 15.By Proposition 14, the obvious isomorphism of algebras between H * (Ω 2 BG) and H * (ΩG) is an isomorphism of Batalin-Vilkovisky algebra.By Corollary 23, the inclusion of algebras is also a morphism of H * (S 1 )-modules.By Theorem 1, H * (Θ G,M ) is a morphism of algebras.By Proposition 10 ii), H * (Θ G,M ) is also a morphism of H * (S 1 )-modules.

some computations
Using Hepworth's definition of the homology suspension σ * (Definition 17), Lemma 11 of [13] becomes the well-known fact.
Lemma 28.Let X be a pointed topological space.Let n ≥ 0. Denote by hur X : π n (X) → H n (X) the Hurewicz map.We have the commutative diagram where π n (ΩX) ∼ = π n+1 (X) is the adjunction map.
Proposition 29.Let M be a compact oriented manifold equipped with an action of the circle S 1 .Denote by x a generator of Z. Then the Batalin-Vilkovisky algebra Proof.By applying Lemma 28, to the degree i map S 1 → S 1 , we obtain that σ * (x i ) = i[S 1 ].By applying Proposition 18 and Lemma 19, we conclude.
The following Proposition generalises the computation of the Batalin-Vilkovisky algebra H * (LS 3 ) due independently to the author [13, Theorem 16] and to Tamanoi [18].
Proposition 30.Let M be a compact oriented manifold equipped with an action of the sphere S 3 .Then the Batalin-Vilkovisky algebra Proof.Let ad : S 2 → ΩS In [10, Proposition 10], Hepworth gave a new proof for the computation of the Batalin-Vilkovisky algebra H * (LS 3 ).This proof of Hepworth gives also a proof of Proposition 30.
is a morphism of graded Lie algebras, i.e. sH * (M) is a module over the Lie algebra π ≥2 (G).Therefore by the definition of the semidirect product sH • {x, y} := 0 for x and y ∈ H * (M), Theorem 33.Let G be a path-connected topological monoid with an homotopy inverse acting on a smooth compact oriented manifold M. Assume that H * (ΩG) is torsion free.Let Γ1 : s −1 π ≥2 (G)⊗k → H * (LM) be the composite of the adjunction map, the Hurewicz morphism and of the map considered in Theorem 24 Then the k-linear morphism is a morphism of Lie algebras between the Lie bracket from example 32 and the loop bracket of LM.
Recall that s : M ֒→ LM denotes the trivial section.
Proof.Recall from the proof of Proposition 14, that there exists a homomorphism of H-spaces h : G ≃ → ΩBG which is also a weak equivalence and that H * (Ωh) : H * (ΩG) O O H n (ΩG) By [3, Remark 1.2 p. 214-5], the top line is a morphim of Lie algebras between the Samelson bracket and the Browder bracket.Therefore, the bottom line is also a morphism of Lie algebras.By Theorem 24, the composite is a morphism of Lie algebras.Therefore, by composition, Γ1 : Theorem 34.If k is a field of characteristic 0 and G is aut 1 M, the monoid of self-equivalences homotopic to the identity, the morphism of Lie algebras considered in Theorem 33 is injective.
Proof.Felix and Thomas [5, Theorem 2] showed that for n ≥ 1, Γ1 : is also injective.Our morphism of Lie algebras coincides with Γ1 in positive degree and with H * (s) in non-positive degree.Therefore, we have proved the theorem.
Here is a example due to Yves Felix showing that the Lie algebras considered in Theorem 34 are not abelian even for a very simple manifold M.
Theorem 36 tells us in particular that if we know the Batalin-Vilkovisky algebra H * (ΩG; Q) and the action of the spherical elements of H * (G) on H * (M), we can compute the Batalin-Vilkovisky algebra H * (ΩG; Q) ⊗ H * (M; Q).In [6, Theorem 4.4], Gerald Gaudens and the author computed the Batalin-Vilkovisky algebra H * (ΩG; Q) ∼ = H * (Ω 2 BG; Q) assuming that G is simply-connected.As we have already seen in Lemma 19, the Batalin-Vilkovisky algebra H * (ΩG) is also known when G is a pathconnected Lie group.Therefore, we have: Corollary 38.Let G be a path-connected Lie group acting on a smooth compact oriented manifold M. Then the Batalin-Vilkovisky algebra Note that there are no signs in this Corollary.
Proof.The rational homotopy groups π * (G) ⊗ Q are concentrated in odd degrees.Therefore s −1 f i are all in even degree and the Corollary follows from Theorem 36.
Let x i 's be the generators defined by Theorem 39.Let Θ be the isomorphim of algebras from • for 1 ≤ i ≤ l, the elements x n i i ⊗ 1 (respectively x n i i ⊗ x ∨ i ) in the i th factor of H * (LS 1 ; Q) ⊗dim π 1 (G)⊗Q to x n i i ⊗1⊗1 (respectively x n i i ⊗1⊗x ∨ i ), • for each y ∈ π 1 (G) tor , the element y to y ⊗ 1 ⊗ 1 and •for l < i ≤ r, the elements s −1 x n i i ⊗ 1 (respectively s −1 x n i i ⊗ x ∨ i ) in the (i−l) th factor of +∞ k=1 H * (LS 2k+1 ; Q) ⊗dim π 2k+1 (G)⊗Q to 1⊗s −1 x n i i ⊗1 (respectively 1 ⊗ s −1 x n i i ⊗ x ∨ i ).Explicitly Θ is the linear isomorphism mapping the element to x n 1 1 . . .x n l l y ⊗ s −1 x n l+1 l+1 . . .s −1 x nr r ⊗ x ∨ j 1 . . .x ∨ jp .In the tensor product of Batalin-Vilkovisky algebras H * (LS 2i+1 ; Q) ⊗dim π 2i+1 (G)⊗Q , the operator B is given by

6 .
A sub Lie algebra of H * (LM; Q) Let M be a smooth compact oriented manifold.We show that the rational free loop homology on M, H * (LM; Q), equipped with the loop bracket contains a sub Lie algebra.Definition 31.([7, p. 272], [19, 7.4.9],[11, p. 235]) Let L be a graded Lie algebra.Let N be a left L-module.Then the direct product of graded k-modules N × L can be equipped with a Lie bracket defined by {(m, l), (m ′ , l ′ )} := (l.m ′ − (−1) |l ′ ||m| l ′ .m,{l, l ′ }) for m, m ′ ∈ N and l, l ′ ∈ L. This graded Lie algebra is denoted N ⋊ L and is called the semidirect product of N and L or trivial extension of L by N. Example 32.(The Lie bracket of degree +1 on s −1 π ≥2 (G)⊗k⊕H * (M)) Let G be a path-connected topological monoid acting on a smooth compact oriented manifold M. The associative algebra H * (G) acts on H * (M) and therefore on sH * (M) by a.sx := (−1) |a|−1 s(a.x) for a ∈ H * (G) and x ∈ H * (M).Since G is a path-connected topological monoid, by [20, X.6.3], the Hurewicz morphism hur G : π * (G) → H * (G) is a morphism of graded Lie algebras from the Samelson product to the commutator associated to the associative algebra H * (G).The truncated homotopy groups of G, π ≥2 (G) is a sub-Lie algebra of π * (G).So finally, the composite an isomorphism of Batalin-Vilkovisky algebras.Since h : G ≃ → ΩBG is a homomorphism of H-spaces between two path-connected homotopy associative H-spaces, π * (h) : π * (G) ∼ = → π * (ΩBG) is a morphism of graded Lie algebras with respect to the Samelson brackets on G and on ΩBG.For n ≥ 1, consider the commutative diagram of graded k-modules is a morphism of Lie algebras between the Samelson bracket of G and the loop bracket of LM.The commutative graded algebra H * (M) equipped with the trivial operator B can be considered as a Batalin-Vilkovisky algebra.Since the trivial section s : M ֒→ LM is S 1 -equivariant with respect to the trivial action on M, H * (s) : H * (M) ֒→ H * (LM) is an inclusion of Batalin-Vilkovisky algebras and so an inclusion of Lie algebras.Denote by f ∈ π n (ΩG), the adjoint of f ∈ π n+1 (G).By Lemma 28,σ * • hur ΩG f = hur G f .Since n ≥ 1, a := hur ΩG f isprimitive and so by Corollary 26, {hur ΩG f ⊗ [M], 1 ⊗ x} = (−1) n 1 ⊗ hur G f.x.Since by Theorem 15, H * (Θ G,M ) : H * (ΩG) ⊗ H * (M) → H * (LM) is a morphism of Lie algebras, finally we have

7 .
The Batalin-Vilkovisky algebra H * (LG; Q) Theorem 36.Let G be a path-connected topological monoid with a homotopy inverse acting on a smooth compact oriented manifold M. Then the Batalin-Vilkovisky algebra H * (ΩG; Q and of H * (M; Q) equipped with the intersection product.