Bitopological BV-Algebras

1. Introduction

The interaction between topology and algebra is a central theme in modern mathematics. One important example is string topology, initiated by Chas and Sullivan [1]. For a closed, oriented manifold M, the homology of its free loop space carries a product and a degree-one operator which together give rise, after the appropriate grading convention, to a Batalin–Vilkovisky algebra.

The BV structure connects loop-space topology with Gerstenhaber algebras, Hochschild cohomology, and cyclic homology. Cohen and Jones developed a homotopy-theoretic approach to string topology [2], while Cohen and Voronov provided a systematic account of its constructions and applications [5]. Félix and Thomas established a relationship between string topology and Hochschild cohomology in the simply connected, closed-manifold setting [6].

A different direction in topology is the study of bitopological spaces. Instead of equipping a set with a single topology, one considers a pair $({\tau }_1,{\tau }_2)$ of topologies on the same set. This permits the simultaneous study of two notions of convergence, continuity, and separation.

The purpose of this paper is to formulate an algebraic framework in which two BV structures can be compared. The framework is motivated by the possibility that two topologies on the same geometric space induce different loop-space homology theories.

Our main contributions are the following:

(i)

We define a bitopological BV-algebra as a pair of BV-algebras together with specified comparison maps.

(ii)

We give compatibility conditions for the products and BV operators, and derive compatibility of the associated Gerstenhaber brackets.

(iii)

We describe diagonal and common-subalgebra constructions.

(iv)

We formulate a string-topological realization problem for manifolds carrying two topologies.

The algebraic definitions do not require the two topologies to be comparable. Geometric applications, however, require additional hypotheses ensuring that the relevant loop products and BV operators exist.

2. Rudimentary Background on BV-Algebras

Throughout, $\mathbb{k}$ denotes a field, and all graded algebras are $\mathbb{Z}$-graded.

Definition 2.1.

A BV-algebra over $\mathbb{k}$ is a graded-commutative algebra

$$(A,·,1)$$

together with a linear operator

$$D:A^n⟶A^{n-1}$$

such that

$$D^2=0$$

and the derived bracket

$$\{a,b\}={(-1)}^{\left|a\right|}\left(D(a·b)-\left(Da\right)·b-{(-1)}^{\left|a\right|}a·Db\right)$$

defines a Gerstenhaber bracket of degree $-1$.

Here graded commutativity means

$$a·b={(-1)}^{\left|a\right|\left|b\right|}b·a$$

for homogeneous elements $a,b\in A$.

The Gerstenhaber bracket is graded antisymmetric, satisfies the graded Jacobi identity, and is a graded derivation in each variable with respect to the product.

This definition is equivalent to the usual formulation of a BV-algebra as a Gerstenhaber algebra equipped with a compatible second-order operator. See [6,7].

Definition 2.2.

Let $(A,{·}_A,D_A)$ and $(B,{·}_B,D_B)$ be BV-algebras. A BV-algebra morphism is a degree-preserving linear map

$$f:A⟶B$$

such that

$$f\left(a{·}_{A}b\right)=f\left(a\right){·}_{B}f\left(b\right),f\left(1_A\right)=1_B,$$

and

$$f\left(D_{A}a\right)=D_{B}f\left(a\right).$$

Proposition 2.3.

Every BV-algebra morphism preserves the induced Gerstenhaber bracket.

Proof. 

Let $a,b\in A$ be homogeneous. By definition of the Gerstenhaber bracket, we have

$$\begin{array}{cc}\hfill f\left({\{a,b\}}_A\right)& ={(-1)}^{\left|a\right|}f\left(D_A\left(a{·}_{A}b\right)-\left(D_{A}a\right){·}_{A}b-{(-1)}^{\left|a\right|}a{·}_A{D}_{A}b\right)\hfill \\ & ={(-1)}^{\left|a\right|}(D_B\left(f\left(a\right){·}_{B}f\left(b\right)\right)-D_{B}f\left(a\right){·}_{B}f\left(b\right)\hfill \\ & -{(-1)}^{\left|a\right|}f\left(a\right){·}_B{D}_{B}f\left(b\right))\hfill \\ & ={\{f\left(a\right),f\left(b\right)\}}_B.\hfill \end{array}$$

Thus, f preserves the Gerstenhaber bracket. □

3. Bitopological Spaces and Comparison Maps

Definition 3.1.

A bitopological space is a triple

$$\mathcal{X}=(X,{\tau }_1,{\tau }_2),$$

where ${\tau }_1$ and ${\tau }_2$ are topologies on X.

A map

$$f:(X,{\tau }_1,{\tau }_2)⟶(Y,{\sigma }_1,{\sigma }_2)$$

is called bicontinuous if

$$f:(X,{\tau }_i)⟶(Y,{\sigma }_i)$$

is continuous for $i=1,2$.

The algebraic framework below is designed to accommodate two possibly different algebraic invariants associated with the same underlying bitopological space.

Definition 3.2.

A pair of BV-algebras associated with a bitopological space $\mathcal{X}$ consists of

$$(A_1,{·}_1,D_1),(A_2,{·}_2,D_2),$$

together with degree-preserving linear comparison maps

$$\varphi :A_1⟶A_2,\psi :A_2⟶A_1.$$

The maps are not assumed to be isomorphisms.

The comparison maps may arise, for example, from continuous identity maps between the two topological structures, provided the chosen homology or cohomology theory is functorial for those maps.

Definition 3.3.

Abitopological BV-algebrais a tuple

$$\mathcal{A}=(A_1,A_2,\varphi ,\psi )$$

such that:

• $(A_i,{·}_i,D_i)$ is a BV-algebra for $i=1,2$;
• ϕ and ψ are BV-algebra morphisms;
• the comparison maps satisfy any additional specified coherence relations, such as $\psi \varphi ={\mathrm{id}}_{A_1}$ or $\varphi \psi ={\mathrm{id}}_{A_2}$, when these are required.

The coherence relations are optional structural conditions, not part of the minimal definition. In particular, the identity maps need not be isomorphisms.

Remark 3.4.

The terminology refers to the presence of two BV structures associated with a bitopological space. It does not assert that every bitopological space canonically determines such structures. Their existence depends on the invariant being used and on the geometric or algebraic hypotheses.

4. Compatibility of Products and BV Operators

Definition 4.1.

Let $A_1$ and $A_2$ be BV-algebras. A degree-preserving linear map $\varphi :A_1\to A_2$ is called a compatible BV comparison map if

$$\varphi \left(a{·}_{1}b\right)=\varphi \left(a\right){·}_2\varphi \left(b\right)$$

and

$$\varphi \left(D_{1}a\right)=D_2\varphi \left(a\right)$$

for all homogeneous $a,b\in A_1$, and if $\varphi \left(1_{A_1}\right)=1_{A_2}$.

Theorem 4.2.

Let

$$\mathcal{A}=(A_1,A_2,\varphi ,\psi )$$

be a bitopological BV-algebra. Then the comparison maps preserve the associated Gerstenhaber brackets:

$$\varphi \left({\{a,b\}}_1\right)={\{\varphi \left(a\right),\varphi \left(b\right)\}}_2$$

and

$$\psi \left({\{u,v\}}_2\right)={\{\psi \left(u\right),\psi \left(v\right)\}}_1.$$

Proof. 

The first identity follows from the fact that $\varphi$ preserves the product and the BV operator. Indeed,

$$\begin{array}{cc}\hfill \varphi \left({\{a,b\}}_1\right)& ={(-1)}^{\left|a\right|}[\varphi \left(D_1\left(a{·}_{1}b\right)\right)-\varphi \left(\left(D_{1}a\right){·}_{1}b\right)\hfill \\ & -{(-1)}^{\left|a\right|}\varphi \left(a{·}_1{D}_{1}b\right)]\hfill \\ & ={(-1)}^{\left|a\right|}[D_2\left(\varphi \left(a\right){·}_2\varphi \left(b\right)\right)-D_2\varphi \left(a\right){·}_2\varphi \left(b\right)\hfill \\ & -{(-1)}^{\left|a\right|}\varphi \left(a\right){·}_2{D}_2\varphi \left(b\right)]\hfill \\ & ={\{\varphi \left(a\right),\varphi \left(b\right)\}}_2.\hfill \end{array}$$

The proof for $\psi$ is identical. □

Corollary 4.3.

If $\varphi :A_1\to A_2$ is an isomorphism of BV-algebras, then $A_1$ and $A_2$ have isomorphic Gerstenhaber algebras.

Proof. 

The inverse of a BV-algebra isomorphism preserves the product and BV operator. The preceding theorem therefore gives an isomorphism of the induced Gerstenhaber algebras. □

5. Diagonal and Common BV Structures

There are two elementary ways to construct a single BV-algebra from a pair.

Proposition 5.1.

Let A be a BV-algebra. Then

$$(A,A,{\mathrm{id}}_A,{\mathrm{id}}_A)$$

is a bitopological BV-algebra.

Proof. 

Both components carry the same BV structure, and the identity map preserves the product, unit, and BV operator. □

Proposition 5.2.

Let $A_1$ and $A_2$ be BV-algebras, and let

$$A=A_1\oplus A_2$$

with componentwise product and BV operator. Then A is a BV-algebra.

Proof. 

Graded commutativity and the unit property follow componentwise. The operator

$$D(a_1,a_2)=(D_1{a}_1,D_2{a}_2)$$

satisfies $D^2=0$. The derived bracket is

$$\{(a_1,a_2),(b_1,b_2)\}=({\{a_1,b_1\}}_1,{\{a_2,b_2\}}_2),$$

so the Gerstenhaber identities hold componentwise. □

Remark 5.3.

The direct sum construction is an algebraic combination of two BV-algebras. It does not by itself encode the topology of a bitopological space. To obtain a topologically meaningful invariant, the two summands and their comparison maps must be constructed functorially from the space.

6. Bitopological Free Loop Spaces

Let

$$\mathcal{M}=(M,{\tau }_1,{\tau }_2)$$

be a bitopological space. Write

$$LM=C(S^1,M)$$

for the set of continuous loops, with the topology specified below.

For $i=1,2$, let

$$L_{i}M=C((S^1,{\tau }_{S^1}),(M,{\tau }_i))$$

be the free loop space associated with ${\tau }_i$. The common loop space is

$$L_{\mathrm{bi}}M=L_{1}M\cap L_{2}M.$$

Let ${\tau }_i^L$ be the compact-open topology induced on $L_{\mathrm{bi}}M$ by ${\tau }_i$. This gives

$$\mathcal{L}M=(L_{\mathrm{bi}}M,{\tau }_1^L,{\tau }_2^L).$$

For each i, evaluation at the base point defines

$${\mathrm{ev}}_i:(L_{\mathrm{bi}}M,{\tau }_i^L)⟶(M,{\tau }_i),{\mathrm{ev}}_i\left(\gamma \right)=\gamma \left(0\right).$$

The circle acts on each loop space by rotation:

$$\rho :S^1\times LM⟶LM,\rho (\theta ,\gamma )\left(t\right)=\gamma (t+\theta ),$$

where the parameter is taken modulo 1.

When this action is continuous and the relevant homology theory is defined, it induces an operator

$$D_i:H_n(L_{\mathrm{bi}}M;\mathbb{k})⟶H_{n+1}(L_{\mathrm{bi}}M;\mathbb{k}).$$

The degree convention here is homological. After the usual shift and the corresponding convention for the loop product, this operator is the string-topological BV operator.

The existence of a compatible loop product requires additional geometric input, including an appropriate intersection or umkehr construction associated with the diagonal of M.

7. String Topology and the BV Structure

Suppose that M is a closed, smooth, oriented manifold of dimension d. Under the classical string topology hypotheses, the shifted loop homology

$${\mathbb{H}}_{*}(LM;\mathbb{k})=H_{*+d}(LM;\mathbb{k})$$

carries a BV-algebra structure [1,2,5].

If two topologies ${\tau }_1$ and ${\tau }_2$ are placed on the underlying set of M, one may attempt to construct two shifted homology theories

$${\mathbb{H}}_{*}^{\left(i\right)}=H_{*+d}(L_{i}M;\mathbb{k}),i=1,2.$$

To obtain a bitopological BV-algebra, one needs:

1.

a well-defined loop product on each ${\mathbb{H}}_{*}^{\left(i\right)}$;

2.

a continuous circle action inducing a square-zero BV operator;

3.

comparison maps between the two loop-space invariants;

4.

compatibility of the comparison maps with both the product and the BV operator.

These conditions are not automatic for arbitrary topologies. In particular, a topology on the underlying set of a smooth manifold need not be compatible with its smooth structure.

Theorem 7.1 (Recovery of the classical structure).

Let M be a closed, smooth, oriented manifold of dimension d, and suppose

$${\tau }_1={\tau }_2={\tau }_M,$$

where ${\tau }_M$ is the topology induced by the smooth structure. Then the two loop-space homology theories agree, and the resulting bitopological BV-algebra is the ordinary string topology BV-algebra.

Proof. 

Since the two topologies agree, their compact-open topologies on the free loop space agree. Thus

$$L_{1}M=L_{2}M=LM.$$

The evaluation maps, circle action, and geometric diagonal construction are the classical ones. Consequently, the products and BV operators agree with those of the Chas–Sullivan construction. □

Remark 7.2.

The theorem is a consistency result. It does not establish the existence of a new BV structure when ${\tau }_1\ne {\tau }_2$. That case requires a separate analysis of the two loop-space constructions and their comparison maps.

8. Examples and Special Cases

Example 8.1 (The equal-topology case).

Let M be a closed, oriented manifold and set ${\tau }_1={\tau }_2={\tau }_M$. The associated pair of string topology BV-algebras is diagonal:

$$({\mathbb{H}}_{*}\left(LM\right),{\mathbb{H}}_{*}\left(LM\right),\mathrm{id},\mathrm{id}).$$

Example 8.2 (A pair of abstract BV-algebras).

Let $A_1$ and $A_2$ be any BV-algebras, and suppose there are BV morphisms

$$\varphi :A_1\to A_2,\psi :A_2\to A_1.$$

Then $(A_1,A_2,\varphi ,\psi )$ is a bitopological BV-algebra in the algebraic sense of Definition 3.3. To realize it geometrically, one must identify a bitopological space whose chosen invariants yield these algebras and maps.

Example 8.3 (Comparison induced by a map).

Suppose a continuous map between two spaces induces a map between their chosen loop-space homology groups. If the induced map preserves the loop product and commutes with the BV operator, it is a BV-algebra morphism. Such a map can serve as one of the comparison maps in a bitopological BV-algebra.

9. Natural Hypotheses for Compatible Bitopological String Topology BV-Algebras

In this section, we investigate sufficient conditions under which a bitopological manifold

$$\mathcal{M}=(M,{\tau }_1,{\tau }_2)$$

admits two compatible string topology Batalin–Vilkovisky (BV) algebras.

The principal difficulty is that a bitopological space need not possess a smooth structure compatible with both of its topologies. Moreover, even when both topologies arise from smooth manifolds, the existence of two BV-algebra structures does not automatically imply that they are compatible.

Our approach is to impose geometric hypotheses that guarantee the existence of the two classical string topology structures and then formulate conditions under which the corresponding comparison maps preserve the loop products and BV operators.

The classical starting point is the Chas–Sullivan construction, which equips the shifted homology of the free loop space of a closed, oriented manifold with a BV-algebra structure [1,2]. The homotopy invariance of the loop product and related string topology operations was subsequently established under appropriate hypotheses by Cohen, Klein, and Sullivan [?].

9.1. The Geometric Setting

Let M be a set equipped with two topologies ${\tau }_1$ and ${\tau }_2$. We write

$$\mathcal{M}=(M,{\tau }_1,{\tau }_2).$$

We first distinguish the topological structures from the smooth structures used to construct the string topology operations.

Definition 9.1.

A smooth realization of $\mathcal{M}$ consists of two smooth manifolds

$$M_1=(M,{\tau }_1,{\mathcal{S}}_1),M_2=(M,{\tau }_2,{\mathcal{S}}_2),$$

where ${\mathcal{S}}_i$ is a smooth atlas compatible with ${\tau }_i$.

The realization is called closed and oriented of dimension d if both $M_1$ and $M_2$ are compact, without boundary, oriented, and of dimension d.

The following hypotheses provide a natural sufficient condition for the existence of two string topology BV-algebras.

Definition 9.2

(Admissible bitopological manifold). A bitopological manifold

$$\mathcal{M}=(M,{\tau }_1,{\tau }_2)$$

is called admissible for classical string topology if the following conditions hold:

(H1)

For $i=1,2$, the space $(M,{\tau }_i)$ admits a smooth manifold structure of dimension d.

(H2)

Both smooth manifolds are compact and without boundary.

(H3)

Both manifolds are oriented, and the coefficient ring is a field $\mathbb{k}$.

(H4)

For each i, the free loop space

$$L{M}_i=C^{\infty }(S^1,M_i)$$

is equipped with a topology and a homology theory for which the classical loop product and the circle-action operator are defined.

(H5)

The orientations and grading conventions are fixed consistently, so that both loop homology groups are shifted by the same dimension d.

Remark 9.3.

The use of smooth loops is convenient for the geometric construction. Equivalent continuous-loop models may be used when the relevant comparison theorems apply.

Compactness and absence of boundary are standard hypotheses for the classical closed-manifold construction. There are extensions to manifolds with boundary and to other geometric settings, but these require additional choices and hypotheses.

9.2. Existence of the Two BV-Algebra Structures

For $i=1,2$, define the shifted loop homology by

$${\mathbb{H}}_n(L{M}_i;\mathbb{k})=H_{n+d}(L{M}_i;\mathbb{k}).$$

The Chas–Sullivan loop product has degree $-d$ on unshifted homology:

$${★}_i:H_p(L{M}_i;\mathbb{k})\otimes H_q(L{M}_i;\mathbb{k})⟶H_{p+q-d}(L{M}_i;\mathbb{k}).$$

After shifting the grading, it becomes a product of degree zero:

$${★}_i:{\mathbb{H}}_p(L{M}_i;\mathbb{k})\otimes {\mathbb{H}}_q(L{M}_i;\mathbb{k})⟶{\mathbb{H}}_{p+q}(L{M}_i;\mathbb{k}).$$

The rotation action

$${\rho }_i:S^1\times L{M}_i⟶L{M}_i,{\rho }_i(\theta ,\gamma )\left(t\right)=\gamma (t+\theta ),$$

induces the operator

$${\mathsf{\Delta }}_i:{\mathbb{H}}_n(L{M}_i;\mathbb{k})⟶{\mathbb{H}}_{n+1}(L{M}_i;\mathbb{k}).$$

The operator ${\mathsf{\Delta }}_i$ satisfies

$${\mathsf{\Delta }}_i^2=0$$

and is compatible with the loop product in the BV sense.

Theorem 9.4

(Existence of two string topology BV-algebras). Let

$$\mathcal{M}=(M,{\tau }_1,{\tau }_2)$$

be an admissible bitopological manifold. Then each of the two smooth manifolds $M_i$ determines a string topology BV-algebra

$${\mathcal{B}}_i=\left({\mathbb{H}}_{*}(L{M}_i;\mathbb{k}),{★}_i,{\mathsf{\Delta }}_i\right),i=1,2.$$

Proof. 

By (H1)–(H3), each $M_i$ is a closed, oriented, smooth manifold of dimension d.

The classical string topology construction therefore provides a loop product on $H_{*}(L{M}_i;\mathbb{k})$. The circle action on $L{M}_i$ induces the operator ${\mathsf{\Delta }}_i$.

By the Chas–Sullivan theorem, the shifted homology

$${\mathbb{H}}_{*}(L{M}_i;\mathbb{k})=H_{*+d}(L{M}_i;\mathbb{k})$$

is a BV-algebra. Applying this construction separately to $M_1$ and $M_2$ gives the two asserted BV-algebras. □

Remark 9.5.

The theorem establishes existence, but not compatibility. The two BV-algebras may have different products, different BV operators, and different underlying loop spaces. A comparison map is needed to relate them.

9.3. Compatibility Through an Orientation-Preserving Diffeomorphism

The most natural strong compatibility hypothesis is the existence of an orientation-preserving diffeomorphism between the two smooth realizations.

Definition 9.6.

Let $M_1$ and $M_2$ be closed, oriented, smooth manifolds of the same dimension. A map

$$f:M_1⟶M_2$$

is an orientation-preserving diffeomorphism if it is a diffeomorphism and preserves the chosen orientations.

Such a map induces a map of free loop spaces

$$Lf:L{M}_1⟶L{M}_2,\left(Lf\right)\left(\gamma \right)=f\circ \gamma .$$

Since f is a diffeomorphism, $Lf$ is a homeomorphism in the standard smooth-loop topologies. It also commutes with the circle action:

$$Lf\left({\rho }_1(\theta ,\gamma )\right)={\rho }_2\left(\theta ,Lf\left(\gamma \right)\right).$$

Consequently, the induced homology map commutes with the BV operators.

Theorem 9.7

(BV compatibility under orientation-preserving diffeomorphisms). Let $M_1$ and $M_2$ be closed, oriented, smooth manifolds of the same dimension d. Suppose that

$$f:M_1⟶M_2$$

is an orientation-preserving diffeomorphism.

Then the induced map

$${\left(Lf\right)}_{*}:{\mathbb{H}}_{*}(L{M}_1;\mathbb{k})⟶{\mathbb{H}}_{*}(L{M}_2;\mathbb{k})$$

is an isomorphism of BV-algebras. In particular,

$${\left(Lf\right)}_{*}\left(a{★}_{1}b\right)={\left(Lf\right)}_{*}\left(a\right){★}_2{\left(Lf\right)}_{*}\left(b\right)$$

and

$${\left(Lf\right)}_{*}\left({\mathsf{\Delta }}_{1}a\right)={\mathsf{\Delta }}_2\left({\left(Lf\right)}_{*}a\right).$$

Proof. 

The map f induces a map of free loop spaces by composition. Since f is a diffeomorphism, the induced map $Lf$ is a homeomorphism.

The Chas–Sullivan product is defined using evaluation at a marked point, the intersection construction associated with the diagonal, and loop concatenation. The diffeomorphism f preserves these constructions, including the orientation data that determine the umkehr map.

Naturality of the loop product therefore gives

$${\left(Lf\right)}_{*}\left(a{★}_{1}b\right)={\left(Lf\right)}_{*}\left(a\right){★}_2{\left(Lf\right)}_{*}\left(b\right).$$

Moreover, $Lf$ commutes with rotation:

$$Lf\circ {\rho }_1={\rho }_2\circ ({\mathrm{id}}_{S^1}\times Lf).$$

Passing to homology yields

$${\left(Lf\right)}_{*}{\mathsf{\Delta }}_1={\mathsf{\Delta }}_2{\left(Lf\right)}_{*}.$$

Thus ${\left(Lf\right)}_{*}$ preserves both the product and the BV operator. Its inverse is induced by $f^{-1}$, so it is an isomorphism of BV-algebras. □

Corollary 9.8.

Under the hypotheses of Theorem 9.7, the two string topology BV-algebras are isomorphic:

$${\mathcal{B}}_1\cong {\mathcal{B}}_2.$$

In particular, their induced Gerstenhaber algebras are isomorphic.

Proof. 

The first assertion follows from Theorem 9.7. The second follows because a BV-algebra isomorphism preserves the derived Gerstenhaber bracket. □

9.4. The Bitopological Case and the Identity Map

An important point is that the identity map on the underlying set need not be a homeomorphism between the two topologies.

Proposition 9.9.

Suppose that

$${\mathrm{id}}_M:(M,{\tau }_1)⟶(M,{\tau }_2)$$

is a homeomorphism. Then

$${\tau }_1={\tau }_2.$$

Proof. 

Continuity of the identity map implies

$${\tau }_2\subseteq {\tau }_1.$$

Continuity of its inverse, which is also the identity map, implies

$${\tau }_1\subseteq {\tau }_2.$$

Hence the two topologies are equal. □

Thus, if ${\tau }_1\ne {\tau }_2$, compatibility cannot be obtained by assuming that the identity map is a homeomorphism. Instead, one may use a nonidentity diffeomorphism between the two smooth realizations.

Corollary 9.10.

Let

$$\mathcal{M}=(M,{\tau }_1,{\tau }_2),{\tau }_1\ne {\tau }_2,$$

and suppose that there is an orientation-preserving diffeomorphism

$$f:(M,{\tau }_1,{\mathcal{S}}_1)⟶(M,{\tau }_2,{\mathcal{S}}_2).$$

Then the two associated string topology BV-algebras are isomorphic.

Proof. 

Apply Theorem 9.7 to the two smooth realizations. □

Remark 9.11.

This result allows the topologies to be distinct as collections of open subsets of the same set. The diffeomorphism need not be the identity function.

The conclusion is an isomorphism of the resulting BV-algebras. It does not assert that the two topologies coincide or that their identity map is continuous in both directions.

9.5. Compatibility Under Weaker Comparison Hypotheses

A diffeomorphism is a strong sufficient condition. For some purposes, one may seek weaker hypotheses.

Definition 9.12.

A continuous map

$$f:M_1⟶M_2$$

between closed, oriented, smooth manifolds of the same dimension is called string-topology compatible if its induced map

$${\left(Lf\right)}_{*}:{\mathbb{H}}_{*}(L{M}_1;\mathbb{k})⟶{\mathbb{H}}_{*}(L{M}_2;\mathbb{k})$$

is defined and satisfies

$${\left(Lf\right)}_{*}\left(a{★}_{1}b\right)={\left(Lf\right)}_{*}\left(a\right){★}_2{\left(Lf\right)}_{*}\left(b\right)$$

and

$${\left(Lf\right)}_{*}{\mathsf{\Delta }}_1={\mathsf{\Delta }}_2{\left(Lf\right)}_{*}.$$

Proposition 9.13.

If $f:M_1\to M_2$ is string-topology compatible, then ${\left(Lf\right)}_{*}$ is a morphism of BV-algebras. If ${\left(Lf\right)}_{*}$ is also an isomorphism of graded vector spaces, then it is an isomorphism of BV-algebras.

Proof. 

The two compatibility identities state precisely that ${\left(Lf\right)}_{*}$ preserves the product and the BV operator. The map is therefore a BV-algebra morphism. If it is bijective, its inverse also preserves the product and BV operator, and hence it is a BV-algebra isomorphism. □

Remark 9.14.

A homotopy equivalence between closed, oriented manifolds of the same dimension can induce isomorphisms of string topology operations under the hypotheses of the relevant homotopy-invariance theorems; see [?].

For a precise application to BV-algebras, one should verify that the chosen homotopy-invariance result applies to the BV operator as well as to the loop product. The diffeomorphism hypothesis in Theorem 9.7 avoids this additional issue.

9.6. A Common Loop Space with Two BV Operators

A different situation arises when the two topologies yield the same underlying loop space but potentially different BV structures.

Suppose that a common graded vector space A carries two products and two operators:

$$(A,{★}_1,{\mathsf{\Delta }}_1),(A,{★}_2,{\mathsf{\Delta }}_2).$$

The structures are compatible in the strongest sense if the identity map is a BV-algebra isomorphism.

Proposition 9.15.

Suppose that

$${★}_1={★}_2\mathrm{and}{\mathsf{\Delta }}_1={\mathsf{\Delta }}_2.$$

Then the identity map on A is a BV-algebra isomorphism between the two structures.

Proof. 

The identity map preserves the product and the BV operator by the assumed equalities. □

More generally, one may seek a degree-preserving isomorphism

$$T:A⟶A$$

satisfying

$$T\left(a{★}_{1}b\right)=T\left(a\right){★}_{2}T\left(b\right),T{\mathsf{\Delta }}_1={\mathsf{\Delta }}_{2}T.$$

These identities provide an algebraic criterion for equivalence even when the two constructions are not identical.

9.7. Summary of Sufficient Hypotheses

The preceding results give the following useful sufficient conditions.

Theorem 9.16

(Sufficient conditions for a compatible pair). Let

$$\mathcal{M}=(M,{\tau }_1,{\tau }_2)$$

be a bitopological space. Suppose that:

1.

Each $(M,{\tau }_i)$ admits a closed, oriented, smooth manifold structure of dimension d.

2.

The classical string topology BV-algebra is defined for each resulting manifold.

3.

There exists an orientation-preserving diffeomorphism between the two smooth realizations.

Then $\mathcal{M}$ determines two string topology BV-algebras

$${\mathcal{B}}_1,{\mathcal{B}}_2,$$

and the diffeomorphism induces a BV-algebra isomorphism

$$\mathsf{\Phi }:{\mathcal{B}}_1\stackrel{\cong }{⟶}{\mathcal{B}}_2.$$

Consequently,

$$\mathsf{\Phi }\left(a{★}_{1}b\right)=\mathsf{\Phi }\left(a\right){★}_2\mathsf{\Phi }\left(b\right),\mathsf{\Phi }{\mathsf{\Delta }}_1={\mathsf{\Delta }}_2\mathsf{\Phi }.$$

Proof. 

The first two hypotheses give the existence of the two BV-algebras by Theorem 9.4. The third hypothesis and Theorem 9.7 give the required BV-algebra isomorphism. □

Remark 9.17

(Scope of the theorem). The hypotheses above are sufficient, not necessary. They provide a direct geometric criterion that avoids having to construct comparison maps at chain level.

The genuinely new problem is to find natural conditions under which distinct topologies produce compatible string topology BV-algebras without being related by an orientation-preserving diffeomorphism.

10. Distinct Bitopologies and Non-Isomorphic String Topology BV-Algebras

We now address a fundamental question for the theory developed in this paper. Suppose that $\mathcal{M}=(M,{\tau }_1,{\tau }_2)$ is a bitopological manifold, where ${\tau }_1$ and ${\tau }_2$ are distinct manifold topologies on the same underlying set M. Can the two topologies give rise to non-isomorphic string topology Batalin–Vilkovisky algebras?

The answer is subtle. Distinctness of the topologies alone is not sufficient. Indeed, string topology is remarkably insensitive to many changes of geometric structure. In particular, the loop product and the associated BV structure are homotopy invariant under suitable orientation-preserving homotopy equivalences of closed oriented manifolds. Consequently, two distinct manifold topologies may produce isomorphic string topology BV-algebras.

This observation leads to two complementary problems:

• determine rigidity conditions under which the two BV-algebras must be isomorphic;
• construct bitopological manifolds for which the resulting BV-algebras are genuinely non-isomorphic.

10.1. Two Smooth Realizations of the Same Set

Let

$$M_1=(M,{\tau }_1,{\mathcal{S}}_1),M_2=(M,{\tau }_2,{\mathcal{S}}_2)$$

be two closed, connected, oriented smooth manifolds having the same underlying set M. Assume that both manifolds have dimension d.So $H_{*+d}(L{M}_i;\mathbb{k})$ for the shifted loop homology over a field $\mathbb{k}$.

$$\left({\mathbb{H}}_{*}^{\left(i\right)},{★}_i,{\mathsf{\Delta }}_i\right),$$

where

$${★}_i:{\mathbb{H}}_p^{\left(i\right)}\otimes {\mathbb{H}}_q^{\left(i\right)}⟶{\mathbb{H}}_{p+q}^{\left(i\right)}$$

is the shifted Chas–Sullivan product and

$${\mathsf{\Delta }}_i:{\mathbb{H}}_n^{\left(i\right)}⟶{\mathbb{H}}_{n+1}^{\left(i\right)}$$

is the BV operator induced by the $S^1$-action on the free loop space.

The question is whether ${\mathcal{B}}_1\cong {\mathcal{B}}_2$ must hold.

10.2. A Rigidity Theorem

The strongest general rigidity principle comes from homotopy invariance.

Theorem 10.1

(String-topology rigidity). Let

$$M_1=(M,{\tau }_1,{\mathcal{S}}_1),M_2=(M,{\tau }_2,{\mathcal{S}}_2)$$

be closed, connected, oriented smooth manifolds of the same dimension d. Suppose that there exists an orientation-preserving homotopy equivalence

$$f⟶M_2.$$

Then the induced map

$$Lf⟶L{M}_2,Lf\left(\gamma \right)=f\circ \gamma ,$$

induces an isomorphism

$$\left(Lf\right)*:\mathbb{H}(L{M}_1;\mathbb{k})⟶{\mathbb{H}}_{(}L{M}_2;\mathbb{k})$$

of BV-algebras.

$${\mathsf{\Delta }}_2\left({\left(Lf\right)}_{*}a\right).$$

Proof. 

The map f induces a map of free loop spaces

$$Lf⟶L{M}_2.$$

Since f is a homotopy equivalence, so is $Lf$.

$$\left(Lf\right)*\left(a\right){★}_2\left(Lf\right)*\left(b\right).$$

$${\mathsf{\Delta }}_2{\left(Lf\right)}_{*}.$$

Thus $\left(Lf\right)*$ preserves both the loop product and the BV operator. Since f has a homotopy inverse, $\left(Lf\right)*$ is an isomorphism. Therefore it is an isomorphism of BV-algebras. □

The theorem is a particularly strong rigidity statement. It shows that the string topology BV-algebra cannot distinguish arbitrary changes of manifold structure that remain within the same oriented homotopy type.

The homotopy-invariance result of Cohen, Klein, and Sullivan states that an orientation-preserving homotopy equivalence of closed oriented manifolds induces an isomorphism of the loop homology algebra; their result also identifies the corresponding BV-algebra structure as homotopy invariant.

10.3. Consequences for Bitopological Manifolds

The preceding theorem immediately gives the following.

Corollary 10.2.

Let $\mathcal{M}=(M,{\tau }_1,{\tau }_2)$ be a bitopological manifold whose two smooth realizations $M_1$ and $M_2$ are closed, connected, oriented, and have the same dimension.

If there exists an orientation-preserving homotopy equivalence

$$f:(M,{\tau }_1)⟶(M,{\tau }_2),$$

then the two associated string topology BV-algebras are isomorphic:

$${\mathcal{B}}_1\cong {\mathcal{B}}_2.$$

Proof. 

This is an immediate consequence of Theorem 13.3. □

This result shows that the condition ${\tau }_1\ne {\tau }_2$ is not sufficient to produce distinct BV-algebras.

Indeed, the two topologies may be different while the corresponding manifolds remain oriented-homotopy equivalent.

10.4. A Criterion for Non-isomorphic BV-Algebras

We next obtain a useful contrapositive principle.

Theorem 10.3

(Homotopy obstruction). Let

$$M_1=(M,{\tau }_1,{\mathcal{S}}_1),M_2=(M,{\tau }_2,{\mathcal{S}}_2)$$

be closed, connected, oriented manifolds of dimension d.

Suppose that an invariant of the string topology BV-algebra distinguishes the two spaces. Then

$${\mathcal{B}}_1\neg \cong {\mathcal{B}}_2.$$

In particular, if the underlying graded vector spaces

$${\mathbb{H}}_{(}L{M}_1;\mathbb{k})\mathrm{and}{\mathbb{H}}_{(}L{M}_2;\mathbb{k})$$

are not isomorphic, then the corresponding BV-algebras are not isomorphic.

Proof. 

An isomorphism of BV-algebras is, in particular, an isomorphism of the underlying graded vector spaces. Therefore, if the graded vector spaces are not isomorphic, no BV-algebra isomorphism can exist.

More generally, any invariant preserved by BV-algebra isomorphisms must take the same value on ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$. Consequently, different values imply non-isomorphism. □

The theorem gives a practical strategy. Instead of attempting to compare the complete BV operations directly, one can search for a simpler invariant of the underlying loop homology.

For example, if

$${\dim }_{\mathbb{k}}{\mathbb{H}}_r(L{M}_1;\mathbb{k})\ne {\dim }_{\mathbb{k}}{\mathbb{H}}_r(L{M}_2;\mathbb{k})$$

for some degree r, then

$${\mathcal{B}}_1\neg \cong {\mathcal{B}}_2.$$

10.5. Fundamental Groups as an Obstruction

A particularly useful source of examples comes from the fundamental group.

For a connected manifold M, the free loop space decomposes according to free homotopy classes:

$$LM\simeq \underset{\left[\gamma \right]\in \left[{\pi }_1\left(M\right)\right]}{⨆}{L}_{\left[\gamma \right]}M.$$

Thus, information about ${\pi }_1\left(M\right)$ is reflected in the component structure of $LM$.

This suggests the following obstruction.

Proposition 10.4.

Suppose that two closed connected manifolds $M_1$ and $M_2$ have free loop spaces whose sets of path components are not equivalent in a manner compatible with the induced loop-space homology. Then their string topology BV-algebras cannot be isomorphic as BV-algebras with component decomposition.

Proof. 

The decomposition of $LM$ into components is preserved by any homeomorphism of free loop spaces. In particular, a BV-algebra isomorphism induced by a map of free loop spaces must preserve the corresponding decomposition of homology. If the component structures are incompatible, no such isomorphism can exist. □

For aspherical manifolds, this observation can be especially useful because the free loop space is closely related to conjugacy classes in the fundamental group. Thus, manifolds with sufficiently different fundamental groups provide natural candidates for non-isomorphic string topology algebras.

10.6. Betti-Number Obstruction

Another elementary obstruction comes from the homology of the free loop spaces.

Proposition 10.5.

Let $M_1$ and $M_2$ be closed oriented manifolds. If there exists an integer r such that

$${\dim }_{\mathbb{k}}{H}_r(L{M}_1;\mathbb{k})\ne {\dim }_{\mathbb{k}}{H}_r(L{M}_2;\mathbb{k}),$$

then

$${\mathcal{B}}_1\neg \cong {\mathcal{B}}_2.$$

Proof. 

The shifted grading changes the index but not the dimensions. An isomorphism of BV-algebras must induce an isomorphism of the underlying graded vector spaces. Therefore all graded Betti numbers must agree. The assumed inequality gives a contradiction. □

$$\sum _{r\ge 0}{\dim }_{\mathbb{k}}{H}_r(LM;\mathbb{k})t^r.$$

If

$$P_{L{M}_1}\left(t\right)\ne P_{L{M}_2}\left(t\right),$$

then the two string topology BV-algebras cannot be isomorphic.

10.7. An Important Limitation

It is important to distinguish two different statements:

• the two manifold topologies are distinct;
• the two string topology BV-algebras are non-isomorphic.

The implication

$${\tau }_1\ne {\tau }_2⟹{\mathcal{B}}_1\neg \cong {\mathcal{B}}_2$$

is false in general.

The reason is precisely the homotopy invariance described above. If there exists an orientation-preserving homotopy equivalence

$$f⟶M_2,$$

then

$${\mathcal{B}}_1\cong {\mathcal{B}}_2$$

even when ${\tau }_1\ne {\tau }_2$.

Thus the relevant equivalence relation is not merely equality of topologies. The string topology BV-algebra is insensitive to many changes of the underlying smooth or topological realization.

10.8. A Method for Constructing Genuine Examples

To construct a bitopological manifold with non-isomorphic string topology BV-algebras, one should proceed as follows.

Let X and Y be closed connected oriented manifolds satisfying

$${\mathbb{H}}_{(}LX;\mathbb{k})\neg \cong {\mathbb{H}}_{(}LY;\mathbb{k})$$

as graded vector spaces.

Choose a set M together with bijections

$${\phi }_1⟶X,{\phi }_2⟶Y.$$

$${\tau }_2=\left\{{\phi }_2^{-1}\left(V\right)\subseteq M|V\subseteq Y\mathrm{is}\mathrm{open}\right\}.$$

Then

$$(M,{\tau }_1)\cong X,(M,{\tau }_2)\cong Y$$

as topological manifolds.

Consequently, the two string topology BV-algebras are

$${\mathcal{B}}_1\cong {\mathbb{H}}_{(}LX;\mathbb{k}),{\mathcal{B}}_2\cong {\mathbb{H}}_{(}LY;\mathbb{k}).$$

Hence, whenever

$${\mathbb{H}}_{(}LX;\mathbb{k})\neg \cong {\mathbb{H}}_{(}LY;\mathbb{k}),$$

we obtain ${\mathcal{B}}_1\neg \cong {\mathcal{B}}_2$.

This construction is conceptually simple but important: the same underlying set can carry two manifold topologies whose associated manifolds have different homotopy types.

Theorem 10.6

(Transport construction). Let X and Y be closed, connected, oriented smooth manifolds of the same dimension. Suppose that

$${\mathbb{H}}_{(}LX;\mathbb{k})\neg \cong {\mathbb{H}}_{(}LY;\mathbb{k})$$

as graded vector spaces.

Then there exists a set M and two manifold topologies ${\tau }_1,{\tau }_2$ on M such that

$$(M,{\tau }_1)\cong X,(M,{\tau }_2)\cong Y,$$

but the associated string topology BV-algebras are non-isomorphic.

Proof. 

Choose a set M with cardinality equal to the common cardinality of X and Y, and choose bijections

$${\phi }_1\to X,{\phi }_2\to Y.$$

Define ${\tau }_1$ and ${\tau }_2$ by transporting the manifold topologies of X and Y, respectively.

Then ${\phi }_1$ and ${\phi }_2$ are homeomorphisms with respect to the corresponding transported topologies. Hence

$$(M,{\tau }_1)\cong X,(M,{\tau }_2)\cong Y.$$

The associated free loop spaces therefore have the same homotopy types as $LX$ and $LY$, respectively. Thus their shifted loop homologies have the same graded dimensions as those of $LX$ and $LY$. By assumption these graded vector spaces are not isomorphic. Consequently the resulting BV-algebras cannot be isomorphic. □

10.9. A Concrete Family: Spheres Versus Products

The transport construction gives many possible examples. For instance, consider manifolds of the same dimension such as

$$S^n\mathrm{and}{S}^p\times S^q,p+q=n.$$

If the corresponding free loop homologies have different graded dimensions, then transporting the two manifold topologies to the same underlying set produces a bitopological manifold with non-isomorphic string topology BV-algebras.

More precisely, choose

$$X=S^n,Y=S^p\times S^q,p+q=n.$$

Let $M=X{\bigsqcup }_{\mathrm{set}}Y$ denote an abstract set equipped with bijections to X and Y. Transport the two topologies to M.

The resulting bitopological space

$$(M,{\tau }_X,{\tau }_Y)$$

has two manifold realizations

$$(M,{\tau }_X)\cong S^n,(M,{\tau }_Y)\cong S^p\times S^q.$$

Whenever one verifies that

$$H_{(}L{S}^n;\mathbb{k})\neg \cong H_{(}L(S^p\times S^q);\mathbb{k})$$

as graded vector spaces, Theorem 13.3 gives

$${\mathcal{B}}_{S^n}\neg \cong {\mathcal{B}}_{S^p\times S^q}.$$

This provides a systematic source of examples rather than relying on an arbitrary pair of topologies.

10.10. A Stronger BV-Level Obstruction

The previous arguments use only the underlying graded vector spaces. One can obtain stronger obstructions directly from the algebraic operations.

Suppose

$${\mathcal{B}}_i=({\mathbb{H}}_{*}^{\left(i\right)},{★}_i,{\mathsf{\Delta }}_i).$$

$${\mathsf{\Delta }}_2\mathsf{\Phi }.$$

Consequently, the following are invariants under BV-algebra isomorphism:

• the graded dimensions of $\left({\mathbb{H}}_{*}^{\left(i\right)}\right)$;
• the dimensions of the kernels and images of $\mathsf{\Delta }$;
• the homology of the complex $\left({\mathbb{H}}_{,}\mathsf{\Delta }\right)$;
• the induced Gerstenhaber algebra;
• the ranks of multiplication maps $L_a⟼a★b$;
• the nilpotence properties of homogeneous elements;
• the dimensions of graded centers and annihilators.

Thus, if for example

$${\dim }_{\mathbb{k}}\ker \left({\mathsf{\Delta }}_1\right|{\mathbb{H}}_r^{\left(1\right)})\ne \dim \mathbb{k}\ker \left({\mathsf{\Delta }}_2{|}_{{\mathbb{H}}_r^{\left(2\right)}}\right)$$

for some degree r, then

$${\mathcal{B}}_1\neg \cong {\mathcal{B}}_2.$$

Similarly, if the Gerstenhaber brackets satisfy

$${\dim }_{\mathbb{k}}\mathrm{im}\left(-,-1:{\mathbb{H}}_p^{\left(1\right)}\otimes {\mathbb{H}}_q^{\left(1\right)}⟶\mathbb{H}{p+q-1}^{\left(1\right)}\right)$$

different from the corresponding dimension for ${\mathcal{B}}_2$, then the BV-algebras are non-isomorphic.

10.11. A Rigidity Versus Detection Principle

The discussion can be summarized in the following theorem.

Theorem 10.7

(Rigidity–detection principle). Let $\mathcal{M}=(M,{\tau }_1,{\tau }_2)$ be a bitopological manifold with closed oriented smooth realizations $M_1$ and $M_2$ of the same dimension.

Then:

1.

If $M_1$ and $M_2$ are orientation-preserving homotopy equivalent, their string topology BV-algebras are isomorphic.

2.

If their free loop homologies have different graded dimensions, their string topology BV-algebras are non-isomorphic.

3.

More generally, if any invariant preserved by BV-algebra isomorphisms takes different values for the two loop-space constructions, then the two BV-algebras are non-isomorphic.

Proof. 

Part (1) follows from Theorem 13.3.

Part (2) follows from the fact that a BV-algebra isomorphism is in particular an isomorphism of graded vector spaces.

Part (3) follows directly from the definition of an isomorphism of BV-algebras. □

10.12. Interpretation for Bitopological String Topology

The preceding results suggest that the correct question is not simply whether ${\tau }_1\ne {\tau }_2$. Rather, one should ask whether the two topologies determine different homotopy-theoretic string structures.

There are therefore three important levels of equivalence:

$$\mathrm{Same}\mathrm{topology}⟹\mathrm{Same}\mathrm{string}\mathrm{BV}-\mathrm{algebra}$$

and

$$\mathrm{Oriented}\mathrm{homotopy}\mathrm{equivalence}⟹\mathrm{Isomorphic}\mathrm{string}\mathrm{BV}-\mathrm{algebras},$$

while

$$\mathrm{Different}\mathrm{loop}-\mathrm{space}\mathrm{invariants}⟹\mathrm{non}-\mathrm{isomorphic}\mathrm{BV}-\mathrm{algebras}.$$

The first two implications are rigidity statements, whereas the third is a detection principle.

In particular, a genuinely nontrivial bitopological string topology theory should seek pairs $(M,{\tau }_1,{\tau }_2)$ for which the two manifold realizations are not orientation-preserving homotopy equivalent and for which the induced free loop homologies, products, or BV operators are algebraically distinguishable.

We have shown that distinct manifold topologies on the same underlying set can indeed give rise to non-isomorphic string topology BV-algebras, provided that the corresponding manifold realizations have distinguishable string-topological invariants.

At the same time, there is a strong rigidity phenomenon: an orientation-preserving homotopy equivalence between the two realizations forces the corresponding string topology BV-algebras to be isomorphic. Therefore, the inequality ${\tau }_1\ne {\tau }_2$ alone is not enough to produce different BV-algebra structures.

The transport construction provides a general method for producing examples. Given two closed oriented manifolds X and Y of the same dimension with

$${\mathbb{H}}_{(}LX;\mathbb{k})\neg \cong {\mathbb{H}}_{(}LY;\mathbb{k}),$$

one may transport their topologies to the same underlying set. This produces a bitopological manifold whose two associated string topology BV-algebras are non-isomorphic.

A particularly interesting direction for future work is to determine whether the full BV-algebra contains information strictly finer than the oriented homotopy type of the manifold, and to identify which additional string-topology operations can distinguish bitopologies that have the same ordinary loop homology algebra.

11. Weaker Conditions for Preservation of the Loop Product and BV Operator

The usual naturality statement in string topology is often formulated under the assumption that two closed oriented manifolds are related by an orientation-preserving diffeomorphism. This hypothesis is, however, considerably stronger than what is required for the basic Chas–Sullivan string topology operations. In particular, the loop product and the Batalin–Vilkovisky (BV) operator are invariant under suitable orientation-preserving homotopy equivalences. This substantially enlarges the class of maps under which the string topology algebra is functorial.

Let M and N be closed, connected, oriented smooth manifolds of the same dimension n. Denote their free loop spaces by

$$LM=\mathrm{Map}(S^1,M),LN=\mathrm{Map}(S^1,N).$$

The shifted loop homology is

$${\mathbb{H}}_{(}LM)=H_{+n}\left(LM\right),$$

and similarly for (N). The Chas–Sullivan loop product is a degree-zero operation

$${★}_M:{\mathbb{H}}_p\left(LM\right)\otimes {\mathbb{H}}_q\left(LM\right)⟶{\mathbb{H}}_{p+q}\left(LM\right).$$

Equivalently, in the unshifted grading it has degree (-n).

11.1. Orientation-Preserving Homotopy Equivalences

Suppose that Suppose that

$$f:M⟶N$$

is a homotopy equivalence. It induces a map on free loop spaces

$$Lf:LM⟶LN,Lf\left(\gamma \right)=f\circ \gamma .$$

The important observation is that the construction of (Lf) does not require (f) to be differentiable or invertible. The additional issue is whether the map induced by (Lf) on homology respects the intersection-theoretic part of the loop product.

$$\left(Lf\right)*\left(a\right){★}_N\left(Lf\right)\left(b\right)$$

for all

$$a,b\in {\mathbb{H}}_{(}LM).$$

Consequently, an orientation-preserving homotopy equivalence is already sufficient for preservation of the Chas–Sullivan loop product [3].

This gives the implication

$$\begin{array}{c}\mathrm{Orientation}-\mathrm{preserving}\mathrm{diffeomorphism}⟹\mathrm{orientation}-\mathrm{preserving}\mathrm{homotopy}\mathrm{equivalence}⟹\\ \mathrm{preservation}\mathrm{of}\mathrm{the}\mathrm{loop}\mathrm{product}.\end{array}$$

The first implication is immediate, whereas the second expresses the homotopy invariance theorem for the string topology product [3].

11.2. Why the Orientation Condition Is Important

The orientation hypothesis cannot simply be discarded when the loop product is defined with untwisted integral coefficients. The loop product is constructed using the intersection theory of the diagonal

$${\mathsf{\Delta }}_M⟶M\times M,x⟼(x,x).$$

The Thom class of the diagonal and the fundamental class of (M) enter the definition. Therefore, the construction depends on the chosen orientation.

If $f_{[}M]=\left[N\right]$, then f identifies the relevant fundamental classes and hence the intersection-theoretic ingredients of the product. If instead $f_{[}M]=-\left[N\right]$, the orientation data are reversed. In such a situation one should not automatically expect the untwisted integral string topology product to be preserved in exactly the same form.

Thus the natural hypothesis is not necessarily smoothness or invertibility of (f), but rather compatibility of (f) with the orientation data used to define the intersection product.

11.3. The BV Operator Requires No Additional Smoothness

The BV operator is obtained from the circle action on the free loop space. In homology, it can be written schematically as

$$\mathsf{\Delta }\left(a\right)={\rho }_{*}\left(\left[S^1\right]\times a\right),$$

with the appropriate degree conventions.

Now observe that

$$Lf\left(\rho (t,\gamma )\right)=f\circ \rho (t,\gamma )=\rho (t,f\circ \gamma )=\rho \left(t,Lf\left(\gamma \right)\right).$$

Hence the following diagram commutes:

$$\begin{array}{ccc}{S}^1\times LM& \stackrel{{\rho }_M}{\to }& LM\\ {\mathrm{id}}_{S^1}\times Lf\downarrow & & \downarrow Lf\\ S^1\times LN& \stackrel{{\rho }_N}{\to }& LN.\end{array}$$

Therefore, $Lf$ is $S^1$-equivariant. Passing to homology gives

$${\left(Lf\right)}_{*}\circ {\mathsf{\Delta }}_M={\mathsf{\Delta }}_N\circ {\left(Lf\right)}_{*}.$$

This part of the argument does not require f to be a diffeomorphism. Indeed, the $S^1$-action is defined purely by reparametrization of loops, and composition with any continuous map $f:M\to N$ commutes with this action. The substantive hypothesis for obtaining an isomorphism of BV algebras is therefore the homotopy-equivalence condition, together with the orientation compatibility needed for the loop product.

11.4. Main Theorem

The preceding discussion can be summarized in the following theorem.

Theorem 11.1

(Homotopy invariance of the BV algebra).

Let M and N be closed, connected, oriented smooth n-manifolds, and let

$$f:M⟶N$$

be an orientation-preserving homotopy equivalence.

The induced map

$$Lf:LM⟶LN$$

induces an isomorphism

$${\left(Lf\right)}_{*}:{\mathbb{H}}_{*}\left(LM\right)\stackrel{\cong }{\to }{\mathbb{H}}_{*}\left(LN\right)$$

which preserves both the Chas–Sullivan loop product and the BV operator. Thus

$${\left(Lf\right)}_{*}\left(a{★}_{M}b\right)={\left(Lf\right)}_{*}\left(a\right){★}_N{\left(Lf\right)}_{*}\left(b\right)$$

and

$${\left(Lf\right)}_{*}{\mathsf{\Delta }}_M\left(a\right)={\mathsf{\Delta }}_N\left({\left(Lf\right)}_{*}\left(a\right)\right)$$

for all

$$a,b\in {\mathbb{H}}_{*}\left(LM\right).$$

Consequently,

$${\left(Lf\right)}_{*}:{\mathbb{H}}_{*}\left(LM\right)\stackrel{\cong }{\to }{\mathbb{H}}_{*}\left(LN\right)$$

is an isomorphism of BV algebras.

Proof. 

Since f is a homotopy equivalence, the induced map

$$Lf:LM⟶LN$$

is also a homotopy equivalence. The orientation-preserving condition ensures compatibility with the fundamental classes and the Thom classes associated with the diagonal embeddings. Therefore, the homotopy invariance of the Chas–Sullivan loop product gives

$${\left(Lf\right)}_{*}\left(a{★}_{M}b\right)={\left(Lf\right)}_{*}\left(a\right){★}_N{\left(Lf\right)}_{*}\left(b\right).$$

For the BV operator, the $S^1$-actions on the two free loop spaces satisfy

$$Lf\left({\rho }_M(t,\gamma )\right)={\rho }_N\left(t,Lf\left(\gamma \right)\right).$$

Thus $Lf$ is $S^1$-equivariant. Passing to homology and using the definition of the BV operator from the $S^1$-action gives

$${\left(Lf\right)}_{*}{\mathsf{\Delta }}_M\left(a\right)={\mathsf{\Delta }}_N\left({\left(Lf\right)}_{*}\left(a\right)\right).$$

Hence $Lf$ induces an isomorphism of BV algebras. □

11.5. A Weaker Formulation Using Oriented Homotopy Type

The preceding theorem shows that the BV algebra of a closed oriented manifold depends only on its oriented homotopy type, rather than on its particular smooth structure. In particular, if $M\simeq N$ through an orientation-preserving homotopy equivalence, then ${\mathbb{H}}_{(}LM)\cong {\mathbb{H}}_{(}LN)$ as BV algebras. This is substantially weaker than requiring $M\cong N$ by an orientation-preserving diffeomorphism. Thus the smooth structure of the manifold is not essential for the basic BV-algebra structure of loop homology.

It is useful to distinguish three levels of hypotheses:

$$\begin{array}{c}\mathrm{orientation}-\mathrm{preserving}\mathrm{diffeomorphism}\\ \Downarrow \\ \mathrm{orientation}-\mathrm{preserving}\mathrm{homotopy}\mathrm{equivalence}\\ \Downarrow \\ \mathrm{BV}-\mathrm{algebra}\mathrm{isomorphism}\mathrm{on}\mathrm{loop}\mathrm{homology}.\end{array}$$

The second implication is the substantive homotopy-invariance statement.

11.6. Degree-One Maps and the Loop Product

One can weaken the hypotheses even further if the objective is only the loop product. In the literature, a degree-one map

Suppose that

$$f:M⟶N$$

is a map between oriented manifolds. The degree-one condition is closely related to preservation of the fundamental class:

$$f_{*}\left[M\right]=\left[N\right].$$

For suitable hypotheses, degree-one homotopy equivalences already suffice for the preservation of the Chas–Sullivan product. In particular, the loop product is more homotopy-theoretically robust than many other string topology operations. A useful modern survey explicitly records that a degree-one homotopy equivalence induces an algebra isomorphism for the loop product, whereas the loop coproduct is not homotopy invariant in general [4].

This distinction is important. One should not conclude that every string topology operation is invariant under the same weak hypotheses. The homotopy-invariance behavior of the loop product and the BV operator is stronger than that of certain coproduct and higher operations.

11.7. Generalized Homology Theories

The result admits a more general formulation. Let $h_{*}$ be a multiplicative generalized homology theory for which the manifolds M and N are $h_{*}$-oriented. An $h_{*}$-orientation is represented by a fundamental class ${\left[M\right]}_h\in h_n\left(M\right)$ giving the appropriate Poincaré duality isomorphism.

Suppose that $f:M⟶N$ is an $h_{*}$-orientation-preserving homotopy equivalence, meaning that $f_{*}{\left[M\right]}_h={\left[N\right]}_h$. Then the same argument gives an isomorphism ${\left(Lf\right)}_{*}:h_{*}\left(LM\right)⟶h_{*}\left(LN\right)$.

preserving the generalized loop product. The BV compatibility follows from the natural $\left(S^1\right)$-action on the free loop spaces. Cohen, Klein, and Sullivan formulated the homotopy-invariance theorem in precisely this generalized setting [3].

Consequently, the fundamental mechanism is not specifically tied to ordinary singular homology. What is required is a multiplicative homology theory equipped with the appropriate orientation and Poincar’e duality data.

11.8. A Categorical Interpretation

Let $\left({\mathbf{Man}}^{\mathrm{or}}n\right)$ denote the category of closed oriented n-manifolds and orientation-preserving maps. Let $\mathbf{BVAlg}$ denote the category of BV algebras. The assignment $M⟼\mathbb{H}*\left(LM\right)$ may be viewed, at the level of oriented homotopy types, as a functor

$$\mathcal{L}:\mathbf{Ho}\left({\mathbf{Man}}_n^{\mathrm{or}}\right)⟶\mathbf{BVAlg},$$

where $\left(\mathbf{Ho}\left({\mathbf{Man}}_n^{\mathrm{or}}\right)\right)$ denotes the appropriate oriented homotopy category.

For an orientation-preserving homotopy equivalence, let

$$f:M⟶N$$

be an orientation-preserving homotopy equivalence. The induced morphism is

$$\mathcal{L}\left(f\right)={\left(Lf\right)}_{*}.$$

is an isomorphism in $\left(\mathbf{BVAlg}\right)$. Thus the construction factors through oriented homotopy type rather than depending essentially on the smooth diffeomorphism type.

11.9. Limits of the Weakening

The weakening should nevertheless be formulated carefully. The condition “homotopy equivalence” by itself is not sufficient if the orientation data used in defining the integral loop product are not preserved. Moreover, the homotopy invariance of the loop product and BV operator should not be automatically extended to every operation in string topology.

For example, the loop coproduct has different homotopy-invariance behavior. There are examples of homotopy equivalent manifolds for which the loop coproduct is not preserved. Thus the statement

$$\mathrm{homotopy}\mathrm{invariance}\mathrm{of}\mathrm{the}\mathrm{loop}\mathrm{product}\nRightarrow \mathrm{homotopy}\mathrm{invariance}\mathrm{of}\mathrm{every}\mathrm{string}\mathrm{topology}\mathrm{operation}.$$

This distinction is essential when generalizing the present result to higher string topology operations.

The existence of an orientation-preserving diffeomorphism is therefore far from optimal for the basic Chas–Sullivan BV structure. The central replacement is

$$\begin{array}{c}f:M\to N\mathrm{is}\mathrm{an}\mathrm{orientation}-\mathrm{preserving}\mathrm{homotopy}\mathrm{equivalence}\\ ⟹\\ {\left(Lf\right)}_{*}:{\mathbb{H}}_{*}\left(LM\right)\stackrel{\cong }{\to }{\mathbb{H}}_{*}\left(LN\right)\\ \mathrm{is}\mathrm{a}\mathrm{BV}-\mathrm{algebra}\mathrm{isomorphism}.\end{array}$$

More generally, for a multiplicative generalized homology theory $h_{*}$, the appropriate condition is that f be an $h_{*}$-orientation-preserving homotopy equivalence. The loop product is controlled by homotopy-invariant intersection and Poincaré duality data, while the BV operator is controlled by the natural $S^1$-action on free loop spaces. Hence neither a smooth inverse nor an orientation-preserving diffeomorphism is required for the basic BV structure.

12. Two BV Operators Compatible with a Common Loop Product

Let (M) be a closed, oriented, smooth manifold of dimension (d), and let $LM=\mathrm{Map}(S^1,M)$ denote its free loop space. The Chas–Sullivan construction equips the shifted loop homology ${\mathbb{H}}_{(}LM):=H_{+d}\left(LM\right)$ with a graded-commutative associative product

$$\circ :{\mathbb{H}}_p\left(LM\right)\otimes {\mathbb{H}}_q\left(LM\right)⟶{\mathbb{H}}_{p+q}\left(LM\right).$$

The product is obtained from intersection theory together with concatenation of loops. The Cohen–Jones construction gives a homotopy-theoretic realization of this product and of the associated BV structure. [1,2]

$${\rho }_{*}\left(\left[S^1\right]\times \alpha \right),$$

and, after the standard degree shift, $\mathsf{\Delta }$ has degree (1). The operator satisfies ${\mathsf{\Delta }}^2=0$ and is compatible with the loop product in the sense that it is a second-order differential operator with respect to ∘. Consequently, it determines the loop Gerstenhaber bracket. [1,2]

The purpose of this section is to investigate what happens if the same loop-product algebra admits two different BV operators ${\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2$.

12.1. Common Loop-Product Algebra

Let $A={\mathbb{H}}_{*}\left(LM\right)$ and write the loop product simply as juxtaposition: $xy=x\circ y$.

We assume throughout that (A) is a graded-commutative algebra:

$$xy={(-1)}^{\left|x\right|\left|y\right|}yx.$$

A degree-one linear operator

$$\mathsf{\Delta }⟶A,\left|\mathsf{\Delta }\right|=1,$$

is called a BV operator for $\left(\right(A,\circ \left)\right)$ if the following conditions hold:

1.

$${\mathsf{\Delta }}^2=0;$$

2.

$\mathsf{\Delta }$ is a differential operator of order at most two with respect to the product;

3.

the bracket defined by

$${x,y}_{\mathsf{\Delta }}={(-1)}^{\left|x\right|}\left[\mathsf{\Delta }\left(xy\right)-\left(\mathsf{\Delta }x\right)y-{(-1)}^{\left|x\right|}x\mathsf{\Delta }y\right]$$

satisfies the Gerstenhaber identities.

Equivalently, one may formulate the second-order condition through the seven-term identity. With an appropriate sign convention, this identity is

$$\begin{array}{cc}\hfill \mathsf{\Delta }\left(xyz\right)=& \mathsf{\Delta }\left(xy\right)z+{(-1)}^{\left|x\right|}x\mathsf{\Delta }\left(yz\right)+{(-1)}^{\left(\right|x|+1)\left|y\right|}y\mathsf{\Delta }\left(xz\right)\hfill \\ & -\mathsf{\Delta }\left(x\right)yz-{(-1)}^{\left|x\right|}x\mathsf{\Delta }\left(y\right)z-{(-1)}^{\left|x\right|+\left|y\right|}xy\mathsf{\Delta }\left(z\right).\hfill \end{array}$$

The precise signs depend on the grading convention, but the essential point is that the failure of $\mathsf{\Delta }$ to be a derivation is itself a biderivation, giving the Gerstenhaber bracket. A standard characterization of BV algebras is precisely that a Gerstenhaber algebra is equipped with a square-zero degree-one operator generating its bracket. [7,9]

12.2. Definition of a Double BV Structure

We now introduce the central definition.

Definition 12.1.

Let $\left(\right(A,\circ \left)\right)$ be a graded-commutative algebra. Adouble BV structureon (A) is a triple

$$(A,\circ ,{\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2)$$

such that each

$${\mathsf{\Delta }}_i⟶A,i=1,2,$$

is a degree-one BV operator for the same product ∘. Thus

$${\mathsf{\Delta }}_1^2={\mathsf{\Delta }}_2^2=0,$$

and both operators are second order with respect to ∘.

The operators are called distinct if

$${\mathsf{\Delta }}_1\ne {\mathsf{\Delta }}_2$$

as linear operators on (A).

The existence of two BV operators does not imply that they determine the same Gerstenhaber bracket. We therefore distinguish several cases.

12.3. The Two Induced Gerstenhaber Brackets

Each BV operator determines a bracket

$$-,-1:=-,-{\mathsf{\Delta }}_1$$

and

$$-,-2:=-,-{\mathsf{\Delta }}_2.$$

Consequently, the same graded-commutative algebra may carry

$$(A,\circ ,{-,-}_1)$$

and

$$(A,\circ ,{-,-}_2)$$

as two possibly different Gerstenhaber algebras.

This leads to the first classification.

Definition 12.2.

A double BV algebra

$$(A,\circ ,{\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2)$$

is called:

1.

bracket-distinctif

$${-,-}_1\ne {-,-}_2;$$

2.

bracket-compatibleif

$${-,-}_1={-,-}_2;$$

3.

strongly compatibleif, in addition,

$${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0.$$

The distinction between bracket compatibility and strong compatibility is important. Equality of the induced brackets only controls the second-order symbols of the two operators; it does not by itself imply that the operators anticommute.

12.4. The Difference of Two BV Operators

Suppose that

$${\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2$$

are two BV operators on the same algebra A. Define

$$D={\mathsf{\Delta }}_2-{\mathsf{\Delta }}_1.$$

Then (D) has degree one.

A fundamental observation is that if the two BV operators induce the same Gerstenhaber bracket, then D is a derivation of the product. More precisely, suppose that

$${-,-}_{{\mathsf{\Delta }}_2}.$$

Then

$$D\left(x\right)y+{(-1)}^{\left|x\right|}xD\left(y\right)$$

for all homogeneous elements $(x,y\in A)$, up to the chosen sign convention for the BV generator. Thus we obtain the following proposition.

Proposition 12.3.

Let $\left((A,\circ ,{\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2)\right)$ be a double BV algebra. Suppose that the two BV operators induce the same Gerstenhaber bracket, that is,

$${-,-}_{{\mathsf{\Delta }}_2}.$$

Then

$$D={\mathsf{\Delta }}_2-{\mathsf{\Delta }}_1$$

is a degree-one derivation of (A). In particular, for all homogeneous $(x,y\in A)$,

$$D\left(x\right)y+{(-1)}^{\left|x\right|}xD\left(y\right).$$

Proof. 

$${-,-}_{{\mathsf{\Delta }}_1}{-,-}_{{\mathsf{\Delta }}_2-{\mathsf{\Delta }}_1}{-,-}_D.$$

If the two brackets coincide, then

$${-,-}_D=0.$$

The vanishing of the bracket generated by (D) means precisely that the second-order part of (D) vanishes. Hence (D) is a differential operator of order at most one, i.e. a derivation of the graded-commutative algebra (A). □

This gives a useful classification principle:

$$\begin{array}{|c|}\hline {\mathsf{\Delta }}_2={\mathsf{\Delta }}_1+D\\ \hline\end{array}$$

where D is a degree-one derivation, subject to the condition that ${\mathsf{\Delta }}_2^2=0$.

12.5. The Square-Zero Condition

$${({\mathsf{\Delta }}_1+D)}^2{\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_{1}D+D{\mathsf{\Delta }}_1+D^2.$$

Since

$${\mathsf{\Delta }}_1^2=0,$$

the condition

$${\mathsf{\Delta }}_2^2=0$$

is equivalent to

$$\begin{array}{|c|}\hline {\mathsf{\Delta }}_{1}D+D{\mathsf{\Delta }}_1+D^2=0.\\ \hline\end{array}$$

Therefore, two BV operators generating the same Gerstenhaber bracket are classified by degree-one derivations (D) satisfying this Maurer–Cartan-type equation.

A particularly important special case occurs when $D^2=0$ and ${\mathsf{\Delta }}_{1}D+D{\mathsf{\Delta }}_1=0$. Then ${\mathsf{\Delta }}_2={\mathsf{\Delta }}_1+D$ is automatically square-zero.

Proposition 12.4.

Let $(A,\circ ,{\mathsf{\Delta }}_1)$ be a BV algebra, and let

$$D:A⟶A$$

be a degree-one derivation satisfying

$$D^2=0.$$

and

$${\mathsf{\Delta }}_{1}D+D{\mathsf{\Delta }}_1=0.$$

Then

$${\mathsf{\Delta }}_2={\mathsf{\Delta }}_1+D$$

defines another BV operator for the same product. Moreover,

$${-,-}_{{\mathsf{\Delta }}_1}.$$

Proof. 

Since D is a derivation, adding (D) to the second-order operator ${\mathsf{\Delta }}_1$ does not change its second-order symbol. Therefore,

$${-,-}_{{\mathsf{\Delta }}_1}.$$

Furthermore,

$$\begin{array}{cccc}\hfill {\mathsf{\Delta }}_2^2& ={({\mathsf{\Delta }}_1+D)}^2\hfill & \hfill ={\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_{1}D+D{\mathsf{\Delta }}_1+D^2& =0.\hfill \end{array}$$

Here we have used

$${\mathsf{\Delta }}_1^2=0,{\mathsf{\Delta }}_{1}D+D{\mathsf{\Delta }}_1=0,D^2=0.$$

Thus ${\mathsf{\Delta }}_2$ is a square-zero second-order operator generating the same Gerstenhaber bracket as ${\mathsf{\Delta }}_1$. Hence $((A,\circ ,{\mathsf{\Delta }}_2)$) is a BV algebra with the same underlying product and Gerstenhaber bracket as $\left((A,\circ ,{\mathsf{\Delta }}_1)\right)$. □

12.6. Compatible Pairs and Linear Combinations

A stronger structure arises when the two BV operators anticommute:

$${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0.$$

In this situation, every linear combination

$${\mathsf{\Delta }}_{a,b}=a{\mathsf{\Delta }}_1+b{\mathsf{\Delta }}_2$$

satisfies

$${\mathsf{\Delta }}_{a,b}^2=0.$$

Indeed,

$$\begin{array}{cccc}\hfill {\mathsf{\Delta }}_{a,b}^2& ={(a{\mathsf{\Delta }}_1+b{\mathsf{\Delta }}_2)}^2\hfill & \hfill =a^2{\mathsf{\Delta }}_1^2+ab({\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1)+b^2{\mathsf{\Delta }}_2^2& =0.\hfill \end{array}$$

Here we have used

$${\mathsf{\Delta }}_1^2={\mathsf{\Delta }}_2^2=0$$

and

$${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0.$$

Since the space of second-order differential operators is closed under linear combinations, every

$${\mathsf{\Delta }}_{a,b}$$

is again a second-order operator. Consequently, every nonzero linear combination ${\mathsf{\Delta }}_{a,b}$ defines a BV operator for the common product ∘.

The corresponding Gerstenhaber bracket is given by

$$a{-,-}_1+b{-,-}_2.$$

Thus a strongly compatible pair naturally produces a pencil of BV structures parametrized by the projective line

$$\left[a\right]\in {\mathbb{P}}^1.$$

Definition 12.5.

Let $\left((A,\circ ,{\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2)\right)$ be a double BV algebra such that

$${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0.$$

The family

$${\mathsf{\Delta }}_{a,b}=a{\mathsf{\Delta }}_1+b{\mathsf{\Delta }}_2,\left[a\right]\in {\mathbb{P}}^1,$$

is called aBV pencil.

For each $(\left[a\right]\in {\mathbb{P}}^1)$, the operator ${\mathsf{\Delta }}_{a,b}$ is a degree-one square-zero second-order differential operator, and hence determines a BV algebra structure

$$(A,\circ ,{\mathsf{\Delta }}_{a,b}).$$

12.7. Four Algebraic Types

The preceding discussion leads to four useful types of double BV structures.

1.

Independent BV structure.

The operators

$${\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2$$

generate different Gerstenhaber brackets and satisfy no additional compatibility relation. The structure is simply

$$(A,\circ ,{\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2).$$

2.

Common-bracket double BV structure.

The operators are distinct but

$${\{-,-\}}_1={\{-,-\}}_2.$$

In this case

$${\mathsf{\Delta }}_2-{\mathsf{\Delta }}_1$$

is a derivation.

3.

Compatible double BV structure.

The operators satisfy

$${\mathsf{\Delta }}_1^2={\mathsf{\Delta }}_2^2=0,{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0.$$

The pair behaves like two anticommuting differentials.

4.

BV-pencil structure.

A compatible pair gives a whole family

$${\mathsf{\Delta }}_{a,b}=a{\mathsf{\Delta }}_1+b{\mathsf{\Delta }}_2$$

of BV operators. The induced Gerstenhaber brackets form the corresponding pencil

$${\{-,-\}}_{a,b}=a{\{-,-\}}_1+b{\{-,-\}}_2.$$

12.8. Geometric Interpretation on Loop Spaces

$${\rho }_{*}\left(\left[S^1\right]\times \alpha \right).$$

This is the standard Chas–Sullivan BV operator. [1,2]

$${\left({\rho }_2\right)}_{*}\left(\left[S^1\right]\times \alpha \right).$$

However, for the ordinary free loop space of a manifold there is a crucial restriction: the standard construction naturally supplies one canonical rotation action. Therefore, the mere fact that (LM) carries a loop product does not automatically produce a second independent geometric BV operator.

A second operator would generally require additional structure, for example:

• a second circle action;
• an additional symmetry of the loop space;
• a second cyclic or framed-operadic structure;
• an auxiliary geometric operation on loops;
• a generalized or equivariant string-topology construction;
• or a modified setting such as stacks, orbifolds, or bitopological loop spaces.

The operadic description is particularly relevant here. Cohen and Jones showed that the BV structure can be realized through an action of an operad related to framed little disks, providing a homotopy-theoretic explanation of the compatibility between the loop product and the BV operator. [2]

Thus two geometric BV operators should not merely be regarded as two arbitrary linear maps: one should ask whether they arise from genuinely distinct circle actions or from two compatible operadic structures.

12.9. Two Commuting Circle Actions

Suppose that (LM) carries an action of

$$T^2=S^1\times S^1$$

given by

$${\rho }^2\times LM⟶LM.$$

Let

$$S_1^1,S_2^1\subset T^2$$

be the two circle factors. They induce operators

$${\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2⟶A$$

of degree one.

At the chain or equivariant-homological level, the two circle directions behave as odd operators. Under the appropriate compatibility assumptions, one obtains

$${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0.$$

Therefore, a $T^2$-action is a natural geometric source of a compatible pair of BV operators.

If both operators are compatible with the same loop product, the resulting structure is naturally interpreted as a

$$\mathrm{bi}-\mathrm{BV}\mathrm{algebra}$$

or, when all linear combinations remain BV, a

$$\mathrm{BV}-\mathrm{pencil}\mathrm{algebra}.$$

12.10. A Model Example

$${(-1)}^{|a_2\left|\right|b_1|}\left(a_1{\circ }_1{b}_1\right)\otimes \left(a_2{\circ }_2{b}_2\right).$$

Define

$${\mathsf{\Delta }}_1={\delta }_1\otimes 1,{\mathsf{\Delta }}_2=1\otimes {\delta }_2.$$

With the standard graded tensor-product convention, these operators satisfy

$${\mathsf{\Delta }}_1^2=0,{\mathsf{\Delta }}_2^2=0,$$

and

$${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0.$$

Hence

$${\mathsf{\Delta }}_{a,b}=a{\mathsf{\Delta }}_1+b{\mathsf{\Delta }}_2$$

is a BV operator for every (a,b).

This example shows algebraically that two distinct BV operators can coexist naturally on one product algebra.

12.11. Relation with the Loop Product

For a string-topology algebra, the relevant structure can therefore be written schematically as

$$\left({\mathbb{H}}_{*}\left(LM\right),\circ ,{\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2\right).$$

$$\left({\mathsf{\Delta }}_{i}x\right)y$$

$${(-1)}^{\left|x\right|}x{\mathsf{\Delta }}_{i}y$$

encodes the (i)-th Gerstenhaber bracket.

Thus the common loop product can coexist with two distinct infinitesimal symmetries:

$${\mathsf{\Delta }}_1\mathrm{and}{\mathsf{\Delta }}_2.$$

There are then two associated brackets

$${-,-}_1,{-,-}_2.$$

If these brackets coincide, the two operators differ only by a derivation. If they differ, the same loop-product algebra carries two genuinely different Gerstenhaber structures.

12.12. Classification Theorem

The preceding observations can be summarized in the following theorem.

Theorem 12.6.

Let

$$(A,\circ )$$

be a graded-commutative loop-product algebra and suppose that

$${\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2\to A$$

are degree-one BV operators. Then exactly the following alternatives describe their basic algebraic relationship:

1.

The induced Gerstenhaber brackets

$${-,-}_1,{-,-}_2$$

are different. The algebra carries two distinct Gerstenhaber structures compatible with the same product.

2.

The induced brackets coincide:

$${\{-,-\}}_1={\{-,-\}}_2.$$

Then

$$D={\mathsf{\Delta }}_2-{\mathsf{\Delta }}_1$$

is a degree-one derivation, and

$${\mathsf{\Delta }}_2={\mathsf{\Delta }}_1+D$$

with

$${\mathsf{\Delta }}_{1}D+D{\mathsf{\Delta }}_1+D^2=0.$$

3.

If, in addition,

$${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0,$$

then every linear combination

$$a{\mathsf{\Delta }}_1+b{\mathsf{\Delta }}_2$$

is a BV operator. Hence the pair determines a BV pencil.

4.

If two operators arise from two compatible $S^1$-actions on the underlying loop space, the resulting double BV structure has a geometric interpretation. In particular, a compatible $T^2$-action naturally provides a candidate for such a structure.

Proof. 

The first two alternatives follow from the linearity of the BV-derived bracket in the generating operator. If the brackets coincide, their difference is generated by

$$D={\mathsf{\Delta }}_2-{\mathsf{\Delta }}_1,$$

whose derived bracket vanishes; hence (D) is a derivation.

Since

$${\mathsf{\Delta }}_2={\mathsf{\Delta }}_1+D,$$

the equation

$${\mathsf{\Delta }}_2^2=0$$

is equivalent to

$${\mathsf{\Delta }}_{1}D+D{\mathsf{\Delta }}_1+D^2=0.$$

$$a^2{\mathsf{\Delta }}_1^2+ab({\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1)+b^2{\mathsf{\Delta }}_2^2=0.$$

Since a linear combination of second-order operators is again second order, every

$$a{\mathsf{\Delta }}_1+b{\mathsf{\Delta }}_2$$

is a BV operator. The geometric statement follows by applying the fundamental classes of the two circle factors to the corresponding actions. □

12.13. Consequences for Bitopological and Generalized String Topology

The double-BV phenomenon becomes particularly interesting when the loop space carries two natural structures. Suppose, for example, that a space (X) has two topologies ${\tau }_1,{\tau }_2$ and one considers a loop object associated with the resulting bitopological space. If the two structures generate two natural circle-type actions ${\rho }_1,{\rho }_2$, then their induced operators ${\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2$ may coexist on a common homology algebra.

The resulting object can be represented schematically by

$$\left({\mathbb{H}}_{*}\left(LX\right),\circ ,{\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2\right).$$

There are then several levels of structure:

$$\begin{array}{c}\mathrm{common}\mathrm{loop}\mathrm{product}\\ \Downarrow \\ \mathrm{two}\mathrm{BV}\mathrm{operators}\\ \Downarrow \\ \mathrm{two}\mathrm{Gerstenhaber}\mathrm{brackets}\\ \Downarrow \\ \mathrm{compatible}\mathrm{pair}\mathrm{or}\mathrm{BV}\mathrm{pencil}.\end{array}$$

$${\mathsf{\Delta }}_1+{\mathsf{\Delta }}_2,$$

which satisfies

$${\mathsf{\Delta }}_{\mathrm{tot}}^2=0.$$

This suggests a potentially useful notion of bitopological string topology: the two topological structures could contribute two independent unary operations while sharing one intersection/concatenation product.

The existence of two BV operators on a common loop-product homology algebra is algebraically consistent and leads to a meaningful hierarchy:

$$\begin{array}{|c|}\hline \mathrm{common}\mathrm{product}⟶\mathrm{double}\mathrm{BV}\mathrm{algebra}⟶\mathrm{bi}-\mathrm{BV}\mathrm{algebra}⟶\mathrm{BV}\mathrm{pencil}.\\ \hline\end{array}$$

The essential distinction is between the following situations:

$${\mathsf{\Delta }}_1\ne {\mathsf{\Delta }}_2,$$

$${-,-}_1\ne {-,-}_2,$$

and

$${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\mathsf{\Delta }}_2{\mathsf{\Delta }}_1=0.$$

These are independent conditions and should not be conflated.

For ordinary string topology, the canonical BV operator is supplied by the rotation of free loops. Thus a second BV operator on the same $H_{*}\left(LM\right)$ should normally be accompanied by additional geometric data rather than being assumed to exist automatically. The most natural geometric mechanism is an additional compatible circle action or a richer operadic/string-topology structure. The algebraic framework above nevertheless provides a precise classification of what can happen once two such operators are available. [2,?]

13. Chain-Level Comparison of String Topology Structures in a Bitopological Setting

Let (M) be a smooth manifold equipped with two topologies ${\tau }_1,{\tau }_2$, and write $M_{\mathrm{bi}}=(M,{\tau }_1,{\tau }_2)$.

We investigate how the two topologies determine different models of the free loop space and, consequently, different chain-level realizations of string topology. The central problem is to construct comparison maps between these chain complexes and to determine when such maps preserve the loop product, the BV operator, the Gerstenhaber bracket, and, more generally, the action of the relevant string-topology operad.

The motivation comes from the ordinary string-topology situation. If (M) is a closed oriented manifold, the free loop space $LM=\mathrm{Map}(S^1,M)$ carries the Chas–Sullivan string-topology operations. At homology level these include the loop product and the BV operator, while at chain level they can be organized by an action of a dg model of the framed little disks operad or by equivalent cactus-type operads. Chain-level BV constructions using de Rham chains with marked points provide an explicit realization of this principle [?]. Similarly, cactus-operad constructions give a chain-level realization of the BV structure [?,?].

13.1. Two Loop-Space Topologies

For (i=1,2), let $L_{i}M=\mathrm{Map}(S^1,(M,{\tau }_i))$ be the free loop space equipped with the compact-open topology induced by ${\tau }_i$. We emphasize that the underlying set $\mathrm{Map}(S^1,M)$ is the same for $L_{1}M$ and $L_{2}M$; only its topology is different.

Suppose first that ${\tau }_1\subseteq {\tau }_2$. Then the identity map

$$\mathrm{id}M:(M,{\tau }_2)⟶(M,{\tau }_1)$$

is continuous. By functoriality of the compact-open construction, this induces a continuous map

$$I21⟶L_{1}M,I_{21}\left(\gamma \right)=\gamma .$$

Hence, there is an induced map on singular chains

$$C_{(}{I}_{21}):C_{(}{L}_{2}M)⟶C_{(}{L}_{1}M).$$

We shall denote this map simply by

$${\mathsf{\Phi }}_{21}:C_{*}\left(L_{1}M\right)⟶C_{*}\left(L_{2}M\right).$$

If, in addition, ${\tau }_2\subseteq {\tau }_1$, then we obtain a map in the opposite direction

$${\mathsf{\Phi }}_{12}:C_{*}^{\left(1\right)}\left(LM\right)⟶C_{*}^{\left(2\right)}\left(LM\right).$$

Thus, when the two topologies are equal, the two chain complexes coincide canonically, whereas when they are merely comparable, one obtains a canonical comparison map in the direction determined by the inclusion of topologies.

More generally, if $f:(M,{\tau }_1)⟶(M,{\tau }_2)$ is continuous, then composition induces

$$Lf⟶L_{2}M,Lf\left(\gamma \right)=f\circ \gamma ,$$

and hence

$$C_{*}\left(Lf\right):C_{*}^{\left(1\right)}\left(LM\right)⟶C_{*}^{\left(2\right)}\left(LM\right).$$

This formulation is preferable when the two topologies live on different underlying manifolds.

13.2. The Two Chain Complexes

Let

$$C_{*}^{\left(i\right)}\left(LM\right)=C_{*}^{\mathrm{sing}}(L_{i}M;R),i=1,2.$$

where (R) is a commutative coefficient ring. The boundary operators will be denoted by

$${\partial }_i^i\left(LM\right)⟶C_{n-1}^i\left(LM\right).$$

$${\mathsf{\Phi }}_{21}{\partial }_2.$$

Hence

$${\mathsf{\Phi }}_{21}:\left(C_{*}^2\left(LM\right),{\partial }_2\right)⟶\left(C_{*}^1\left(LM\right),{\partial }_1\right).$$

If the superscripts are labels rather than powers, I recommend using parentheses:

$${\mathsf{\Phi }}_{21}:\left(C_{*}^{\left(2\right)}\left(LM\right),{\partial }_2\right)⟶\left(C_{*}^{\left(1\right)}\left(LM\right),{\partial }_1\right).$$

The second version is clearer because $C_{*}^{\left(1\right)}$ and $C_{*}^{\left(2\right)}$ denote two different chain complexes rather than suggesting powers of $C_{*}$.

The induced homology map is

$$\left({\mathsf{\Phi }}_{21}\right)*:H(L_{2}M;R)⟶H_{(}{L}_{1}M;R).$$

If the identity maps in both directions are continuous, then the two loop-space topologies are identical. More generally, if the two identity maps are only homotopy equivalences, then the corresponding chain complexes are chain-homotopy equivalent. This distinction is important: equality of chain-level operations should not be expected merely from the existence of a homology isomorphism.

13.3. Evaluation Maps and the Comparison Problem

The compatibility of the evaluation maps with the induced map on free loop spaces is expressed by

$$f\left(\gamma \left(0\right)\right)=f\left({\mathrm{ev}}_0\left(\gamma \right)\right)={\mathrm{ev}}_0\left(\left(Lf\right)\left(\gamma \right)\right),$$

and similarly,

$$f\left({\mathrm{ev}}_1\left(\gamma \right)\right)={\mathrm{ev}}_1\left(\left(Lf\right)\left(\gamma \right)\right).$$

This compatibility is essential. The loop product is defined geometrically by imposing the matching condition

$${\gamma }_1\left(0\right)={\gamma }_2\left(0\right),$$

which is encoded by the fiber product

$$L_{i}M{\times }_M{L}_{i}M.$$

Therefore, a comparison map can preserve the loop product only if it is compatible with the corresponding fiber-product constructions. At chain level, one therefore requires a diagram of the form

$$\begin{array}{ccccccccc}{C}_{(}{L}_{1}M)\otimes C_{(}{L}_{1}M)& \stackrel{{\mu }_1}{\to }& C_{-d}\left(L_{1}M\right)\mathsf{\Phi }\otimes \mathsf{\Phi }\downarrow \phantom{\mathsf{\Phi }\otimes \mathsf{\Phi }}& & \phantom{\mathsf{\Phi }}\downarrow \mathsf{\Phi }& & C_{(}{L}_{2}M)\otimes C_{(}{L}_{2}M)& \stackrel{{\mu }_2}{\to }& C_{-d}\left(L_{2}M\right),\end{array}$$

where $d=\dim M$ and ${\mu }_i$ denotes a chosen chain-level model of the loop product.

$$\mathsf{\Phi }\circ {\mu }_2={\mu }_2(\mathsf{\Phi }\otimes \mathsf{\Phi }).$$

If this equality holds strictly, $\mathsf{\Phi }$ is a morphism of chain-level loop-product algebras. If it holds only up to coherent chain homotopy, then $\mathsf{\Phi }$ should instead be regarded as a morphism of $\left(A_{\infty }\right)$- or $E_2$-algebras.

13.4. A Chain-Level Model Using an Operad

Let $\left(\mathcal{O}\right)$ be a dg operad modeling the framed little (2)-disks operad. Alternatively, one may use a dg cactus operad or another quasi-isomorphic chain-level model of the same homotopy type. The use of such an operad is natural because the string-topology operations form a BV-type structure and the framed little disks operad governs BV algebras at the homological level [?,?].

For each $(i=1,2)$, suppose that

$${\theta }_i:\mathcal{O}⟶\mathrm{End}{C}^i\left(LM\right)$$

is an operad action. Thus, for every $r\ge 0$ and every

$$o\in \mathcal{O}\left(r\right),$$

we have a chain operation

$${\theta }_i\left(o\right):C_{*}^{\left(i\right)}{\left(LM\right)}^{\otimes r}⟶C_{*}^{\left(i\right)}\left(LM\right).$$

We obtain two chain-level string-topology algebras

$$\left(C_{*}^{\left(1\right)}\left(LM\right),{\theta }_1\right),\left(C_{*}^{\left(2\right)}\left(LM\right),{\theta }_2\right).$$

$${\theta }_2\left(o\right)\circ {\mathsf{\Phi }}^{\otimes r}.$$

for every

$$o\in \mathcal{O}\left(r\right).$$

Equivalently, the following diagram commutes:

$$\begin{array}{ccc}\mathcal{O}\left(r\right)\otimes C_{*}^{\left(1\right)}{\left(LM\right)}^{\otimes r}& \stackrel{{\theta }_1}{\to }& C_{*}^{\left(1\right)}\left(LM\right)\\ 1\otimes {\mathsf{\Phi }}^{\otimes r}\downarrow \phantom{1\otimes {\mathsf{\Phi }}^{\otimes r}}& & \phantom{\mathsf{\Phi }}\downarrow \mathsf{\Phi }& & \\ \mathcal{O}\left(r\right)\otimes C_{*}^{\left(2\right)}{\left(LM\right)}^{\otimes r}& \stackrel{{\theta }_2}{\to }& C_{*}^{\left(2\right)}\left(LM\right).\end{array}$$

This condition provides the basic definition of a bitopological string-topology comparison map.

$${\theta }_2\left(o\right){\mathsf{\Phi }}^{\otimes r}$$

for every $o\in \mathcal{O}\left(r\right)$.

Definition 13.1.

If the equality is replaced by a coherent family of chain homotopies, we call Φ ahomotopy $\left(\mathcal{O}\right)$-compatible comparison map.

13.5. The BV Operator

The $S^1$-action on the free loop space is given by

$${\rho }^1\times LM⟶LM,\rho (t,\gamma )\left(s\right)=\gamma (s+t).$$

This action is independent of the choice of topology on M at the set-theoretic level, but its continuity must be checked with respect to the chosen loop-space topology.

For each (i), suppose that

$${\rho }_i^1\times L_{i}M⟶L_{i}M$$

is continuous. Then the fundamental class

$$\left[S^1\right]\in H_1(S^1;R)$$

determines a degree-one operator

$${\mathsf{\Delta }}_i:C_{*}^{\left(i\right)}\left(LM\right)⟶C_{*+1}^{\left(i\right)}\left(LM\right).$$

in a suitable chain model. At the homology level, this gives the usual BV operator

$${\mathsf{\Delta }}_2\mathsf{\Phi }.$$

A sufficient geometric condition is that the diagram

$$\begin{array}{ccc}{S}^1\times L_{1}M& \stackrel{1\times \mathsf{\Phi }}{\to }& S^1\times L_{2}M\\ {\rho }_1\downarrow \phantom{{\rho }_1}& & \phantom{{\rho }_2}\downarrow {\rho }_2& & \\ L_{1}M& \stackrel{\mathsf{\Phi }}{\to }& L_{2}M\end{array}$$

commutes.

For an identity comparison map this commutativity is automatic at the level of underlying maps. The nontrivial requirement is that all four maps be continuous and that the selected chain models realize the circle action functorially.

13.6. Preservation of the Gerstenhaber Bracket

Recall that the Gerstenhaber bracket induced by the BV operator ${\mathsf{\Delta }}_i$ is given, for homogeneous elements $x,y$, by

$${\{x,y\}}_i={(-1)}^{\left|x\right|}\left({\mathsf{\Delta }}_i\left(xy\right)-\left({\mathsf{\Delta }}_{i}x\right)y-{(-1)}^{\left|x\right|}x{\mathsf{\Delta }}_i\left(y\right)\right),$$

up to the convention chosen for the grading and signs. Suppose that $\mathsf{\Phi }$ preserves both the loop product and the BV operator. Then

$$\mathsf{\Phi }\left({\{x,y\}}_1\right)={\{\mathsf{\Phi }\left(x\right),\mathsf{\Phi }\left(y\right)\}}_2.$$

Thus, preservation of the product and BV operator automatically implies preservation of the induced Gerstenhaber bracket. Consequently, it is not necessary to impose the bracket condition independently when the entire framed-operad action is already preserved.

13.7. Geometric Conditions for Operadic Compatibility

The preceding discussion leads to a useful hierarchy of sufficient conditions.

Proposition 13.2.

Let (M) be a closed oriented smooth manifold equipped with two topologies ${\tau }_1$ and ${\tau }_2$. Suppose that:

1.

the identity map

$${\mathrm{id}}_M:(M,{\tau }_1)⟶(M,{\tau }_2)$$

is continuous;

2.

the induced map

$$L\mathrm{id}⟶L_{2}M$$

is continuous;

3.

the evaluation maps

$${\mathrm{ev}}_i⟶(M,{\tau }_i)$$

are continuous and the comparison map commutes with evaluation;

4.

the circle actions

$$S^1\times L_{i}M⟶L_{i}M$$

are continuous and the comparison map is $S^1$-equivariant;

5.

the chain-level intersection or fiber-product constructions used to define the loop product are natural with respect to the comparison map;

6.

the orientation or R-orientation of M is preserved by the comparison map;

7.

the selected chain-level operad actions

$${\theta }_i:\mathcal{O}⟶\mathrm{End}C{*}^i\left(LM\right)$$

are constructed functorially with respect to these maps.

Then the induced chain map

$$\mathsf{\Phi }:C_{*}^{\left(1\right)}\left(LM\right)⟶C_{*}^{\left(2\right)}\left(LM\right).$$

is a morphism of $\left(\mathcal{O}\right)$-algebras. In particular, it preserves the loop product, the BV operator, and the induced Gerstenhaber bracket.

Proof. 

Condition (1) gives a continuous map of the underlying manifolds and hence, by composition, a continuous map of free loop spaces. Therefore (2) induces a chain map

$$\mathsf{\Phi }:C_{*}^{\left(1\right)}\left(LM\right)⟶C_{*}^{\left(2\right)}\left(LM\right).$$

$${\mu }_2(\mathsf{\Phi }\otimes \mathsf{\Phi }).$$

$${\mathsf{\Delta }}_2\mathsf{\Phi }.$$

$${\theta }_2\left(o\right){\mathsf{\Phi }}^{\otimes r}.$$

Thus $\mathsf{\Phi }$ is an $\left(\mathcal{O}\right)$-algebra morphism.

The preservation of the Gerstenhaber bracket follows from the fact that it is determined by the product and BV operator. □

13.8. A Weaker Homotopy-Invariant Version

Strict preservation is often stronger than necessary. In many chain-level constructions the operations are defined only up to coherent homotopy. In this situation the appropriate notion is a homotopy-coherent comparison.

Suppose that

$$\mathsf{\Phi }:C_{*}^{\left(1\right)}\left(LM\right)⟶C_{*}^{\left(2\right)}\left(LM\right)$$

satisfies

$$\mathsf{\Phi }{\theta }_1\left(o\right)\simeq {\theta }_2\left(o\right){\mathsf{\Phi }}^{\otimes r}$$

by a coherent family of chain homotopies. Then $\mathsf{\Phi }$ defines a morphism in the homotopy category of $\left(\mathcal{O}\right)$-algebras.

If, in addition, $\mathsf{\Phi }$ is a quasi-isomorphism, the two string-topology chain models determine equivalent $\left(\mathcal{O}\right)$-algebras up to homotopy. In particular,

$$H_{(}\mathsf{\Phi }):H_{(}{L}_{1}M;R)⟶H_{*}(L_{2}M;R)$$

is an isomorphism of the induced BV algebras.

This distinction is important in the bitopological setting. The two topologies need not produce literally identical chain-level operations in order to produce the same string-topology algebra up to homotopy.

13.9. The Role of Comparable Topologies

The simplest situation occurs when

$${\tau }_1\subseteq {\tau }_2.$$

In this case,

$$\mathrm{id}:(M,{\tau }_2)⟶(M,{\tau }_1)$$

is continuous, and hence

$$L_{2}M⟶L_{1}M$$

is continuous. We therefore obtain

$${\mathsf{\Phi }}_{21}:C_{*}^{\left(2\right)}\left(LM\right)⟶C_{*}^{\left(1\right)}\left(LM\right).$$

If the inclusion

$${\tau }_1\subseteq {\tau }_2$$

is a homotopy equivalence at the level of the corresponding spaces, then

$${\mathsf{\Phi }}_{21}$$

is a quasi-isomorphism. If the evaluation, concatenation, circle-action, and operadic constructions are compatible, the resulting quasi-isomorphism is an equivalence of string-topology algebras.

If, on the other hand, the topologies are not comparable, there need not be a canonical identity-induced comparison map. In that situation one may introduce a third topology

$${\tau }_{\vee }={\tau }_1\vee {\tau }_2$$

or the common coarsening

$${\tau }_{\wedge }={\tau }_1\cap {\tau }_2$$

and construct a diagram

$$(M,{\tau }_{\vee })⟶(M,{\tau }_i)⟶(M,{\tau }_{\wedge }).$$

The corresponding loop spaces and chain complexes can then be compared through the resulting zigzag.

13.10. A Two-Colored Operadic Formulation

The preceding construction can be organized more conceptually using a two-colored dg operad. Let the two colors be denoted by

$$1,2,$$

corresponding to the two topologies. Define a colored operad

$${\mathcal{O}}_{\mathrm{bi}}$$

whose pure operations of color (i) are copies of $\left(\mathcal{O}\right)$, while mixed-color operations encode the comparison maps.

$${\theta }_2\left(o\right)\circ u_{12}^{\otimes r}$$

then expresses precisely the requirement that the comparison map preserve the operation represented by $\left(o\right)$.

A representation of this colored operad consists of chain complexes

$$C_{*}^{\left(1\right)}\left(LM\right),C_{*}^{\left(2\right)}\left(LM\right).$$

together with maps

$${\mathsf{\Phi }}_{12}:C_{*}^{\left(1\right)}\left(LM\right)⟶C_{*}^{\left(2\right)}\left(LM\right),{\mathsf{\Phi }}_{21}:C_{*}^{\left(2\right)}\left(LM\right)⟶C_{*}^{\left(1\right)}\left(LM\right).$$

and the two operad actions

$${\theta }_1,{\theta }_2.$$

This formulation has an important advantage: it does not require the two string-topology structures to be literally identical. It records both structures and the comparison maps between them as part of one algebraic object.

13.11. Chain-Level Theorem

The preceding discussion can be summarized in the following statement.

Theorem 13.3

(Bitopological chain-level comparison theorem). Let

$$M_{\mathrm{bi}}=(M,{\tau }_1,{\tau }_2)$$

be a closed oriented smooth manifold equipped with two topologies. Assume that the corresponding free loop spaces $L_{1}M$ and $L_{2}M$ admit chain-level actions of a $dg$ operad $\left(\mathcal{O}\right)$ modeling the framed little (2)-disks operad.

Suppose that

$$F:(M,{\tau }_1)⟶(M,{\tau }_2)$$

is continuous and that the induced map

$$LF⟶L_{2}M$$

is compatible with evaluation, loop concatenation, the $S^1)$-action, the orientation class, and the chosen chain-level models.

If $LF$ is a homotopy equivalence and the compatibility with the operadic actions holds up to coherent homotopy, then Φ is a quasi-isomorphism of homotopy $\left(\mathcal{O}\right)$-algebras. Hence

$$H_{(}{L}_{1}M;R)\cong H_{(}{L}_{2}M;R)$$

as BV algebras and, consequently, as Gerstenhaber algebras.

Proof. 

The continuity of F gives the continuous map $LF$ and hence the chain map

$$\mathsf{\Phi }=C_{*}\left(LF\right).$$

Compatibility with evaluation implies that the correspondence between loops is compatible with the fiber products used in the definition of the loop product. Compatibility with orientation guarantees that the corresponding Thom or intersection maps have the same degree shift and orientation convention. In particular, the induced Gerstenhaber brackets satisfy

$$\mathsf{\Phi }\left({\{x,y\}}_1\right)={\{\mathsf{\Phi }\left(x\right),\mathsf{\Phi }\left(y\right)\}}_2.$$

More generally, if $\mathcal{O}$ is an operad acting on the relevant chain complexes and ${\theta }_i$ denotes the corresponding operadic action, then

$$\mathsf{\Phi }\circ {\theta }_1\left(o\right)={\theta }_2\left(o\right)\circ {\mathsf{\Phi }}^{\otimes r}$$

for every $o\in \mathcal{O}\left(r\right)$. Hence $\mathsf{\Phi }$ is an $\mathcal{O}$-algebra morphism.

If the compatibility is only up to coherent homotopy, the same argument takes place in the homotopy category of $\left(\mathcal{O}\right)$-algebras. If $LF$ is a homotopy equivalence, the induced chain map is a quasi-isomorphism, and therefore the corresponding homology algebras are isomorphic as BV, and hence Gerstenhaber, algebras. □

13.12. Interpretation for Bitopological String Topology

Observe that

$$\left(C_{*}^{\left(1\right)}\left(LM\right),C_{*}^{\left(2\right)}\left(LM\right),{\mathsf{\Phi }}_{12},{\mathsf{\Phi }}_{21}\right),$$

together with the two operadic actions ${\theta }_1$, ${\theta }_2,$ encodes the two chain-level string-topology structures associated with the two topologies. Thus, a bitopological manifold carries two potentially different string-topology structures and comparison morphisms between them. In the strongest situation, ${\tau }_1={\tau }_2$, the construction reduces to ordinary string topology. If the topologies are different but induce equivalent loop-space homotopy types, the two theories may nevertheless determine equivalent BV algebras. If they induce genuinely different homotopy types, the two string-topology algebras need not be equivalent, and the comparison maps become additional invariants of the bitopological structure.

This gives a precise way to distinguish three levels of comparison:

$$\begin{array}{|c|}\hline \mathrm{continuous}\mathrm{comparison};⟹;\mathrm{chain}\mathrm{comparison};⟹;\mathrm{operadic}\mathrm{comparison}.\\ \hline\end{array}$$

The first requires continuity, the second requires functoriality of the chain construction, and the third requires compatibility with evaluation, intersection/concatenation, the $S^1$-action, orientations, and the complete operadic action.

Accordingly, a natural research program is to determine which classes of bitopological manifolds admit canonical $\left(\mathcal{O}\right)$-compatible comparison maps and to identify conditions under which the resulting maps are quasi-isomorphisms. This would provide a chain-level foundation for a genuinely bitopological version of string topology.

14. Conclusion

We have introduced a framework for studying BV-algebras associated with bitopological spaces. The basic object consists of two BV-algebras together with comparison maps preserving their products and BV operators.

This formulation yields immediate algebraic results, including compatibility of the associated Gerstenhaber brackets and recovery of the ordinary BV-algebra in the diagonal case.

The principal geometric problem is to construct such pairs from spaces equipped with two distinct topologies. In string topology, this requires compatible loop-space homology theories, circle actions, intersection products, and comparison maps. The equal-topology case recovers the classical Chas–Sullivan structure, while the genuinely bitopological case remains a setting for further investigation.