Homotopy Non-Invariance of the String Cobracket and the Failure of the Lie Bialgebra Structure

Abstract: We prove that the string cobracket is not a homotopy invariant. Adapting Naef's method arXiv:2106.11307 for computing the string coproduct, we show that the string cobrackets on the three-dimensional lens spaces $L(9;1)$ and $L(9;4)$ differ. We further relate the string cobracket to the Whitehead torsion, analogously to the case of the string coproduct. In addition, we show that the string bracket and the string cobracket do not endow the $S<sup>1$-equivariant homology of the free loop space with a Lie bialgebra structure. These findings indicate that the analogy with the Turaev cobracket breaks down in higher-dimensional string topology.

• The paper shows that the string cobracket distinguishes homotopy-equivalent lens spaces by exhibiting different counts of nonzero components.
• It establishes that under homotopy equivalence, the transformation of the string cobracket is linked to Whitehead torsion via the Dennis trace map.
• It demonstrates that the string bracket and cobracket fail to satisfy Drinfeld’s compatibility, refuting a Lie bialgebra structure in this context.

Homotopy Non-Invariance of the String Cobracket and Incompatibility with Lie Bialgebra Structures

Background and Motivation

String topology investigates algebraic operations on the homology of free loop spaces $LM = \mathrm{Map}(S^1, M)$, inspired by the analogues with the topology of surfaces, particularly the Goldman bracket and Turaev cobracket. The core string topology operations are the string product and string coproduct, originally introduced by Chas and Sullivan, and their equivariant incarnations, the string bracket and string cobracket, acting on $S^1$-equivariant homology.

While the string product is a homotopy invariant for all oriented manifolds and the string coproduct has known invariance properties in simply connected and rational settings, computations by Naef have demonstrated non-invariance of the string coproduct for certain three-dimensional lens spaces. Analogues of the Turaev cobracket for higher-dimensional manifolds thus display increased structural subtlety. This paper examines whether the string cobracket shares this sensitivity to homotopy type and investigates its compatibility with Lie bialgebra structures on $H_*^{S^1}(LM)$.

Main Results

The paper presents three major results, each with strong implications for the structure of string topology in higher dimensions:

• The string cobracket is not a homotopy invariant: Explicit calculations for the lens spaces $L(9;1)$ and $L(9;4)$, which are homotopy equivalent but not homeomorphic, show that the number of components with nonzero string cobracket differ between the two. This demonstrates that, as with the string coproduct, the string cobracket can distinguish between non-homeomorphic, homotopy-equivalent 3-manifolds.
• Relationship to Whitehead torsion: The transformation of the string cobracket under a homotopy equivalence can be expressed via the Whitehead torsion, extending known relationships for the string coproduct. The string cobracket thus detects Whitehead torsion through the Dennis trace map, indicating a subtle sensitivity to simple homotopy type rather than just homotopy type.
• The string bracket and string cobracket do not form a Lie bialgebra on $S^1$-equivariant homology: For certain higher-dimensional manifolds with integer coefficients, the Drinfeld compatibility between the bracket and cobracket fails. This breaks the analogy with the surface case (Goldman bracket and Turaev cobracket) and the rational string topology of simply connected manifolds.

Technical Approach

Computation on Lens Spaces

The focus is on the three-dimensional lens spaces $L(9;1)$ and $L(9;4)$. Both possess fundamental group $\mathbb{Z}/9\mathbb{Z}$, leading to a decomposition of the free loop space $LM$ into nine path-connected components indexed by $S^1$0. Homological computations employ Leray–Serre spectral sequences for the $S^1$1-bundle

$S^1$2

along with fiberwise algebraic considerations. The explicit cycles and classes in homology are tracked through the spectral sequence differentials and their relations to $S^1$3-actions. For these non-prime cyclic lens spaces, richer torsion phenomena arise in equivariant homology, crucially affecting the structure and computation of the string cobracket.

Non-Invariance and Component Analysis

Assessing the string cobracket involves tracking nonzero contributions in each component of $S^1$4, specifically counting the $S^1$5 for which the cobracket does not vanish. The calculations reveal that this count is $S^1$6 for $S^1$7 and $S^1$8 for $S^1$9, certifying non-invariance under homotopy equivalence. This difference is robust under further algebraic projections given the coassociativity of the cobracket.

Whitehead Torsion and Transformation

For a homotopy equivalence $H_*^{S^1}(LM)$0, the paper derives a transformation formula:

$H_*^{S^1}(LM)$1

where $H_*^{S^1}(LM)$2 is the Whitehead torsion and $H_*^{S^1}(LM)$3 is its Dennis trace. The precise alteration of homology classes induced by torsion underlies the detected discrepancies between the two lens spaces. This explicitly connects algebraic invariance failures in string topology to classical simple-homotopy invariants.

Failure of Lie Bialgebra Structure

The string bracket $H_*^{S^1}(LM)$4 and string cobracket $H_*^{S^1}(LM)$5 fail to satisfy Drinfeld’s compatibility condition that defines a Lie bialgebra:

$H_*^{S^1}(LM)$6

Explicit counterexamples are given for $H_*^{S^1}(LM)$7, with specific elements in $H_*^{S^1}(LM)$8 violating the required relation. In particular, computations demonstrate cancellation on the left-hand side and nontriviality on the right-hand side of the compatibility equation. This non-Lie bialgebra behavior contrasts sharply with classical cases and marks a divergence between string topology in higher dimensions versus the surface case or the rational, simply-connected setting.

Implications and Future Directions

The demonstrated non-invariance of the string cobracket under homotopy equivalence, together with its ability to detect Whitehead torsion, reveals that string topological operations are highly sensitive probes of manifold structure, particularly in the presence of torsion and beyond simple homotopy equivalence. The failure of Lie bialgebra compatibility for the bracket/cobracket pair exposes deeper subtleties in the algebraic structure of $H_*^{S^1}(LM)$9-equivariant homologies of free loop spaces.

These results imply several key theoretical consequences:

• Refinement of string topology invariants: The string cobracket is not governed by homotopy or even simple-homotopy equivalence in general, making it a finer invariant than previously assumed for higher-dimensional or non-simply connected settings.
• Obstruction to quantization approaches relying on Lie bialgebraic structures: Quantization techniques invoking such algebraic structures must be revised for manifolds with torsion or complex fundamental groups.
• Sensitivity to Whitehead torsion: The connection to classical invariants such as Whitehead torsion opens the way for a unified perspective linking geometric topology and string-topological operations.

Future developments should investigate:

• The precise boundary between cases where Lie bialgebra structures are present or absent, depending on manifold dimension, fundamental group, or (co)homological coefficients.
• Characterizations of the full algebraic structure induced by string topology in the presence of torsion, potentially leading to new classes of topological invariants or finer classifications of high-dimensional manifolds.
• The role of these findings in string field theory and related physical models, especially where algebraic structures on free loop space homology enter in the enumeration of topological invariants.

Conclusion

This work establishes that the string cobracket in string topology is not a homotopy invariant, with explicit computational evidence derived from lens spaces with nontrivial torsion. The string cobracket encodes information about Whitehead torsion and can distinguish manifolds that are otherwise homotopy equivalent. Moreover, the incompatibility of the string bracket and cobracket with a Lie bialgebra structure for these manifolds signals important limits to algebraic analogies with the two-dimensional surface case and necessitates new theoretical frameworks for understanding string topology in higher dimensions (2605.10273).