[Paper Review] Invariant Hochschild cohomology of smooth functions

Lukas Miaskiwskyi|arXiv (Cornell University)|Aug 24, 2018

Homotopy and Cohomology in Algebraic Topology17 references4 citations

TL;DR

This paper investigates invariant Hochschild cohomology of smooth functions on a manifold under a Lie group action, distinguishing between invariants of cochains and invariants of cohomology classes. It proves that for proper group actions, these two notions coincide, establishing an invariant Hochschild-Kostant-Rosenberg theorem identifying invariant cohomology with invariant multivector fields.

ABSTRACT

Given an action of a Lie group on a smooth manifold, we discuss the induced action on the Hochschild cohomology of smooth functions, and notions of invariance on this space. Depending on whether one considers invariance of cochains or invariance of cohomology classes, two different spaces of invariants arise. We perform a general comparison of these notions, give an interpretation of the lower orders of the invariant cohomology spaces and conclude as our main result that for proper group actions both spaces are isomorphic. As a corollary and a geometric interpretation, an invariant version of the Hochschild-Kostant-Rosenberg theorem is given, identifying the cohomology of invariant cochains with invariant multivector fields. Using this theorem, we shortly discuss the invariant Hochschild cohomology in the case of homogeneous spaces.

Motivation & Objective

• To clarify the distinction between invariance of Hochschild cochains and invariance of Hochschild cohomology classes under a Lie group action.
• To investigate the relationship between the cohomology of invariant cochains and the space of invariant cohomology classes.
• To establish conditions under which these two spaces of invariants are isomorphic, particularly for proper group actions.
• To provide a geometric interpretation via an invariant version of the Hochschild-Kostant-Rosenberg theorem.
• To apply the results to homogeneous spaces and relate invariant cohomology to invariant multivector fields.

Proposed method

• Introduces two notions of invariance: invariance of cochains (via group action on cochain complex) and invariance of cohomology classes (via induced action on cohomology).
• Defines the natural morphism $\iota: \mathrm{HH}^{\bullet}_G(\mathscr{A}, \mathsf{N}) \to \mathrm{HH}^{\bullet}(\mathscr{A}, \mathsf{N})^G$ between the two spaces of invariants.
• Uses the existence of invariant partitions of unity for proper actions to construct an averaging operator on cochains, ensuring injectivity of $\iota$.
• Applies the classical Hochschild-Kostant-Rosenberg theorem to deduce surjectivity of $\iota$ without restrictions on the group action.
• Employs seminorm estimates and locally finite partitions of unity to prove continuity of the averaging map and locally finite sums of cochains.
• Derives a geometric interpretation by identifying $\mathrm{HH}^{\bullet}_G(C^\infty(M))$ with the cohomology of invariant multivector fields on $M$.

Experimental results

Research questions

• RQ1 Under what conditions do the space of invariant cochains and the space of invariant cohomology classes coincide?
• RQ2 How does the group action affect the Hochschild cohomology of smooth functions on a manifold?
• RQ3 What is the geometric meaning of the cohomology of invariant cochains in terms of differential geometry?
• RQ4 Can an invariant version of the Hochschild-Kostant-Rosenberg theorem be established for proper group actions?
• RQ5 How does the invariant cohomology relate to the geometry of homogeneous spaces?

Key findings

• For a proper Lie group action on a smooth manifold $M$, the natural map $\iota: \mathrm{HH}^{\bullet}_G(C^\infty(M)) \to \mathrm{HH}^{\bullet}(C^\infty(M))^{G}$ is an isomorphism.
• The isomorphism is established via an averaging operator on cochains, made possible by the existence of invariant partitions of unity under proper actions.
• The cohomology of invariant cochains is isomorphic to the cohomology of invariant multivector fields, yielding an invariant Hochschild-Kostant-Rosenberg theorem.
• Surjectivity of $\iota$ follows from the classical HKR theorem and does not require properness, while injectivity relies on properness.
• The continuity of the averaging map and locally finite sums of cochains is rigorously established using seminorm estimates and compact exhaustion techniques.
• The results provide a finite-dimensional classification of invariant deformations in deformation quantization for symmetric phase spaces.