Rational Analytic Novikov Conjecture with Coefficients

Updated 21 January 2026

The rational analytic Novikov conjecture with coefficients extends the classical conjecture by rationalizing analytic assembly maps to secure homotopy invariance of higher signatures.
It employs tools like twisted Roe algebras, Mayer–Vietoris sequences, and Bott periodicity to guarantee permanence under various group extensions and coefficient systems.
The framework has practical implications for both manifolds and singular spaces, linking operator K-theory with geometric invariants such as positive scalar curvature control.

The rational analytic Novikov conjecture with coefficients is an extension of the Novikov conjecture, formulated in the language of higher index theory, assembly maps in operator $K$ -theory, and topological invariants of manifolds and singular spaces. This version rationalizes the assembly map and incorporates arbitrary coefficients in a $G$ - $C^*$ -algebra or more general Banach algebras, yielding powerful permanence and extension properties. The conjecture underpins the study of homotopy invariance of higher signatures, provides control over positive scalar curvature on spin manifolds, and connects deeply with group-theoretic properties such as coarse embeddability, amenability, and group extensions.

1. Analytic Assembly Maps and Formulation with Coefficients

For a countable discrete group $G$ and a $G$ – $C^*$ –algebra $B$ , the equivariant $K$ -homology with coefficients is defined as

$K^G_*(\underline E G;B)=\varinjlim_{\Delta\subset \underline{E}G\ \text{%%%%8%%%%–cocompact}}\ KK^G_*\big(C_0(\Delta),B\big)$

where $\underline{E}G$ is the universal proper $G$ 0–space and $G$ 1 denotes Kasparov’s equivariant $G$ 2-theory. The corresponding reduced crossed product algebra is $G$ 3.

The analytic assembly map with coefficients,

$G$ 4

is constructed using a sequence of steps involving Roe algebras, boundary maps in exact sequences, and the passage to inductive limits over finite-propagation representatives.

The rational analytic Novikov conjecture with coefficients asserts that the tensor product

$G$ 5

is injective for all $G$ 6– $G$ 7–algebras $G$ 8, i.e., higher signatures with coefficients are homotopy-invariant after rationalization (Deng, 2019, Zhang, 14 Jan 2026).

2. Extension Properties and Main Theorems

A distinguishing feature of the rational analytic Novikov conjecture with coefficients is its behavior under group extensions. For an exact sequence $G$ 9, if both $C^*$ 0 and $C^*$ 1 admit coarse embeddings into Hilbert space, then for any $C^*$ 2– $C^*$ 3–algebra $C^*$ 4, the analytic assembly map is rationally injective, i.e.,

$C^*$ 5

(Deng, 2019). This result extends to cases where $C^*$ 6 satisfies the rational Baum–Connes conjecture (with or without coefficients) and $C^*$ 7 satisfies the rational analytic Novikov conjecture with coefficients in a cross-sectional algebra built from $C^*$ 8 and $C^*$ 9 (Zhang, 14 Jan 2026).

The following table summarizes critical group-theoretic closure properties:

Extension Type	Sufficient Hypotheses	Preservation of RANC
Direct product $G$ 0	$G$ 1, $G$ 2 satisfy RANC with $G$ 3 (Künneth holds)	Yes
Central extension	Central subgroup $G$ 4 is a-T-menable	Yes
Extension by finite group	$G$ 5 satisfies rational BC with $G$ 6	Yes
Non-coarse-embeddable extension	$G$ 7 in BCC, $G$ 8-type group	Yes (by permanence)

This permanence, verified through Mayer–Vietoris arguments and the use of Green’s imprimitivity theorem, substantiates the robustness of the rational version over a wide class of groups (Zhang, 14 Jan 2026).

3. Technical Approach: Assembly Maps, Localization, and Bott Periodicity

The technical construction proceeds via several advanced tools:

Twisted Roe and Localization Algebras: For coarse-geometric contexts (e.g., where group actions admit compactifications or proper transformation groupoids), one passes to twisted Roe algebras $G$ 9 and their localization counterparts.
Evaluation-at-zero Isomorphism: The key isomorphism in $G$ 0-theory (the twisted Baum–Connes theorem) is established by demonstrating that the evaluation-at-zero map $G$ 1 from the twisted localization algebra to the twisted Roe algebra is an isomorphism in $G$ 2-theory for models built from the coarse embedding data (Deng, 2019, Guo et al., 2024).
Geometric Bott Map: Utilizing the Higson–Kasparov–Trout infinite-dimensional Bott periodicity, a geometric Bott map is constructed to relate the $G$ 3-theory of ordinary and twisted algebras, inducing isomorphism after rationalization.
Reduction to Trivial Coefficients: In many cases, the inclusion $G$ 4 induces $G$ 5-theory isomorphisms, reducing analysis to the trivial-coefficient case via Eilenberg–Moore arguments.

For group extensions, direct-product reductions and Mayer–Vietoris sequences in the $G$ 6-theory of (equivariant) localization algebras, alongside Green’s imprimitivity, are consistently utilized to establish injectivity of the rationalized assembly map (Zhang, 14 Jan 2026).

4. Core Examples and Classes of Groups

The rational analytic Novikov conjecture with coefficients has been established for several classes:

Extensions of Coarsely Embeddable Groups: If both $G$ 7 and $G$ 8 embed coarsely into Hilbert space, then $G$ 9 satisfies the RANC with coefficients (Deng, 2019).
Central and Finite-Group Extensions: Central extensions (with abelian $C^*$ 0), extensions by finite groups, and direct products all preserve the rational injectivity of the assembly map, provided the relevant subgroups satisfy analogous properties (Zhang, 14 Jan 2026).
Non-Embeddable Targets: By combining permanence results with non-coarse-embeddable groups known to satisfy rational BC (e.g., Arzhantseva–Tessera groups), the conjecture is shown to hold in contexts not governed by geometric group-embeddability (Zhang, 14 Jan 2026).
Cheeger Spaces: In the context of singular stratified spaces where $C^*$ 1–de Rham theories are defined (Cheeger spaces), if the fundamental group satisfies the strong Novikov conjecture, the rational analytic Novikov conjecture (with coefficients in the Mishchenko bundle $C^*$ 2) guarantees stratified homotopy invariance of higher signatures (Albin et al., 2013).

The rational analytic Novikov conjecture with coefficients admits analogous formulations in Banach (e.g., $C^*$ 3 (Ogle, 2010)) and coarse assembly settings. For Banach algebraic versions, polynomially bounded cohomology is used to transfer arguments, showing rational injectivity for finitely presented discrete groups.

In the coarse category, for proper $C^*$ 4–spaces $C^*$ 5 with equivariant bounded geometry admitting equivariant coarse embeddings into admissible Hilbert–Hadamard spaces, the analytic equivariant coarse Novikov conjecture posits that the Mishchenko–Kasparov assembly map is rationally injective. Twisted Roe and localization algebras with nontrivial coefficients, together with Bott periodicity and Mayer–Vietoris decompositions, achieve the required rational injectivity under very general geometric circumstances (Guo et al., 2024).

The rationalized versions are essential for circumventing torsion phenomena and for making the use of Bott periodicity and $C^*$ 6-theoretical splitting arguments (depending crucially on the Künneth formula and the vanishing of all finite-group torsion upon tensoring with $C^*$ 7).

6. Implications, Applications, and Homotopy Invariance

Injectivity of the rational assembly map enforces the homotopy invariance of higher signatures, both in the manifold and Cheeger space settings. This homotopy invariance, for instance, ensures that for any closed manifold $C^*$ 8 with fundamental group $C^*$ 9 in the prescribed class (e.g., extension of coarsely embeddable groups), higher signatures of $B$ 0 are invariant under orientation-preserving homotopy equivalence (Deng, 2019, Albin et al., 2013). Similar implications hold for the existence of positive scalar curvature metrics on spin manifolds with the relevant fundamental group—through the Gromov–Lawson–Rosenberg framework.

In singular spaces, e.g., Cheeger spaces with self-dual mezzoperversities, rational injectivity of the assembly map with coefficients secures the invariance of the analytic signature class under stratified homotopy equivalences and cobordism, extending the classical manifold theory to stratified spaces (Albin et al., 2013). Here, twisting with the Mishchenko bundle and the analytic signature operators $B$ 1 establish a direct link between geometric invariants and operator $B$ 2-theory indices.

The general structure of permanence phenomena, extension properties, and the stability of the conjecture under various group-theoretic constructions—direct products, finite extensions, and central extensions—makes the rational analytic Novikov conjecture with coefficients a flexible and profound tool for the study of rigidity, higher index theory, and novelties in topological invariants.

References:

(Deng, 2019): "The Novikov conjecture and extensions of coarsely embeddable groups" (Zhang, 14 Jan 2026): "The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions" (Ogle, 2010): "Polynomially bounded cohomology and the Novikov Conjecture" (Guo et al., 2024): "Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture" (Albin et al., 2013): "The Novikov conjecture on Cheeger spaces"