Geometric topology · higher signatures · surgery theory · group-ring L-theory

Novikov Conjecture

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Are the higher signatures of an oriented manifold invariant under oriented homotopy equivalence for every discrete fundamental group?

\ker A_{π, n} = 0 (π any discrete group, n \in ℤ)

Known results and sources

The Novikov conjecture asks whether higher signatures survive every oriented homotopy equivalence, or equivalently whether the rational L-theory assembly map is injective.

Research problem

Exact mathematical statement

For every discrete group $\pi$ and every integer $n$ , the rational quadratic $L$ -theory assembly map $A_{\pi,n}$ is injective. In the notation used by the source packet, the exact assertion is

\ker A_{\pi,n}=0 \qquad \text{for every discrete }\pi\text{ and every }n\in\mathbb Z.

Geometrically, this is the Novikov assertion that the higher signatures obtained from the Hirzebruch $L$ -class and cohomology classes on $B\pi$ are invariant under oriented homotopy equivalence. the source reports a package-by-package reduction to a reduced Cappell boundary map, but it marks the general conjecture as unsolved.

Problem infographic

Problem at a glance

A scientific explainer compares oriented homotopy-equivalent manifolds, forms higher signatures from the L-class and a classifying map to B pi, and connects the invariance question to injectivity of the rational L-theory assembly map for arbitrary discrete groups, clearly marked open. — Higher signatures pair the manifold's L-class with cohomology pulled back from its fundamental group's classifying space; Novikov predicts that these numbers depend only on the oriented homotopy type.

Current mathematical picture

Where work on Novikov Conjecture stands

Partially resolved

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Main reductionAssembly kernel as a boundary image

Package by package, the assembly kernel is identified with the image of the reduced Cappell boundary from the acyclic amalgam D.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeConstruct and source-audit a representative-level reduced relative-fiber square with exact signs, models, and the full vertex-null-bordism choice action.Task status · Work already reported in progress

Work mapped so far

Novikov Conjecture in numbers

3.9kretained lines of mathematical investigation3,867 in the current working snapshot

Argument development: 3,284 · 85%
Explored or eliminated routes: 155 · 4%
Computational analysis: 33 · 1%
Open obligations: 102 · 3%
Definitions and setup: 293 · 8%

9selected mapped statements3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed bySergei P. NovikovProposed1970Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Prove rational vanishing of every edge class over the integral group ring rather than only metabolicity after scalar extension to Q[H].

Suggested move: Clear the initial denominator and construct an N-fold stable H-null-cobordism in the ultimate decoration over Z[H], allowing vertex-loop changes but not silently assuming a change-of-rings injection.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

12 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove rational vanishing of every edge class over the integral group ring rather than only metabolicity after scalar extension to Q[H].Suggested move: Clear the initial denominator and construct an N-fold stable H-null-cobordism in the ultimate decoration over Z[H], allowing vertex-loop changes but not silently assuming a change-of-rings injection.

Ready to work on

Supply the genuinely quadratic mechanism that kills the primary quotient class while surviving the folded-primitive and augmentation falsification tests.Suggested move: Derive the pairing and quadratic refinement from algebraic transversality, test candidate Lagrangians and quotient-level symmetries, and compute how all quadratic correction terms interact with the ordinary folded-primitive diagnostic.

Ready to work on

Construct and source-audit a representative-level reduced relative-fiber square with exact signs, models, and the full vertex-null-bordism choice action.Suggested move: Carry out Work Orders 0 and 1 in a nonconnective stable Poincare model, verify that the fiber projection is the Cappell boundary, and define the quotient class independently of representative choices.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusPartially resolved

The higher signatures are known to be homotopy invariant for many major classes of fundamental groups, including word-hyperbolic groups, groups of finite asymptotic dimension under stated finiteness hypotheses, a-T-menable groups, and countable linear groups. The conjecture for arbitrary discrete groups remains open; stronger Baum--Connes, Farrell--Jones, and Borel formulations are not silently counted as equivalent solutions.

[8][9]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2019
Authoritative summaryYu's survey reviewed the large body of affirmative cases while retaining the unrestricted higher-signature statement as a conjecture.[9]
2005
Peer reviewedGuentner, Higson, and Weinberger proved that countable linear groups uniformly embed into Hilbert space and deduced the Novikov conjecture for every countable subgroup of GL(n,K); the paper's stronger Baum--Connes conclusion is confined to its stated rank-two setting.[7]
2003
Peer reviewedKasparov and Skandalis proved the conjecture for discrete groups acting properly and isometrically on bolic weakly geodesic bounded-geometry spaces.[6]
2001
Peer reviewedHigson and Kasparov proved Baum--Connes for a-T-menable groups; rational injectivity of that analytic assembly map gives the Novikov conclusion for the corresponding discrete groups.[5][8]

Mathematical neighborhood

Related results and reusable starting points

Current focusNovikov conjecture

Equivalent formulationRational L-theory assembly injectivity

The higher-signature formulation is equivalent to rational injectivity of the appropriate surgery-theoretic L-theory assembly map. This equivalence is specific to the rational injectivity formulation, not to every stronger assembly conjecture.

[8][9]

Logical consequenceBaum--Connes assembly-map injectivity

Rational injectivity of the analytic Baum--Connes assembly map implies the Novikov higher-signature conclusion. The full Baum--Connes conjecture is stronger and is not an equivalent restatement here.

[5][10]

Logical consequenceFarrell--Jones conjecture in L-theory

Rational injectivity of the L-theoretic Farrell--Jones assembly map implies the Novikov conjecture; full Farrell--Jones contains additional information and is not identified with Novikov.

[10]

Solved special caseWord-hyperbolic groups

The higher signatures are homotopy invariant when the fundamental group is word-hyperbolic.

[3]

Solved special caseGroups of finite asymptotic dimension

The conjecture holds for finitely generated groups of finite asymptotic dimension satisfying Yu's finite-CW classifying-space hypothesis.

[4]

Solved special casea-T-menable groups

The analytic assembly theorem for a-T-menable groups yields the Novikov conclusion for discrete groups admitting a proper affine isometric action on Hilbert space.

[5]

Solved special caseCountable linear groups

The conjecture holds for countable linear groups over arbitrary fields through their uniform Hilbert-space embedding theorem.

[7]

Related problemBorel rigidity conjecture

Borel-type rigidity concerns homotopy rigidity of aspherical manifolds and is closely linked through surgery theory, but it is a stronger rigidity program rather than an alias of higher-signature invariance.

[8][9]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA proof-assistant treatment of closed oriented smooth manifolds, fundamental classes, rational cohomology pairings, and orientation-preserving homotopy equivalence at the required level.
Formalization targetFormal classifying spaces for discrete groups, maps to BΓ, group cohomology, and the naturality needed to define higher signatures.
Formalization targetFormal Hirzebruch L-classes and their relation to rational Pontryagin classes and the signature theorem.
Formalization targetFormal surgery spectra, L-theory, and assembly maps sufficient to state the rational injectivity formulation.
Formalization targetFor analytic routes, formal operator K-theory, equivariant KK-theory, group C*-algebras, and the analytic assembly map.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma3 of 93
negative result2 of 92

Selected mathematical clusters2 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementAre higher signatures invariant under oriented homotopy equivalence for every discrete fundamental group?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementRational assembly injectivityintermediate
retained route statementAssembly kernel as a boundary imageintermediate
retained route statementFolded Primitive Isomorphismintermediate
retained route statementPrimary quotient criterionintermediate
retained route statementDo not kill the full relative classintermediate
retained route statementRelative Fiber Image Theoremintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetConstruct and source-audit a representative-level reduced relative-fiber square with exact signs, models, and the full vertex-null-bordism choice action.in progress reported
Research targetProve rational vanishing of every edge class over the integral group ring rather than only metabolicity after scalar extension to Q[H].open
Research targetSupply the genuinely quadratic mechanism that kills the primary quotient class while surviving the folded-primitive and augmentation falsification tests.open

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeConstruct and source-audit a representative-level reduced relative-fiber square with exact signs, models, and the full vertex-null-bordism choice action.

The current research map records this as an open mathematical step.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve rational vanishing of every edge class over the integral group ring rather than only metabolicity after scalar extension to Q[H].

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Research contextPrepared context for any AI agent

Are the higher signatures of an oriented manifold invariant under oriented homotopy equivalence for every discrete fundamental group?

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Sources and references12 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Algebraic construction and properties of Hermitian analogs of K-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. Ioriginal source · Sergei P. Novikov · Mathematics of the USSR-Izvestiya · 1970 · DOI 10.1070/IM1970v004n02ABEH000903 · accessed Aug 7, 2026
2
Algebraic construction and properties of Hermitian analogs of K-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. IIoriginal source · Sergei P. Novikov · Mathematics of the USSR-Izvestiya · 1970 · DOI 10.1070/IM1970v004n03ABEH000916 · accessed Aug 7, 2026
3
Cyclic cohomology, the Novikov conjecture and hyperbolic groupspeer reviewed result · Alain Connes, Henri Moscovici · Topology · 1990 · DOI 10.1016/0040-9383(90)90003-3 · accessed Aug 7, 2026
4
The Novikov conjecture for groups with finite asymptotic dimensionpeer reviewed result · Guoliang Yu · Annals of Mathematics · 1998 · DOI 10.2307/121011 · accessed Aug 7, 2026
5
E-theory and KK-theory for groups which act properly and isometrically on Hilbert spacepeer reviewed result · Nigel Higson, Gennadi Kasparov · Inventiones Mathematicae · 2001 · DOI 10.1007/s002220000118 · accessed Aug 7, 2026
6
Groups acting properly on 'bolic' spaces and the Novikov conjecturepeer reviewed result · Gennadi Kasparov, Georges Skandalis · Annals of Mathematics · 2003 · DOI 10.4007/annals.2003.158.165 · accessed Aug 7, 2026
7
The Novikov conjecture for linear groupspeer reviewed result · Erik Guentner, Nigel Higson, Shmuel Weinberger · Publications Mathématiques de l'IHÉS · 2005 · DOI 10.1007/s10240-005-0030-5 · accessed Aug 7, 2026
8
Novikov's Conjecturesurvey or monograph · Jonathan Rosenberg · Springer · 2016 · ARXIV 1506.05408 · DOI 10.1007/978-3-319-32162-2_11 · accessed Aug 7, 2026
9
The Novikov conjecturesurvey or monograph · Guoliang Yu · Russian Mathematical Surveys · 2019 · DOI 10.1070/RM9882 · accessed Aug 7, 2026
10
L-theory of C*-algebraspeer reviewed result · Markus Land · Proceedings of the London Mathematical Society · 2023 · DOI 10.1112/plms.12564 · accessed Aug 7, 2026
11
Mathlib documentation indexformalization · Lean community · accessed Aug 7, 2026
12
Formal Conjecturesformalization · Google DeepMind · GitHub · accessed Aug 7, 2026

Important qualifications

The proved-class list is representative rather than exhaustive; it does not catalogue every coarse-geometric, group-action, or assembly-map theorem related to Novikov.
Finiteness and action hypotheses are retained where material. No cited theorem is broadened beyond its stated group class.
Rational injectivity of Baum--Connes or Farrell--Jones assembly implies the higher-signature conjecture, but those full conjectures are not treated as equivalent aliases of Novikov.
The scoped recognition review found no current Clay Millennium, Hilbert, Smale, Erdős, FrontierMath/Epoch, or comparable named-prize designation for this exact conjecture. This is not a global nonexistence claim.
The scoped formalization search checked current public Mathlib documentation and Google's Formal Conjectures repository and found no problem-level machine-checked statement or proof of the full conjecture. This does not establish nonexistence in every proof assistant or private project.
No canonical external computation, dataset, or certificate was identified for the general higher-signature statement.
No unreviewed source material, submitted mathematical claim, contributor estimate, attachment, or packet computation was inspected or used as external authority. This record has no proof, novelty, review, acceptance, visibility, publication, or deployment authority.

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