[#P51] Novikov conjecture

Problem. For every discrete group \(\Gamma\), the higher signatures associated with classes in \(H^*(B\Gamma;\mathbb{Q})\) are invariant under oriented homotopy equivalences of closed manifolds.

Agent access

Work in ChatGPT

The agent reads this problem's research brief and asks you before it saves anything to TheoremDB.

Definitions and notation

1Context

The conjecture connects the topology of manifolds with the large-scale geometry and operator algebras of their fundamental groups.

2Problem setup

Remark 1. For a reference map f from M to BG and a rational cohomology class x on BG, the corresponding higher signature pairs f*(x) cup L(M) with the fundamental class of M.

Definition 1 (L(M). L(M) is the Hirzebruch L-class, and BG is a classifying space for G.

3What counts as a solution

Prove homotopy invariance of all higher signatures for every discrete group, or construct an orientation-preserving homotopy equivalence and a classifying-space cohomology class for which the corresponding higher signatures differ.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Assembly-map injectivity, hence the Novikov conclusion, is known for many group classes. Tian and Yu add groups with finite-complexity coarse embeddings into the universal Banach space. Exact unresolved remainder: Prove homotopy invariance of all higher signatures for arbitrary discrete fundamental groups.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Assembly-map injectivity, hence the Novikov conclusion, is known for many group classes. Tian and Yu add groups with finite-complexity coarse embeddings into the universal Banach space. Exact unresolved remainder: Prove homotopy invariance of all higher signatures for arbitrary discrete fundamental groups.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Assembly-map injectivity, hence the Novikov conclusion, is known for many group classes. Tian and Yu add groups with finite-complexity coarse embeddings into the universal Banach space. Exact unresolved remainder: Prove homotopy invariance of all higher signatures for arbitrary discrete fundamental groups.^[1][2] Relevance to this problem For Novikov conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Assembly-map injectivity, hence the Novikov conclusion, is known for many group classes. Tian and Yu add groups with finite-complexity coarse embeddings into the universal Banach space. Exact unresolved remainder: Prove homotopy invariance of all higher signatures for arbitrary discrete fundamental groups. Open full record →
Prove homotopy invariance of all higher signatures for every discrete group, or construct an orientation-preserving homotopy equivalence and a classifying-space cohomology class for which the corresponding higher signatures differ.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove homotopy invariance of all higher signatures for every discrete group, or construct an orientation-preserving homotopy equivalence and a classifying-space cohomology class for which the corresponding higher signatures differ.^[1] Relevance to this problem For Novikov conjecture, record novikov-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove homotopy invariance of all higher signatures for every discrete group, or construct an orientation-preserving homotopy equivalence and a classifying-space cohomology class for which the corresponding higher signatures differ. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove homotopy invariance of all higher signatures for every discrete group, or construct an orientation-preserving homotopy equivalence and a classifying-space cohomology class for which the corresponding higher signatures differ. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
No records match these filters.

Notes and companion material

Original intake status. The cited survey describes the general Novikov conjecture as a central unsolved problem, and current public status was checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

The conjecture is known for many classes of fundamental groups. The general assertion ranges over every discrete group.

Recorded example 1. For the trivial cohomology class, the higher signature reduces to the ordinary signature, whose homotopy invariance is known.

2See also

Torsion-freeness for king-grid independence complexestopology
Integral torsion in scale-four hypercube Rips complexestopology
Hodge conjecturetopology

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Novikov conjecture.” TheoremDB. P51. Problem statement; statement text SHA-256 d0ff26277a3e2d37c57e8568e4a729b5c0e8b07f42135021e171911778231fca. https://theoremdb.org/statement/?ref=P51

BibTeX

@misc{theoremdb-problem-d0ff26277a3e2d37c57e8568e4a729b5c0e8b07f42135021e171911778231fca,
  title = {{Novikov conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 d0ff26277a3e2d37c57e8568e4a729b5c0e8b07f42135021e171911778231fca},
  url = {https://theoremdb.org/statement/?ref=P51}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Jonathan Rosenberg, “Novikov's Conjecture”. "Open Problems in Mathematics", J. F. Nash, Jr., and M. Th. Rassias, eds, Springer, 2016, pp. 377-402. DOI 10.1007/978-3-319-32162-2. arXiv:1506.05408 (2015). Jonathan Rosenberg, arXiv:1506.05408, abstract and survey formulations. ↗preprint · primary source · arXiv:1506.05408, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited survey describes the general Novikov conjecture as a central unsolved problem, and current public status was checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Abstract and survey introduction.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Packet-linked general-status survey.Source named by the research packet.
Geng Tian and Guoliang Yu, “Embedding complexity into the universal Banach space and the strong Novikov conjecture”. arXiv:2605.12930 (2026). Abstract, revision dated 2026-07-23. ↗preprint · primary source · arXiv:2605.12930v4 · checked 2026-08-01Source use: original summary.Current positive result for a new group class.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P51] Novikov conjecture

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance