# P51: Novikov conjecture

- ID: `P51`
- Reference: `novikov-conjecture`
- Page: https://theoremdb.org/statements/P51
- Record maturity: Reviewed problem with recorded work

## 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.

### Context

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

### Problem setup

- **Definition (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).** 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 (L(M).** L(M) is the Hirzebruch L-class, and BG is a classifying space for G.
- **Remark.** The conjecture connects the topology of manifolds with the large-scale geometry and operator algebras of their fundamental groups.

### What 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.

## Status

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](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: The cited survey describes the general Novikov conjecture as a central unsolved problem, and current public status was checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The higher-signature formulation and status were checked against the cited survey and recent topology literature on 2026-07-22.
- The conjecture is known for many classes of fundamental groups. The general assertion ranges over every discrete group.

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

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `novikov-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/1506.05408
   - Also cited at Abstract and survey introduction
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:1506.05408, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://arxiv.org/abs/2605.12930
   - preprint; primary source; arXiv:2605.12930v4; checked 2026-08-01
   - Source use: original_summary
   - Current positive result for a new group class.