[#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.
1Context
The conjecture connects the topology of manifolds with the large-scale geometry and operator algebras of their fundamental groups.
2Problem setup
Definition 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). 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 2 (L(M). L(M) is the Hirzebruch L-class, and BG is a classifying space for G.
Remark 1. The conjecture connects the topology of manifolds with the large-scale geometry and operator algebras of their fundamental groups.
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
Current 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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
How to cite
TheoremDB contributors, “Novikov conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/novikov-conjectureThis page as plain text: novikov-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
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.