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Immersions of manifolds
Open accessAbstract
Immersions of an m-manifold in an n-manifold, n>m, are classified up to regular homotopy by the homotopy classes of sections of a vector bundle E associated to the tangent bundle of M. When N = Rn , the fibre of E is the Stiefel manifold of m-frames in n-space.
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Any noncompact connected orientable 3-manifold immerses in R2+1.
“Any noncompact connected orientable 3-manifold immerses in R 2+1 by the immersion theorem [1], so some restriction is needed.”
From Lorentz spacetimes of constant curvature (2007) · cites this paper as [1]
The Introduction and Theorem 6.3 state that a parallelizable k-manifold immerses in Euclidean space of dimension k+1; Theorem 6.7 concerns compact 3-manifolds immersing in E4. I searched for noncompact orientable 3-manifolds and same-dimensional immersions, but found no passage in our copy establishing the claimed immersion in R³. The E4 results do not contradict it. (LLM, conf=0.76)
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Research works printed but not linked: 15
- Bott, Raoul, 1957, The stable homotopy of the classical groups, Proc. Nat. Acad. Sci. U.S.A., 43, 933, 10.1073/pnas.43.10.933 · DOI
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- Massey, W. S., 1958, On the cohomology ring of a sphere bundle, J. Math. Mech., 7, 265, 10.1512/iumj.1958.7.57020 · DOI
- Milnor, John, 1956, On the immersion of 𝑛-manifolds in (𝑛+1)-space, Comment. Math. Helv., 30, 275, 10.1007/BF02564347 · DOI
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Cited by
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Lorentz spacetimes of constant curvature
Mess, G. (2007) · Geometriae Dedicata · doi:10.1007/s10711-007-9155-7 · Cited by 278
Metadata
- Authors
- Hirsch, M. W.
- Corresponding author
- Hirsch, Morris W.
- ORCID
- Hirsch, Morris W.
- Venue
- Transactions of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 1959
- Volume
- 93
- Issue
- 2
- Pages
- 242-276
- ISSN
- 0002-9947 (ISSN-L)
- 1088-6850
- OpenAlex
- W2056690023
- Cited by
- 491
- References
- 0
- Citations by year
- 2026: 3 · 2025: 6 · 2024: 6 · 2023: 3 · 2022: 3 · 2021: 12 · 2020: 8 · 2019: 11 · 2018: 11 · 2017: 14 · 2016: 9 · 2015: 9 · 2014: 8 · 2013: 11 · 2012: 6
- FWCI
- 13.20
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Transactions of the American Mathematical Society (publishedVersion)
- eScholarship (California Digital Library) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Keywords
- regular homotopy, vector bundles, tangent bundle, Stiefel manifold, frames, classification of immersions
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