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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Groups of embedded manifolds


Author: Max K. Agoston
Journal: Trans. Amer. Math. Soc. 154 (1971), 365-375
MSC: Primary 57.10
MathSciNet review: 0273628
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Abstract: This paper defines a group $ \theta ({M^n},{\nu _\varphi })$ which generalizes the group of framed homotopy n-spheres in $ {S^{n + k}}$. Let $ {M^n}$ be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let $ {\nu _\varphi }$ be the normal bundle of some imbedding $ \varphi :{M^n} \to {S^{n + k}}$, where $ 2k \geqq n + 3$. Then $ \theta ({M^n},{\nu _\varphi })$ is the set of h-cobordism classes of triples $ (F,{V^n},f)$, where $ F:{S^{n + k}} \to T({\nu _\varphi })$ is a map which is transverse regular on M, $ {V^n} = {F^{ - 1}}({M^n})$, and $ f = F\vert{V^n}$ is a homotopy equivalence. ( $ T({\nu _\varphi })$ is the Thom complex of $ {\nu _\varphi }$.) There is a natural group structure on $ \theta ({M^n},{\nu _\varphi })$, and $ \theta ({M^n},{\nu _\varphi })$ fits into an exact sequence similar to that for the framed homotopy n-spheres.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0273628-6
Keywords: Framed homotopy n-spheres in $ {S^{n + k}}$, normally equivalent n-manifolds in $ {S^{n + k}}$, exact sequence
Article copyright: © Copyright 1971 American Mathematical Society