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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Diffeomorphisms almost regularly homotopic to the identity


Author: Robert Wells
Journal: Trans. Amer. Math. Soc. 239 (1978), 279-292
MSC: Primary 57D50
MathSciNet review: 0488089
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Abstract: Let $ f:M \to M$ be a self-map of a closed smooth n-manifold. Does there exist a diffeomorphism $ \varphi :M \to M$ homotopic to f? Define $ \varphi $ to be almost regularly homotopic to the identity if $ \varphi \vert M - {\text{pt}}$. is regularly homotopic to the inclusion $ M - {\text{pt}}. \subset M$. Let $ \psi :M \to M \vee M$ be the result of collapsing the boundary of a smooth n-cell in M, and let $ M \vee M\mathop \to \limits^{\Delta '} M$ be the codiagonal. For $ \xi \in {\pi _n}(M)$ define $ \tau (\xi )$ to be the composition

$\displaystyle M\mathop \to \limits^\psi M \vee M\mathop \to \limits^{1 \vee \xi } M \vee M\mathop \to \limits^{\Delta '} M.$

Theorem. If M is 2-connected, s-parallelizable, and $ n = 2l > 5$ with $ l\;{\nequiv}\;0 \bmod (4)$, then $ \tau (\xi )$ contains a diffeomorphism almost regularly homotopic to the identity iff $ \xi $ is in the kernel of the stabilization map $ {\pi _n}(M) \to \pi _n^s(M)$.


References [Enhancements On Off] (What's this?)

  • [1] R. E. Stong, Lecture notes on cobordism, Princeton Univ. Press, Princeton, N. J., 1969.
  • [2] L. Favaro and R. Wells, The mapping torus construction and concordance of diffeomorphisms, Bol. Soc. Brasil. Mat. 5 (1974), no. 2, 127–146. MR 0423375 (54 #11354)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0488089-8
PII: S 0002-9947(1978)0488089-8
Keywords: Diffeomorphism, Postnikov tower, cobordism, stable homotopy, mapping torus
Article copyright: © Copyright 1978 American Mathematical Society