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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Diffeomorphisms almost regularly homotopic to the identity
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by Robert Wells PDF
Trans. Amer. Math. Soc. 239 (1978), 279-292 Request permission

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 $\varphi |M - \mathrm {pt}$. is regularly homotopic to the inclusion $M - \mathrm {pt}. \subset M$. Let $\psi :M \to M \vee M$ be the result of collapsing the boundary of a smooth $n$-cell in $M$, and let $M \vee M \stackrel {\Delta ’}{\to } M$ be the codiagonal. For $\xi \in {\pi _n}(M)$ define $\tau (\xi )$ to be the composition \[ M \stackrel {\psi }{\rightarrow } M \vee M \stackrel {1 \vee \xi }{\rightarrow } M \vee M \stackrel {\Delta ’}{\rightarrow } M. \] 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
    R. E. Stong, Lecture notes on cobordism, Princeton Univ. Press, Princeton, N. J., 1969.
  • L. Favaro and R. Wells, The mapping torus construction and concordance of diffeomorphisms, Bol. Soc. Brasil. Mat. 5 (1974), no. 2, 127–146. MR 423375, DOI 10.1007/BF02938486
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 239 (1978), 279-292
  • MSC: Primary 57D50
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0488089-8
  • MathSciNet review: 0488089