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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
DOI: https://doi.org/10.1090/S0002-9947-1978-0488089-8
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 $\varphi |M - \mathrm {pt}$. 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 ${\pi _n}(M) \to \pi _n^s(M)$.


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Keywords: Diffeomorphism, Postnikov tower, cobordism, stable homotopy, mapping torus
Article copyright: © Copyright 1978 American Mathematical Society