Diffeomorphisms almost regularly homotopic to the identity

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 ${\pi _n}(M) \to \pi _n^s(M)$.