Homotopy groups of the space of self-homotopy-equivalences

Abstract:

Let $M$ be a connected sum of $r$ closed aspherical manifolds of dimension $n \geqslant 3$, and let $EM$ denote the space of self-homotopy-equivalences of $M$, with basepoint the identity map of $M$. Using obstruction theory, we calculate ${\pi _q}(EM)$ for $1 \leqslant q \leqslant n - 3$ and show that ${\pi _{n - 1}}(EM)$ is not finitely-generated. As an application, for the case $n = 3$ and $r \geqslant 3$ we show that infinitely many generators of ${\pi _1}(E{M^3},{\text {i}}{{\text {d}}_M})$ can be realized by isotopies, to conclude that ${\pi _1}({\text {Homeo}}({M^3}),{\text {i}}{{\text {d}}_M})$ is not finitely-generated.