Homotopy Self-Equivalences of Manifolds

Abstract

Hambleton and Kreck described the group of based homotopy self-equivalences of a closed oriented $4$-manifold by a commutative braid of exact sequences built from bordism groups and the Postnikov $2$-type. We construct the analogous braid for closed oriented $BO⟨n+1⟩$-manifolds $M$ of dimension $2n$ or $2n-1$ whose fundamental group is finite or infinite cyclic and which satisfy one condition on the action of ${\pi }_1$ on the homotopy of the fibre of $M\to P_n(M)$. We compute ${\mathrm{Aut}}_{\bullet }(S^1\times S^{m-1})\cong (\mathbb{Z}/2{)}^2$ for $m\ge 4$, and ${\mathrm{Aut}}_{\bullet }(S^3\times S^3)\cong (\mathbb{Z}/12{)}^2⋊S{L}_2(\mathbb{Z})$ with the standard action modulo $12$. Further, ${\mathrm{Aut}}_{\bullet }(S^3\times S^4)\cong (\mathbb{Z}/2\oplus \mathbb{Z}/24)⋊(\mathbb{Z}/2{)}^2$ has order $192$, that ${\mathrm{Aut}}_{\bullet }(S^4\times S^4)\cong D_4\times \mathbb{Z}/2$ has order $16$, and that for connected sums of $r\ge 2$ copies of $S^3\times S^3$ the kernel contains a subgroup $(\mathbb{Z}/12{)}^{2r}$ on which the permutations of the summands act non-trivially. For $S^p\times S^p$ the kernel is a sum of two copies of ${\pi }_{2p}(S^p)$ modulo Whitehead products with ${\iota }_p$, and the image is determined by an obstruction on the Whitehead square. In particular ${\mathrm{Aut}}_{\bullet }(S^5\times S^5)$ is isomorphic to the theta subgroup of $S{L}_2(\mathbb{Z})$ and ${\mathrm{Aut}}_{\bullet }(S^7\times S^7)$ is an extension of $S{L}_2(\mathbb{Z})$ by $(\mathbb{Z}/120{)}^2$.

arXiv comment30 pages