Year: 2026
Type: preprint
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 $BO\langle n+1\rangle$-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 $π_1$ on the homotopy of the fibre of $M\to P_n(M)$. We compute $\operatorname{Aut}_\bullet(S^1\times S^{m-1})\cong(\mathbb{Z}/2)^2$ for $m\ge4$, and $\operatorname{Aut}_\bullet(S^3\times S^3)\cong(\mathbb{Z}/12)^2\rtimes SL_2(\mathbb{Z})$ with the standard action modulo $12$. Further, $\operatorname{Aut}_\bullet(S^3\times S^4)\cong(\mathbb{Z}/2\oplus\mathbb{Z}/24)\rtimes(\mathbb{Z}/2)^2$ has order $192$, that $\operatorname{Aut}_\bullet(S^4\times S^4)\cong D_4\times\mathbb{Z}/2$ has order $16$, and that for connected sums of $r\ge2$ 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 $π_{2p}(S^p)$ modulo Whitehead products with $ι_p$, and the image is determined by an obstruction on the Whitehead square. In particular $\operatorname{Aut}_\bullet(S^5\times S^5)$ is isomorphic to the theta subgroup of $SL_2(\mathbb{Z})$ and $\operatorname{Aut}_\bullet(S^7\times S^7)$ is an extension of $SL_2(\mathbb{Z})$ by $(\mathbb{Z}/120)^2$.
Source: arXiv (Cornell University)
Author: Pamuk, Mehmetcik
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220426016