Sphere fibrations over highly connected manifolds

Abstract

We construct sphere fibrations over $(n-1)$-connected $2n$-manifolds such that the total space is a connected sum of sphere products. More precisely, for $n$ even, we construct fibrations $S^{n-1} \to \#^{k-1}(S^n \times S^{2n-1}) \to M_k$, where $M_k$ is a $(n-1)$-connected $2n$-dimensional Poincar\'{e} duality complex which satisfies $H_n(M_k)\cong \mathbb{Z}^k$, in a localized category of spaces. The construction of the fibration is proved for $k\geq 2$, where the prime $2$, and the primes which occur as torsion in $\pi_{2n-1}(S^n)$ are inverted. In specific cases, by either assuming $n$ is small, or assuming $k$ is large we can reduce the number of primes that need to be inverted. Integral results are obtained for $n=2$ or $4$, and if $k$ is bigger than the number of cyclic summands in the stable stem $\pi_{n-1}^s$, we obtain results after inverting $2$. Finally, we prove some applications for fibrations over $N\# M_k$, and for looped configuration spaces.