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 Poincaré duality complex that satisfies $H_n\left(M_k\right)\cong Z^k$, in a localised category of spaces. The construction of the fibration is proved for $k⩾2$, where the prime 2, and the primes that occur as torsion in ${\pi }_{2n-1}\left(S^n\right)$ 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.