Alexander duality and Hurewicz fibrations

Abstract:

We explore conditions under which the restriction of the projection map $p:{S^n} \times B \to B$ to an open subset $U \subset S^n \times B$ is a Hurewicz fibration. As a consequence, we exhibit Hurewicz fibrations $p:E \to I$ such that: (i) $p:E \to I$ is not a locally trivial bundle, (ii) $p^{ - 1}(t)$ is an open $n$-manifold for each $t$, and (iii) $p\; \circ \;{\text {proj:E}} \times {R^1} \to I$ is a locally trivial bundle. The fibers in our examples are distinguished by having nonisomorphic fundamental groups at infinity. We also show that when the fibers of a Hurewicz fibration with open $n$-manifold fibers have finitely generated $(n - 1){\text {st}}$ homology, then all fibers have the same finite number of ends. This last shows that the punctured torus and the thrice punctured two-sphere cannot both be fibers of a Hurewicz fibration $p:E \to I$ with open $2$-manifold fibers.