Algebraic Topology · Problem 2B.5

Cohomology — Allen Hatcher

Mathematics — Topology · embedded-sphere-complement

Problem

Let SS be an embedded kk-sphere in SnS^n for which there exists a disk Dn⊂SnD^n \subset S^n intersecting SS in the disk Dk⊂DnD^k \subset D^n defined by the first kk coordinates of DnD^n. Let Dn−k⊂DnD^{n-k} \subset D^n be the disk defined by the last n−kn-k coordinates, with boundary sphere Sn−k−1S^{n-k-1}. Show that the inclusion Sn−k−1↪Sn−SS^{n-k-1} \hookrightarrow S^n - S induces an isomorphism on homology groups.

Browse the problem library