Hatcher-Algebraic-Topology · Problem 2B.5

Cohomology

Mathematics — Graph Theory · embedded-sphere-complement-homology

Problem

Let

S

be an embedded

k

-sphere in

S^n

for which there exists a disk

D^n \subset S^n

intersecting

S

in the disk

D^k \subset D^n

defined by the first

k

coordinates of

D^n

. Let

D^{n-k} \subset D^n

be the disk defined by the last

n-k

coordinates, with boundary sphere

S^{n-k-1}

. Show that the inclusion

S^{n-k-1} \hookrightarrow S^n - S

induces an isomorphism on homology groups.