Algebraic Topology · Problem 2B.4

Cohomology — Allen Hatcher

Mathematics — Topology · subspheres-unit-sphere

Problem

In the unit sphere

S^{p+q-1} \subset \mathbb{R}^{p+q}

let

S^{p-1}

and

S^{q-1}

be the subspheres consisting of points whose last

q

and first

p

coordinates are zero, respectively. (a) Show that

S^{p+q-1} - S^{p-1}

deformation retracts onto

S^{q-1}

, and is in fact homeomorphic to

S^{q-1}\times\mathbb{R}^p

. (b) Show that

S^{p-1}

and

S^{q-1}

are not the boundaries of any pair of disjointly embedded disks

D^p

and

D^q

D^{p+q}

. [The preceding exercise may be useful.]