Hatcher-Algebraic-Topology · Problem 2B.3

Cohomology

Mathematics — Graph Theory · homology-sphere-disk-inclusion

Problem

Let

(D, S) \subset (D^n, S^{n-1})

be a pair of subspaces homeomorphic to

(D^k, S^{k-1})

, with

D \cap S^{n-1} = S

. Show the inclusion

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

induces an isomorphism on homology. [Glue two copies of

(D^n, D)

to the two ends of

(S^{n-1}\times I, S\times I)

to produce a

k

-sphere in

S^n

and look at a Mayer--Vietoris sequence for the complement of this

k

-sphere.]