Hatcher-Algebraic-Topology · Problem 16

The Fundamental Group

Mathematics — Graph Theory · no-retraction

Problem

Show that there are no retractions

r : X \to A

in the following cases: ((a))

X = \mathbb{R}^3

with

A

any subspace homeomorphic to

S^1

. ((b))

X = S^1 \times D^2

with

A

its boundary torus

S^1 \times S^1

. ((c))

X = S^1 \times D^2

and

A

the circle shown in the figure. ((d))

X = D^2 \vee D^2

with

A

its boundary

S^1 \vee S^1

. ((e))

X

a disk with two points on its boundary identified and

A

its boundary

S^1 \vee S^1

. ((f))

X

the M\"obius band and

A

its boundary circle.