Serge Ballif
MATH 527 Homework 6
October 12, 2007
Problem 1.
Does the BorsukUlam theorem hold for the torus? In other words, for every map
f
:
S
1
×
S
1
→
2
must there exist
(
x, y
)
∈
S
1
×
S
1
, such that
f
(
x, y
) =
f
(

x,

y
)
?
No. We can imbed the torus in
3
symmetric about the
z
axis and the
xy
plane.
Then the projection from torus to an annulus in the plane (
f
(
x, y, z
) = (
x, y
))
is a continuous map. Since the map
f
preserves the angles in the
xy
plane the
antipodal points of the torus are mapped to what we might call “antipodal points”
of the annulus. Thus,
f
(
x, y
)
6
=
f
(

x,

y
).
Problem 2.
Compute the fundamental group of
S
2
∨
S
2
.
Let
X
=
S
2
∨
S
2
with common point
p
.
If we remove any point other than
p
from
X
we get an open subspace space homeomorphic to
S
2
∨
D
2
where
D
2
is a closed two dimensional disk.
We can express
X
as the union of two such
subspaces:
U
1
∼
=
S
2
∨
D
2
and
U
2
∼
=
D
2
∨
S
2
. The intersection
U
1
∩
U
2
∼
=
D
2
∨
D
2
is path connected. The space
D
2
is contractible, so
S
2
∨
D
2
’
S
2
. Therefore,
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 Spring '07
 ROTMAN,REGINA
 Math, Topology, Fundamental group, Serge Ballif, U2 D2 D2

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