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Math W4051: Problem Set 11
due Wednesday, December 5
Reading: Munkres, Ch.13. Also take a look at Theorem 54.6 in Ch.9.
1.
Express the following abelian groups in standard form: (a) Ab
h
x, y

x
2
=
y
3
i
; (b)
Ab
h
x, y

x
2
=
y
4
i
.
2.
Let
X
and
Y
be manifolds (possibly of diﬀerent dimensions). Pick
x
0
∈
X
and
y
0
∈
Y.
We deﬁne their
wedge
X
∨
Y
to be the space obtained from the disjoint union of
X
and
Y
after identifying
x
0
with
y
0
:
X
∨
Y
=
X
q
Y / x
0
∼
y
0
.
Show that
π
(
X
∨
Y, x
0
) =
π
1
(
X, x
0
)
*
π
1
(
Y, y
0
)
.
3.
Let
S
1
sit inside
R
3
as
{
(
x, y,
0)
∈
R
3

x
2
+
y
2
= 1
}
.
Denote by
X
the complement of
S
1
in
R
3
.
Compute
π
1
(
X,
0)
.
(Hint: Draw a deformation
retraction from
X
to
S
2
∨
S
1
.
)
4.
We denote by
B
n
the standard open unit ball in
R
n
and by
B
±
⊂
B
n
the smaller open
ball of radius
±
= 1
/
2
.
We also let
S
±
=
∂
B
±
be the sphere of radius
±.
(a) Let
X
be a connected manifold of dimension
n
≥
3
.
Let
h
:
B
n
→
U
be a homeomor
phism from
B
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This note was uploaded on 02/10/2010 for the course MATH 205 taught by Professor Smith during the Spring '05 term at Adrian College.
 Spring '05
 SMITH
 Math

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