Download Homework Assignment # 3, due Sept. 18 1. Show that the connected

Homework Assignment # 3, due Sept. 18
1. Show that the connected sum RP2 #RP2 of two copies of the real projective plane is
homeomorphic to the Klein bottle K.
2. a) Show that the surface of genus g
Σg = T # . . . #T
| {z }
g
and the connected sum
RP2 # . . . #RP2
{z
}
|
k
are both homeomorphic to the topological space Σ(W ) associated to a word W . What is
the word W in these two cases?
b) Calculate the Euler characteristic of these manifolds.
3. Important examples of quotient spaces are orbit spaces of a group G acting on a topological
space X. We recall that a (left) of a group G on a set X is given by a map
G × X −→ X
typically written as
(g, x) 7→ gx,
such that g1 (g2 x) = (g1 g2 )x for g1 , g2 ∈ G, x ∈ X (associativity) and ex = x for e the unit
element of G, x ∈ X (unit property). Given a G-action on a topological space X the orbit
space denoted X/G is the quotient space X/ ∼ where two elements x, y ∈ X are declared
equivalent if and only if there is some g ∈ G with gx = y. In particular, the equivalence
class of x is the subset {gx | g ∈ G}, which is called the orbit of x, and hence X/G is the
space of orbits. Although we haven’t used this terminology, we’ve already encountered orbit
spaces, namely
RPn = S n /{±1}
and
CPn = S 2n+1 /S 1 .
Here the actions are given by
{±1} × S n → S n
(t, (v0 , . . . , vn )) 7→ (tv0 , . . . , tvn )
S 1 × S 2n+1 → S 2n+1
,
(z, (z0 , . . . , zn )) 7→ (zz0 , . . . , zzn )
and
where z ∈ S 1 ⊂ C and (z0 , . . . , zn ) ∈ S 2n+1 ⊂ Cn+1 .
(a) Consider the action Z2 × R2 → R2 , (m, n), (x, y) 7→ (x + m, y + n). Show that the
quotient space R2 /Z2 is homeomorphic to the torus, described as the quotient space
b
a
a
b
Use without proof the fact that this orbit space is Hausdorff (this will come up later this
semester).
1
(b) Consider the action G × R2 → R2 where G is the subgroup of the group of isometries of
the metric space R2 generated by the isometries g, h : R2 → R2 defined by
g(x, y) = (x + 1, y)
and
h(x, y) = (−x, y + 1)
Show that the quotient space R2 /G is homeomorphic to the Klein bottle, described as
the quotient space of the square [0, 1] × [0, 1] with edge identifications
b
a
a
b
Again, use without proof the fact that this orbit space is Hausdorff. Hint: Show that
every orbit can be represented by a point (x, y) ∈ [0, 1] × [0, 1]. To do this, it might be
helpful to argue that the composition ghgh−1 is the identity and to use this to show that
every element of G can be uniquely written in the form g m hn with m, n ∈ Z.
4. a) Show that a subspace of a Hausdorff space is Hausdorff and that a product of Hausdorff
spaces is Hausdorff.
b) Show that a subspace of a second countable space is second countable and that a product
of second countable spaces is second countable.
2