Simplicial Structures in Topology

V.2 Closed Surfaces 185
Before proceeding with this Theorem, we note that the real projective plane may
be obtained by the adjunction of a disk to a Möbius band. In fact, the Möbius band M
is obtained from a square by identifying the vertical sides with opposite orientations:
for instance, M comes from the square I × I through the identifications:
(0,t) ≡ (1,1 −t) , 0 ≤ t ≤ 1 .
Let D2 be the unity disk of R2 and let S1 = ∂ D2 . We define the function
f : S 1 → M, e 2πit
�
(2t,0)
↦→
0 ≤ t ≤ 1 (2t − 1,1)
2
1
2 ≤ t ≤ 1;
the reader may easily verify that RP 2 is given by the pushout
S 1
��
��
D 2
f
¯f
��
M
��
��
��
2
RP
The reader could study the problem also backwards: the real projective plane has
a closed disk D 2 with boundary a 1 a 1 as a model; let Oa be the origin of a and D a
small closed disk contained in D 2 , with centre at Oa; it is easily seen that
D 2 � ˚D ∼ = M
and so RP 2 is homeomorphic to the adjunction space of M and a closed disk. Note
that M is a surface with boundary.
We finally note that, as a consequence of the preceding results, the connected
sum of a torus and a real projective plane is homeomorphic to the connected sum of
three real projective planes; for its proof, it is enough to see that
T 2 #RP 2 ∼ = K#RP 2 .
Initially, we have noted that the connected sum of two surfaces does not depend on
the position of the disks that we remove for gluing the two surfaces; therefore, we
may glue T 2 and K to the component of RP 2 given by the Möbius band M. Inorder
to attach T 2 to M, we have to remove an open disk of T 2 andanopendiskofM,
and glue the two spaces (by means of a pushout!) on the boundaries created by this
removal; as we have seen, this is like attaching a handle to M. However, the space
that we obtain is exactly the same as the one we would get by removing two distinct
open disks from M and gluing a cylinder to M, the two circles that bound the cylinder
being identified with the same orientation to the boundaries formed by the removal
of the two open disks (if we travel clockwise the two circles limiting the cylinder,
both attachments will be made clockwise). For attaching a Klein bottle K to M,