Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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V.2 Closed Surfaces 185<br />
Before proceed<strong>in</strong>g with this Theorem, we note that the real projective plane may<br />
be obta<strong>in</strong>ed by the adjunction of a disk to a Möbius band. In fact, the Möbius band M<br />
is obta<strong>in</strong>ed from a square by identify<strong>in</strong>g the vertical sides with opposite orientations:<br />
for <strong>in</strong>stance, M comes from the square I × I through the identifications:<br />
(0,t) ≡ (1,1 −t) , 0 ≤ t ≤ 1 .<br />
Let D2 be the unity disk of R2 and let S1 = ∂ D2 . We def<strong>in</strong>e the function<br />
f : S 1 → M, e 2πit<br />
�<br />
(2t,0)<br />
↦→<br />
0 ≤ t ≤ 1 (2t − 1,1)<br />
2<br />
1<br />
2 ≤ t ≤ 1;<br />
the reader may easily verify that RP 2 is given by the pushout<br />
S 1<br />
��<br />
��<br />
D 2<br />
f<br />
¯f<br />
��<br />
M<br />
��<br />
��<br />
��<br />
2<br />
RP<br />
The reader could study the problem also backwards: the real projective plane has<br />
a closed disk D 2 with boundary a 1 a 1 as a model; let Oa be the orig<strong>in</strong> of a and D a<br />
small closed disk conta<strong>in</strong>ed <strong>in</strong> D 2 , with centre at Oa; it is easily seen that<br />
D 2 � ˚D ∼ = M<br />
and so RP 2 is homeomorphic to the adjunction space of M and a closed disk. Note<br />
that M is a surface with boundary.<br />
We f<strong>in</strong>ally note that, as a consequence of the preced<strong>in</strong>g results, the connected<br />
sum of a torus and a real projective plane is homeomorphic to the connected sum of<br />
three real projective planes; for its proof, it is enough to see that<br />
T 2 #RP 2 ∼ = K#RP 2 .<br />
Initially, we have noted that the connected sum of two surfaces does not depend on<br />
the position of the disks that we remove for glu<strong>in</strong>g the two surfaces; therefore, we<br />
may glue T 2 and K to the component of RP 2 given by the Möbius band M. Inorder<br />
to attach T 2 to M, we have to remove an open disk of T 2 andanopendiskofM,<br />
and glue the two spaces (by means of a pushout!) on the boundaries created by this<br />
removal; as we have seen, this is like attach<strong>in</strong>g a handle to M. However, the space<br />
that we obta<strong>in</strong> is exactly the same as the one we would get by remov<strong>in</strong>g two dist<strong>in</strong>ct<br />
open disks from M and glu<strong>in</strong>g a cyl<strong>in</strong>der to M, the two circles that bound the cyl<strong>in</strong>der<br />
be<strong>in</strong>g identified with the same orientation to the boundaries formed by the removal<br />
of the two open disks (if we travel clockwise the two circles limit<strong>in</strong>g the cyl<strong>in</strong>der,<br />
both attachments will be made clockwise). For attach<strong>in</strong>g a Kle<strong>in</strong> bottle K to M,