Simplicial Structures in Topology

V.2 Closed Surfaces 181
a
a
b
a
a
d
c
b
b
a b
d
b a d b d a a d d
b
Fig. V.12 Connected sum of two projective planes: the Klein bottle
c
a
a
b
Fig. V.10 Torus less a closed
disk
Fig. V.11 Connected sum of
two tori
chosen vertices; in this manner, we obtain two triangles with boundaries a 1 a 1 c 1 and
b 1 b 1 c −1 ; these triangles are put together by identifying c (in other words, by taking
their connected sum) and getting a square with boundary a 1 a 1 b 1 b 1 ; this square,
with the necessary identifications, is homeomorphic to RP 2 #RP 2 . By cutting the
square along the diagonal d, we are now left with two right triangles, both having a
side a and a side b; we have thus obtained the right triangles a 1 b 1 d −1 and b 1 a 1 d 1 .
Next, we glue the triangles along the (oriented) side a in order to obtain a square
with boundary d 1 b 1 d 1 b −1 which, with the necessary identifications, corresponds to
a Klein bottle. Figure V.12, illustrates the steps for the procedure that we have just
described.
We close by noting that the surfaces
b
nT 2 := T 2 #...#T 2
� �� �
n times
nRP 2 := RP 2 #...#RP 2
� �� �
n times
b
b
b
d