Simplicial Structures in Topology

180 V Triangulable Manifolds
We view the torus T 2 as a square (homeomorphic to a closed disk) whose horizontal
sides, as well as its vertical sides, have been identified. See Fig. V.8.Inother
a
b
b
a
Fig. V.8 Identifying polygon
for the torus T 2
words, we interpret T 2 as a square (closed disk) with a single vertex A and two sides
a, b. If we travel clockwise the boundary of the square that represents T 2 , we read
such boundary as the word a 1 b 1 a −1 b −1 .
The real projective plane RP 2 is viewed as a closed disk with a single vertex A;
its boundary is given by a 1 a 1 (we identify the antipodal points of the boundary), as
in Fig. V.9.
a
A B
a=b
Fig. V.9 Projective plane
Let us now try to interpret the connected sum T 2 #T 2 . We suppose the first torus
to be represented by a square with boundary a 1 b 1 a −1 b −1 and the second one, by
a square with boundary c 1 d 1 c −1 d −1 ; we remove from each square the interior of
(a portion corresponding to) a closed disk that meets the boundary of the square
exactly at one of its vertices, as shown in Fig. V.10. We obtain two closed polygons
with boundaries a 1 b 1 a −1 b −1 e and c 1 d 1 c −1 d −1 f , respectively. By identifying e and
f , we may interpret T 2 #T 2 as an octagon whose vertices are all identified into a
single one, and with boundary a 1 b 1 a −1 b −1 c 1 d 1 c −1 d −1 ,asinFig. V.11.
How should we interpret the connected sum RP 2 #RP 2 ? Weassumethefirst
(respectively, the second) of these projective planes to be a closed disk with two
identified antipodal vertices and two sides a 1 a 1 (respectively, b 1 b 1 ) whose antipodal
points have been identified. We remove, from each of these disks, the interior of
a small closed disk tangent to the boundary of the larger disk, at one of the two