Simplicial Structures in Topology

V.2 Closed Surfaces 179
Fig. V.6 Connected sum of
two tori: the torus of genus 2
their images such that h2 = fh1. The independence from the choice of h1 and h2
derives from the universal property of pushouts. The details of this proof are left to
the reader. �
If S represents the set of all classes of homeomorphisms of closed connected
surfaces, we have the following result whose proof we shall omit:
(V.2.2) Proposition. The connected sum determines an operation
#: S × S → S
which is associative, commutative, and has a neutral element (the sphere S2 ).
The sphere S2 ,thetorusT2 , the real projective plane RP2 , and the connected
sums of these spaces have particularly interesting representations. We begin
with S2 . In Sect. I.1 we have proved that the sphere S2 is homeomorphic to the
quotient space obtained from D2 by identifying each point (x,y) ∈ S1 to (x,−y)
(see Example (I.1.7)). So if, as in Fig. V.7, a denotes the semicircle from (−1,0)
a
A B
a=b
Fig. V.7 Identifying polygon
for S 2
to (1,0) through (0,1) and b, the semicircle from (−1,0) to (1,0) through (0,−1)
(both from (−1,0) to (1,0) ), S 2 maybeviewedasthediskD 2 with two vertices
A =(−1,0), B =(1,0) and two identified arcs (sides), that is to say, a disk with
two vertices and only one side a = b; if the boundary is travelled clockwise from
A, sideb is travelled against the direction we have given it and we may therefore
view S 2 as a disk with boundary a 1 a −1 (we assign the exponent 1 to side a when
following the chosen direction, and the exponent −1 when following the opposite
direction).