Simplicial Structures in Topology

178 V Triangulable Manifolds
and take the closed disks D1 ⊂ S1 and D2 ⊂ S2, together with the homeomorphisms
h1 : D 2 → D1 and h2 : D 2 → D2; letS 1 = ∂D 2 be the boundering circle of the unit
disk D 2 . The restrictions of the homeomorphisms h1 and h2 to S 1 are homeomorphisms
from S 1 onto the boundaries ∂D1 and ∂D2, respectively. We now define the
maps
h ′ h1|S1 1 : S1 −→ ∂D1 ↩→ S1 � IntD1
h ′ 2 : S 1 h 2|S 1
−→ ∂D2 ↩→ S2 � IntD2
and construct the pushout of the pair of morphisms (h ′ 1 ,h′ 2 ) in Top. In this manner,
we obtain the space (unique up to homeomorphism)
�
S1#S2 := S1 � IntD1 S2 � IntD2
named connected sum of S1 and S2. Figures V.4, V.5,andV.6 outline a procedure
Fig. V.4 Two tori less a disk
Fig. V.5 Attaching two tori
for obtaining the connected sum of two tori.
(V.2.1) Theorem. The connected sum S1#S2 of two connected surfaces is a
2-manifold independent (up to homeomorphism) from the choice of the closed
disks D1 and D2, and from the homeomorphisms h1 and h2.
Proof. It is easily proved that S1#S2 is a 2-manifold; we only need to find local
charts for the points in the gluing zone. It does not depend on the choice of the
disks D1 and D2 and on the homeomorphisms h1 : D 2 → D1 and h2 : D 2 → D2 due
to the following result: if S is a surface, then for every pair of homeomorphisms
h1 : D 2 → S and h2 : D 2 → S (on the images), there is a homeomorphism f between
h ′ 1 ,h′ 2