Simplicial Structures in Topology

V.3 Poincaré Duality 189
to the connected sum of two projective planes, and the connected sum is associative,
every surface is the connected sum of S 2 , nT tori, and 2nK + nP projective planes. If
2nK +nP = 0, then clearly S = S 2 when nT = 0andS = T 2 #T 2 #...#T 2 (nT times) if
nT > 0. Finally, if 2nK +nP > 0, we may substitute T 2 #RP 2 with the connected sum
RP 2 #RP 2 #RP 2 , iteratively, and conclude that S is homeomorphic to the connected
sum of 2nK + nP + 2nT projective planes. �
Exercises
1. Prove that, if S1 and S2 are two closed surfaces, the connected sum S1#S2 is a
closed surface.
2. Prove that the attaching of a handle to a surface S is equivalent to the connected
sum of S with either a torus or a Klein bottle.
3. Let the closed surfaces S1 and S2 be given; prove that
V.3 Poincaré Duality
χ(S1#S2)=χ(S1)+χ(S2) − 2 .
The reader must have noticed that the second integral homology group of the torus
T 2 is isomorphic to Z, whereas the corresponding homology group of RP2 is trivial.
On the other hand, in the proof of the Fundamental Theorem of Closed Surfaces,
we have noted that the real projective plane may be obtained by the adjunction
of a 2-disk to the Möbius band, while the Klein bottle is the connected sum of
two real projective planes. Well, everybody knows the story of the little man who
walks on a Möbius band and, having gone around once, found himself upside down
on the starting point (this is why the Möbius band was defined a “nonorientable”
surface5 ). The reader could wonder whether the presence of the Möbius band in the
real projective space – and consequently in the Klein bottle – is responsible for the
“abnormal” behavior of the second homology group of these spaces. The answer
is “yes” and has to do with the orientation of the simplexes of these triangulated
spaces. Thus, we give the following definition.
(V.3.1) Definition. A triangulable manifold V ∼ = |K| is orientable if its n-simplexes
may be oriented coherently, that is to say, each (n − 1)-simplex of K inherits opposite
orientations from its two adjacent n-simplexes (see Theorem (V.1.5)).
The reader is highly advised to draw triangulations for the spaces mentioned
above and to verify that it is not possible to give the Klein bottle and the real projective
plane coherent orientations, while this is possible in the case of the torus.
5 Not in the sense of our definition of 2-manifold; it is really a 2-manifold with boundary.