I am reading James Munkres' book, Elements of Algebraic Topology.

Theorem 6.5 on page 39 concerns the homology groups of the connected sum of two projected planes.

Munkres demonstrates the following:

$ H_1 ( P^2 \# P^2 ) \simeq \mathbb{Z} \oplus \mathbb{Z} / 2 $ ... ... ... (1)

and

$ H_2 ( P^2 \# P^2 ) = 0 $ ... ... ... (2)

I would appreciate some help in understanding how Munkres establishes$ H_2 ( P^2 \# P^2 ) = 0 $. He does this moderately early in the proof after setting up the definitions and notation.

The Theorem and the early part of the proof up to the statement that "it is clear that $ H_2 ( P^2 \# P^2 ) = 0 $" is as follows:


Corrected Post - Homology of Connected Sum of Two Projective Planes, P^2 # P^2-theorem-6.5-beginning-proof-munkres-elements-algebraic-topology.png



I have labelled L in a manner that I think is appropriate as follows:


Corrected Post - Homology of Connected Sum of Two Projective Planes, P^2 # P^2-theorem-6.5-appropriate-labelling-l-munkres.png


Early in the proof (see above) Munkres refers to conditions 1 and 2. These conditions are as follows:


Corrected Post - Homology of Connected Sum of Two Projective Planes, P^2 # P^2-theorem-6.5-conditions-1-2.png

As I mentioned above, Munkres states, early in the proof, that

"It is clear that $ H_2 ( P^2 \# P^2 ) = 0 $"

BUT ... this is anything but clear to me ...

Can anyone explain why this 'clearly' follows:

Would appreciate some help.

Peter