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:

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

and

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

I would appreciate some help in understanding how Munkres establishes $\displaystyle 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 $\displaystyle H_2 ( P^2 \# P^2 ) = 0 $" is as follows:

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

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

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

"It is clear that $\displaystyle 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