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Math Help - Homology of Connected Sum of Two Projective Planes, P^2 # P^2

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    Super Member Bernhard's Avatar
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    Homology of Connected Sum of Two Projective Planes, P^2 # P^2

    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:

    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:


    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:


    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
    Last edited by Bernhard; April 22nd 2014 at 10:49 PM.
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