Algebraic Topology

Hello there. I am trying to find out how to compute 'the first homology group of the connected sum of three projective planes' (i.e. H1(RP2#RP2#RP2) = ?).

I have been reading Massey's book - "A Basic Course In Algebraic Topology" and it says that we can construct the sum by identifying the sides of a hexagon in pairs.

In class, my lecturer defined the first homology group as

"one generator for each group"
H1(X) = ------------------------------------------
"the word given by the boundary letters"

In Massey's book, it says that "the word given by the boundary letters" is <aabbcc>. I *think* "one generator for each group" is equal to <a,b,c,d,e,f>. I am not at all sure though.

I have been told by somebody that H1(RP2#RP2#RP2) is equal to 'is the direct sum of two copies of Z and a copy of Z/2'. Now, firstly I do not understand this. Secondly, what does Z/2 mean - Z/Z2 or '{a/2 in Z: a in Z}'?

I have enquired about this question a number of times and I just feel lost. People are suggesting all sorts of things that I know nothing of - triangulation, fundamental group, ...

Personally I do not think I should've been given this question without the material being covered properly.

I would very much appreciate a clear explanation as to how I can arrive at the answer for this.

Thank you.
.

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