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I am trying to help someone with their prelims, but ended up needing some help myself:

1)Show that every map f: CP^2--->S^1 x S^1 x S^1 is nullhomotopic

2)Is the wedge RP^2 \/RP^2 a retract of RP^2 x RP^2 ?

For 1), I think we need to use the cell decomposition for CP^2 somehow, or just use

the fact that the 3rd homology ( 3 being odd ) is 0 , so that the induced map on

3rd homology must be trivial.

For 2, all I can think is that if the wedge were a retract, then it would be homotopic

to the product , so that all homology/homotopy groups would be equal. I think the

answer is no, since the homology of the wedge is the free product, but the

homology of the product is given by Kunneth's theorem, and we get different groups.

Does this work? Any ideas?

Thanks.

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# Null-homotopic, Retracts: Review for Prelims.

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