Homology of a certain space question

James · science forum Guru Wannabe Joined: 04 Feb 2005 Posts: 154

Hi,

I am puzzled by the following question : Let f : S^5 ---> S^5 be a map of degree d and let X be RP^5 with a 6-cell attached by the map p o f where p : S^5 ----> RP^5 is the natural projection. Determine the homology of X.

I don't think this is completely trivial. I know what the boundary maps are on the lower dimensional cells, so I only need to figure out the boundary of the 6-cell. It is supposed to be equal to

sum deg(r_t o f_{boundary{A}) )
(sum is taken over all 5-cells t)

where f_{boundary{A}} = p o f
and r_t is the map that first mods out the 5-skeleton by the 4-skeleton to get a wedge of S^5's (in this case 1 S^5) and then projects to the cell t (in this case the same cell).

The problem is that I don't know what the degree of the whole map is. I don't think it's just d. I think it is 2d since the degree of p is 2. But what is the degree of the map from the 5-skeleton to the 5-skeleton modded out by the 4-skeleton? Is it 1?

Am I confused?

Sincerely,

James

mariano.suarezalvarez@gma · science forum addict Joined: 28 Apr 2006 Posts: 58

James wrote:

James · science forum Guru Wannabe Joined: 04 Feb 2005 Posts: 154

Thank you!

James

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