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James1118 science forum Guru Wannabe
Joined: 04 Feb 2005
Posts: 154

Posted: Wed Jul 19, 2006 11:38 pm Post subject:
Re: Homology of a certain space question



Thank you!
James 

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

Posted: Wed Jul 19, 2006 9:59 pm Post subject:
Re: Homology of a certain space question



James wrote:
Quote:  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 6cell 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 6cell. It is supposed to be equal to
sum deg(r_t o f_{boundary{A}) )
(sum is taken over all 5cells t)
where f_{boundary{A}} = p o f
and r_t is the map that first mods out the 5skeleton by the 4skeleton 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 5skeleton to the 5skeleton modded out by the 4skeleton? Is it 1?

Call g the composition
S^5 > S^5 > RP^5
Then your space X is the mapping cone of g: it can be seen as the
result of adjoining the cone on S^5 (which is precisely the 6cell)
to RP^5 according to g. Then you can use the long exact sequence
(this is *reduced* homology, any coefficients)
> H_q(S^5) > H_q(RP^5) > H_q(X) > H_{q1}(S^5) >
(you can find it in Greenberg's book, chapter 21, for example)
The maps H(S^5) > H(RP^5) are induced by g; and those
H(RP^5) > H(X) are induced by inclusions; the third one is a
connecting homomorphism.
Can you compute H(X) from this information?
 m 

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James1118 science forum Guru Wannabe
Joined: 04 Feb 2005
Posts: 154

Posted: Wed Jul 19, 2006 9:22 pm Post subject:
Homology of a certain space question



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 6cell 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 6cell. It is supposed to be equal to
sum deg(r_t o f_{boundary{A}) )
(sum is taken over all 5cells t)
where f_{boundary{A}} = p o f
and r_t is the map that first mods out the 5skeleton by the 4skeleton 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 5skeleton to the 5skeleton modded out by the 4skeleton? Is it 1?
Am I confused?
Sincerely,
James 

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