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Homology of a certain space question
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James1118
science forum Guru Wannabe


Joined: 04 Feb 2005
Posts: 154

PostPosted: Wed Jul 19, 2006 11:38 pm    Post subject: Re: Homology of a certain space question Reply with quote

Thank you!

James
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mariano.suarezalvarez@gma
science forum addict


Joined: 28 Apr 2006
Posts: 58

PostPosted: Wed Jul 19, 2006 9:59 pm    Post subject: Re: Homology of a certain space question Reply with quote

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 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?

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 6-cell)
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_{q-1}(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

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