De Rham cohomology of $mathbb{RP^n}$
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1
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I have to calculate the De Rham cohomology of $mathbb{RP^n}$ using the Mayer-Vietoris sequence.
I first started by considering $mathbb{RP^n}=S^n/sim $ where $sim$ is the antipodal identification. Then I wrote $mathbb{RP^n}$ as the union of the sets
$U=S^n- {(0,...0,1)}$
and
$V=S^n- {(1,0,...,0)}$
But when I start using Mayer-Vietoris sequence I don't know how to proceed. Do I need to calculate it by induction on the order of the cohomology? Have you any hints or references in which it is solved?
Thank you.
algebraic-topology projective-space de-rham-cohomology
edited Nov 16 at 19:38
KReiser
9,08711335
asked Nov 16 at 17:44
Phi_24
1248
add a comment |
up vote
1
down vote
favorite
I have to calculate the De Rham cohomology of $mathbb{RP^n}$ using the Mayer-Vietoris sequence.
I first started by considering $mathbb{RP^n}=S^n/sim $ where $sim$ is the antipodal identification. Then I wrote $mathbb{RP^n}$ as the union of the sets
$U=S^n- {(0,...0,1)}$
and
$V=S^n- {(1,0,...,0)}$
But when I start using Mayer-Vietoris sequence I don't know how to proceed. Do I need to calculate it by induction on the order of the cohomology? Have you any hints or references in which it is solved?
Thank you.
algebraic-topology projective-space de-rham-cohomology
edited Nov 16 at 19:38
KReiser
9,08711335
asked Nov 16 at 17:44
Phi_24
1248
1
Well you should try to first figure out the cohomology of $U$, $V$, and $U cap V$. Then you should try to figure out what the induced maps on cohomology are. Then you can hope that can exactness will help to figure out the cohomology groups.
– DKS
Nov 16 at 18:20
@DKS These are precisely the steps I can't do. Is there a place where it's explained?
– Phi_24
Nov 17 at 13:28
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have to calculate the De Rham cohomology of $mathbb{RP^n}$ using the Mayer-Vietoris sequence.
I first started by considering $mathbb{RP^n}=S^n/sim $ where $sim$ is the antipodal identification. Then I wrote $mathbb{RP^n}$ as the union of the sets
$U=S^n- {(0,...0,1)}$
and
$V=S^n- {(1,0,...,0)}$
But when I start using Mayer-Vietoris sequence I don't know how to proceed. Do I need to calculate it by induction on the order of the cohomology? Have you any hints or references in which it is solved?
Thank you.
algebraic-topology projective-space de-rham-cohomology
edited Nov 16 at 19:38
KReiser
9,08711335
asked Nov 16 at 17:44
Phi_24
1248
I have to calculate the De Rham cohomology of $mathbb{RP^n}$ using the Mayer-Vietoris sequence.
I first started by considering $mathbb{RP^n}=S^n/sim $ where $sim$ is the antipodal identification. Then I wrote $mathbb{RP^n}$ as the union of the sets
$U=S^n- {(0,...0,1)}$
and
$V=S^n- {(1,0,...,0)}$
But when I start using Mayer-Vietoris sequence I don't know how to proceed. Do I need to calculate it by induction on the order of the cohomology? Have you any hints or references in which it is solved?
Thank you.
algebraic-topology projective-space de-rham-cohomology
algebraic-topology projective-space de-rham-cohomology
edited Nov 16 at 19:38
KReiser
9,08711335
asked Nov 16 at 17:44
Phi_24
1248
edited Nov 16 at 19:38
KReiser
9,08711335
asked Nov 16 at 17:44
Phi_24
1248
edited Nov 16 at 19:38
KReiser
9,08711335
edited Nov 16 at 19:38
KReiser
9,08711335
edited Nov 16 at 19:38
KReiser
9,08711335
9,08711335
asked Nov 16 at 17:44
Phi_24
1248
asked Nov 16 at 17:44
Phi_24
1248
asked Nov 16 at 17:44
Phi_24
1248
1248
1
Well you should try to first figure out the cohomology of $U$, $V$, and $U cap V$. Then you should try to figure out what the induced maps on cohomology are. Then you can hope that can exactness will help to figure out the cohomology groups.
– DKS
Nov 16 at 18:20
@DKS These are precisely the steps I can't do. Is there a place where it's explained?
– Phi_24
Nov 17 at 13:28
add a comment |
1
Well you should try to first figure out the cohomology of $U$, $V$, and $U cap V$. Then you should try to figure out what the induced maps on cohomology are. Then you can hope that can exactness will help to figure out the cohomology groups.
– DKS
Nov 16 at 18:20
@DKS These are precisely the steps I can't do. Is there a place where it's explained?
– Phi_24
Nov 17 at 13:28
1
1
Well you should try to first figure out the cohomology of $U$, $V$, and $U cap V$. Then you should try to figure out what the induced maps on cohomology are. Then you can hope that can exactness will help to figure out the cohomology groups.
– DKS
Nov 16 at 18:20
Well you should try to first figure out the cohomology of $U$, $V$, and $U cap V$. Then you should try to figure out what the induced maps on cohomology are. Then you can hope that can exactness will help to figure out the cohomology groups.
– DKS
Nov 16 at 18:20
@DKS These are precisely the steps I can't do. Is there a place where it's explained?
– Phi_24
Nov 17 at 13:28
@DKS These are precisely the steps I can't do. Is there a place where it's explained?
– Phi_24
Nov 17 at 13:28
add a comment |
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Well you should try to first figure out the cohomology of $U$, $V$, and $U cap V$. Then you should try to figure out what the induced maps on cohomology are. Then you can hope that can exactness will help to figure out the cohomology groups.
– DKS
Nov 16 at 18:20
@DKS These are precisely the steps I can't do. Is there a place where it's explained?
– Phi_24
Nov 17 at 13:28