Reference for cup product in deRham cohomology
0
$begingroup$
Given a smooth manifold $M$, we have what are called deRham cohomology groups $H^i(M,mathbb{R})$.
deRham cohomology ring $H^*(M,mathbb{R})$ is as a set $bigoplus_{i=0}^{text{dim(M)}} H^i(M,mathbb{R})$. This is made a ring by giving multiplication. Given $1$-form $omega$ and a $2$-form $tau$, the product $omega.tau$ is the wedge product $omegawedgetau$.
Is there any reference where this is mentioned?
differential-geometry smooth-manifolds de-rham-cohomology
asked Dec 23 '18 at 2:47
Praphulla KoushikPraphulla Koushik
203119
$endgroup$
|
show 3 more comments
0
$begingroup$
Given a smooth manifold $M$, we have what are called deRham cohomology groups $H^i(M,mathbb{R})$.
deRham cohomology ring $H^*(M,mathbb{R})$ is as a set $bigoplus_{i=0}^{text{dim(M)}} H^i(M,mathbb{R})$. This is made a ring by giving multiplication. Given $1$-form $omega$ and a $2$-form $tau$, the product $omega.tau$ is the wedge product $omegawedgetau$.
Is there any reference where this is mentioned?
differential-geometry smooth-manifolds de-rham-cohomology
asked Dec 23 '18 at 2:47
Praphulla KoushikPraphulla Koushik
203119
$endgroup$
$begingroup$
See $S$ 3.2 of Hatcher's Algebraic Topology. pi.math.cornell.edu/~hatcher/AT/ATpage.html
$endgroup$
– Travis
Dec 23 '18 at 3:17
$begingroup$
@Travis It does not say anything about deRham cohomology but it does say something about cup product... Thank you :) Any reference where deRham cohomology is specifically mentioned?
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:25
$begingroup$
The notion of cup product is a general feature of cohomology theory. I don't know offhand a reference that exclusively treats the cup product of de Rham cohomology.
$endgroup$
– Travis
Dec 23 '18 at 3:42
$begingroup$
@Travis Yes, I know that cup product is in other cohomology theories.. Some one asked me for reference only for deRham cohomology (may be because it is easier than others).. I could not think of any.. So, asked as question here..
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:45
$begingroup$
Look at Warner's Foundations of Manifolds book or Spivak's Differential Geometry, volume 1.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 6:12
|
show 3 more comments
0
0
0
$begingroup$
Given a smooth manifold $M$, we have what are called deRham cohomology groups $H^i(M,mathbb{R})$.
deRham cohomology ring $H^*(M,mathbb{R})$ is as a set $bigoplus_{i=0}^{text{dim(M)}} H^i(M,mathbb{R})$. This is made a ring by giving multiplication. Given $1$-form $omega$ and a $2$-form $tau$, the product $omega.tau$ is the wedge product $omegawedgetau$.
Is there any reference where this is mentioned?
differential-geometry smooth-manifolds de-rham-cohomology
asked Dec 23 '18 at 2:47
Praphulla KoushikPraphulla Koushik
203119
$endgroup$
Given a smooth manifold $M$, we have what are called deRham cohomology groups $H^i(M,mathbb{R})$.
deRham cohomology ring $H^*(M,mathbb{R})$ is as a set $bigoplus_{i=0}^{text{dim(M)}} H^i(M,mathbb{R})$. This is made a ring by giving multiplication. Given $1$-form $omega$ and a $2$-form $tau$, the product $omega.tau$ is the wedge product $omegawedgetau$.
Is there any reference where this is mentioned?
differential-geometry smooth-manifolds de-rham-cohomology
differential-geometry smooth-manifolds de-rham-cohomology
asked Dec 23 '18 at 2:47
Praphulla KoushikPraphulla Koushik
203119
asked Dec 23 '18 at 2:47
Praphulla KoushikPraphulla Koushik
203119
asked Dec 23 '18 at 2:47
Praphulla KoushikPraphulla Koushik
203119
asked Dec 23 '18 at 2:47
Praphulla KoushikPraphulla Koushik
203119
asked Dec 23 '18 at 2:47
Praphulla KoushikPraphulla Koushik
203119
203119
$begingroup$
See $S$ 3.2 of Hatcher's Algebraic Topology. pi.math.cornell.edu/~hatcher/AT/ATpage.html
$endgroup$
– Travis
Dec 23 '18 at 3:17
$begingroup$
@Travis It does not say anything about deRham cohomology but it does say something about cup product... Thank you :) Any reference where deRham cohomology is specifically mentioned?
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:25
$begingroup$
The notion of cup product is a general feature of cohomology theory. I don't know offhand a reference that exclusively treats the cup product of de Rham cohomology.
$endgroup$
– Travis
Dec 23 '18 at 3:42
$begingroup$
@Travis Yes, I know that cup product is in other cohomology theories.. Some one asked me for reference only for deRham cohomology (may be because it is easier than others).. I could not think of any.. So, asked as question here..
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:45
$begingroup$
Look at Warner's Foundations of Manifolds book or Spivak's Differential Geometry, volume 1.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 6:12
|
show 3 more comments
$begingroup$
See $S$ 3.2 of Hatcher's Algebraic Topology. pi.math.cornell.edu/~hatcher/AT/ATpage.html
$endgroup$
– Travis
Dec 23 '18 at 3:17
$begingroup$
@Travis It does not say anything about deRham cohomology but it does say something about cup product... Thank you :) Any reference where deRham cohomology is specifically mentioned?
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:25
$begingroup$
The notion of cup product is a general feature of cohomology theory. I don't know offhand a reference that exclusively treats the cup product of de Rham cohomology.
$endgroup$
– Travis
Dec 23 '18 at 3:42
$begingroup$
@Travis Yes, I know that cup product is in other cohomology theories.. Some one asked me for reference only for deRham cohomology (may be because it is easier than others).. I could not think of any.. So, asked as question here..
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:45
$begingroup$
Look at Warner's Foundations of Manifolds book or Spivak's Differential Geometry, volume 1.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 6:12
$begingroup$
See $S$ 3.2 of Hatcher's Algebraic Topology. pi.math.cornell.edu/~hatcher/AT/ATpage.html
$endgroup$
– Travis
Dec 23 '18 at 3:17
$begingroup$
See $S$ 3.2 of Hatcher's Algebraic Topology. pi.math.cornell.edu/~hatcher/AT/ATpage.html
$endgroup$
– Travis
Dec 23 '18 at 3:17
$begingroup$
@Travis It does not say anything about deRham cohomology but it does say something about cup product... Thank you :) Any reference where deRham cohomology is specifically mentioned?
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:25
$begingroup$
@Travis It does not say anything about deRham cohomology but it does say something about cup product... Thank you :) Any reference where deRham cohomology is specifically mentioned?
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:25
$begingroup$
The notion of cup product is a general feature of cohomology theory. I don't know offhand a reference that exclusively treats the cup product of de Rham cohomology.
$endgroup$
– Travis
Dec 23 '18 at 3:42
$begingroup$
The notion of cup product is a general feature of cohomology theory. I don't know offhand a reference that exclusively treats the cup product of de Rham cohomology.
$endgroup$
– Travis
Dec 23 '18 at 3:42
$begingroup$
@Travis Yes, I know that cup product is in other cohomology theories.. Some one asked me for reference only for deRham cohomology (may be because it is easier than others).. I could not think of any.. So, asked as question here..
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:45
$begingroup$
@Travis Yes, I know that cup product is in other cohomology theories.. Some one asked me for reference only for deRham cohomology (may be because it is easier than others).. I could not think of any.. So, asked as question here..
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:45
$begingroup$
Look at Warner's Foundations of Manifolds book or Spivak's Differential Geometry, volume 1.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 6:12
$begingroup$
Look at Warner's Foundations of Manifolds book or Spivak's Differential Geometry, volume 1.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 6:12
|
show 3 more comments
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$begingroup$
See $S$ 3.2 of Hatcher's Algebraic Topology. pi.math.cornell.edu/~hatcher/AT/ATpage.html
$endgroup$
– Travis
Dec 23 '18 at 3:17
$begingroup$
@Travis It does not say anything about deRham cohomology but it does say something about cup product... Thank you :) Any reference where deRham cohomology is specifically mentioned?
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:25
$begingroup$
The notion of cup product is a general feature of cohomology theory. I don't know offhand a reference that exclusively treats the cup product of de Rham cohomology.
$endgroup$
– Travis
Dec 23 '18 at 3:42
$begingroup$
@Travis Yes, I know that cup product is in other cohomology theories.. Some one asked me for reference only for deRham cohomology (may be because it is easier than others).. I could not think of any.. So, asked as question here..
$endgroup$
– Praphulla Koushik
Dec 23 '18 at 3:45
$begingroup$
Look at Warner's Foundations of Manifolds book or Spivak's Differential Geometry, volume 1.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 6:12