Mixing
Defining a pseudo-gradient field for a $1$-form
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I'm reading Audin and Damian's Morse Theory and Floer Homology; they say there is an analogous way to define nondegenerate critical points for 1-forms as well as pseudo-gradient fields but don't discuss how. The goal is to define such a vector field for a 1-form $alpha$ and then lift it to a pseudo-gradient field for a function on a covering space.
Here's the context. Suppose I have a smooth closed 1-form $alpha$ on a manifold $M$. If I consider a map $phi:pi_1(M) to mathbb{R}$ which is simply integrating $alpha$ along a loop in $M$, then it is in fact a homomorphism. I can then consider $ker phi subset pi_1(M)$ and find a smooth covering space $p: hat{M} to M$ such that $p_*(pi_1(hat{M}))=ker phi$. This means that for all loops $hat{gamma} in pi_1(hat{M})$,
$$int_{hat{gamma}}p^* alpha = 0
$$
by construction. Thus, $p^* alpha$ is exact ($= df$) for some function $f$. We also observe that $(df)_y = 0 Leftrightarrow alpha_{p(y)}=0$. Thus, they say that $f$ and $p^* alpha$ share the same critical points which themselves share properties such as nondegeneracy and index. It seems a critical point for $alpha$ is simply where it vanishes.
My questions: How are the notions of critical points, nondegeneracy, and pseudogradients defined for a 1-form? Can this be done for $k$-forms?
This paper by Latour is referenced but I can't read French:
http://www.numdam.org/article/PMIHES_1994__80__135_0.pdf
differential-geometry morse-theory
asked Jan 29 at 23:11
inkievoydinkievoyd
638318
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add a comment |
$begingroup$
I'm reading Audin and Damian's Morse Theory and Floer Homology; they say there is an analogous way to define nondegenerate critical points for 1-forms as well as pseudo-gradient fields but don't discuss how. The goal is to define such a vector field for a 1-form $alpha$ and then lift it to a pseudo-gradient field for a function on a covering space.
Here's the context. Suppose I have a smooth closed 1-form $alpha$ on a manifold $M$. If I consider a map $phi:pi_1(M) to mathbb{R}$ which is simply integrating $alpha$ along a loop in $M$, then it is in fact a homomorphism. I can then consider $ker phi subset pi_1(M)$ and find a smooth covering space $p: hat{M} to M$ such that $p_*(pi_1(hat{M}))=ker phi$. This means that for all loops $hat{gamma} in pi_1(hat{M})$,
$$int_{hat{gamma}}p^* alpha = 0
$$
by construction. Thus, $p^* alpha$ is exact ($= df$) for some function $f$. We also observe that $(df)_y = 0 Leftrightarrow alpha_{p(y)}=0$. Thus, they say that $f$ and $p^* alpha$ share the same critical points which themselves share properties such as nondegeneracy and index. It seems a critical point for $alpha$ is simply where it vanishes.
My questions: How are the notions of critical points, nondegeneracy, and pseudogradients defined for a 1-form? Can this be done for $k$-forms?
This paper by Latour is referenced but I can't read French:
http://www.numdam.org/article/PMIHES_1994__80__135_0.pdf
differential-geometry morse-theory
asked Jan 29 at 23:11
inkievoydinkievoyd
638318
$endgroup$
add a comment |
$begingroup$
I'm reading Audin and Damian's Morse Theory and Floer Homology; they say there is an analogous way to define nondegenerate critical points for 1-forms as well as pseudo-gradient fields but don't discuss how. The goal is to define such a vector field for a 1-form $alpha$ and then lift it to a pseudo-gradient field for a function on a covering space.
Here's the context. Suppose I have a smooth closed 1-form $alpha$ on a manifold $M$. If I consider a map $phi:pi_1(M) to mathbb{R}$ which is simply integrating $alpha$ along a loop in $M$, then it is in fact a homomorphism. I can then consider $ker phi subset pi_1(M)$ and find a smooth covering space $p: hat{M} to M$ such that $p_*(pi_1(hat{M}))=ker phi$. This means that for all loops $hat{gamma} in pi_1(hat{M})$,
$$int_{hat{gamma}}p^* alpha = 0
$$
by construction. Thus, $p^* alpha$ is exact ($= df$) for some function $f$. We also observe that $(df)_y = 0 Leftrightarrow alpha_{p(y)}=0$. Thus, they say that $f$ and $p^* alpha$ share the same critical points which themselves share properties such as nondegeneracy and index. It seems a critical point for $alpha$ is simply where it vanishes.
My questions: How are the notions of critical points, nondegeneracy, and pseudogradients defined for a 1-form? Can this be done for $k$-forms?
This paper by Latour is referenced but I can't read French:
http://www.numdam.org/article/PMIHES_1994__80__135_0.pdf
differential-geometry morse-theory
asked Jan 29 at 23:11
inkievoydinkievoyd
638318
$endgroup$
I'm reading Audin and Damian's Morse Theory and Floer Homology; they say there is an analogous way to define nondegenerate critical points for 1-forms as well as pseudo-gradient fields but don't discuss how. The goal is to define such a vector field for a 1-form $alpha$ and then lift it to a pseudo-gradient field for a function on a covering space.
Here's the context. Suppose I have a smooth closed 1-form $alpha$ on a manifold $M$. If I consider a map $phi:pi_1(M) to mathbb{R}$ which is simply integrating $alpha$ along a loop in $M$, then it is in fact a homomorphism. I can then consider $ker phi subset pi_1(M)$ and find a smooth covering space $p: hat{M} to M$ such that $p_*(pi_1(hat{M}))=ker phi$. This means that for all loops $hat{gamma} in pi_1(hat{M})$,
$$int_{hat{gamma}}p^* alpha = 0
$$
by construction. Thus, $p^* alpha$ is exact ($= df$) for some function $f$. We also observe that $(df)_y = 0 Leftrightarrow alpha_{p(y)}=0$. Thus, they say that $f$ and $p^* alpha$ share the same critical points which themselves share properties such as nondegeneracy and index. It seems a critical point for $alpha$ is simply where it vanishes.
My questions: How are the notions of critical points, nondegeneracy, and pseudogradients defined for a 1-form? Can this be done for $k$-forms?
This paper by Latour is referenced but I can't read French:
http://www.numdam.org/article/PMIHES_1994__80__135_0.pdf
differential-geometry morse-theory
differential-geometry morse-theory
asked Jan 29 at 23:11
inkievoydinkievoyd
638318
asked Jan 29 at 23:11
inkievoydinkievoyd
638318
asked Jan 29 at 23:11
inkievoydinkievoyd
638318
asked Jan 29 at 23:11
inkievoydinkievoyd
638318
asked Jan 29 at 23:11
inkievoydinkievoyd
638318
638318
add a comment |
add a comment |
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