Constructing a homotopy of nonzero holomorphic functions using local homotopies
$begingroup$
I'll denote by $mathbb{C}^*$ the punctured complex plane $mathbb{C} setminus {0}$. Say that I've got some open cover ${V_j}_{j in J}$ of the closed unit interval $[0,1]$, and continuous functions $w_j: V_j times mathbb{C}^* to mathbb{C}^*$ such that, for any $t in V_j$,
(a) $w_j(t, cdot)$ is a holomorphic function $mathbb{C}^* to mathbb{C}^*$ and
(b) $w_j(t, cdot)$ has a holomorphic antiderivative, say $F: mathbb{C}^* to mathbb{C}$.
I want to find a function $u: [0,1] times mathbb{C}^* to mathbb{C}^*$ that such that, for fixed $t in [0,1]$, $u(t, cdot)$ is still holomorphic and still has a holomorphic antiderivative.
My first thought was to use partitions of unity with respect to this cover, say ${p_j}_{j in J}$ and to set $u(t, cdot) = sum_{j} p_j(t)w_j(t, cdot)$. Of course, such a $u$ does still have an antiderivative for each $t$ but now we can't be sure whether the image of $u$ still lies in $mathbb{C}^*$.
I get that this is a bit vague, but I'd appreciate any good advice and/or ideas...can someone think of another way of constructing $u$. Or perhaps I can pursue the partitions of unity idea, provided $w_j$ and/or the open cover have some additional features, and if so, what kinds of features would enable this? I'd appreciate any ideas.
complex-analysis algebraic-geometry algebraic-topology complex-geometry riemann-surfaces
asked Dec 20 '18 at 5:38
ActonActon
29619
$endgroup$
add a comment |
$begingroup$
I'll denote by $mathbb{C}^*$ the punctured complex plane $mathbb{C} setminus {0}$. Say that I've got some open cover ${V_j}_{j in J}$ of the closed unit interval $[0,1]$, and continuous functions $w_j: V_j times mathbb{C}^* to mathbb{C}^*$ such that, for any $t in V_j$,
(a) $w_j(t, cdot)$ is a holomorphic function $mathbb{C}^* to mathbb{C}^*$ and
(b) $w_j(t, cdot)$ has a holomorphic antiderivative, say $F: mathbb{C}^* to mathbb{C}$.
I want to find a function $u: [0,1] times mathbb{C}^* to mathbb{C}^*$ that such that, for fixed $t in [0,1]$, $u(t, cdot)$ is still holomorphic and still has a holomorphic antiderivative.
My first thought was to use partitions of unity with respect to this cover, say ${p_j}_{j in J}$ and to set $u(t, cdot) = sum_{j} p_j(t)w_j(t, cdot)$. Of course, such a $u$ does still have an antiderivative for each $t$ but now we can't be sure whether the image of $u$ still lies in $mathbb{C}^*$.
I get that this is a bit vague, but I'd appreciate any good advice and/or ideas...can someone think of another way of constructing $u$. Or perhaps I can pursue the partitions of unity idea, provided $w_j$ and/or the open cover have some additional features, and if so, what kinds of features would enable this? I'd appreciate any ideas.
complex-analysis algebraic-geometry algebraic-topology complex-geometry riemann-surfaces
asked Dec 20 '18 at 5:38
ActonActon
29619
$endgroup$
$begingroup$
You don't seem to require any compatibility relations between this $u$ and the previous data, so it's unclear to me what kind of $u$ do you want to construct. Could you give some additional precisions ?
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:57
$begingroup$
P.S I think your previous idea should work with some additional modifications, for example you can assume that $J$ is countable and fix an arbitrary ordering on it. Now it should be possible to inductively build the $p_j$ so that the function is still $Bbb C^*$ valued.
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:59
$begingroup$
Thanks for your remarks! Is it actually possible to build the partition of unity inductively so that the function is still nonzero? I'm not quite sure if I understand what you mean!
$endgroup$
– Acton
Dec 21 '18 at 2:00
add a comment |
$begingroup$
I'll denote by $mathbb{C}^*$ the punctured complex plane $mathbb{C} setminus {0}$. Say that I've got some open cover ${V_j}_{j in J}$ of the closed unit interval $[0,1]$, and continuous functions $w_j: V_j times mathbb{C}^* to mathbb{C}^*$ such that, for any $t in V_j$,
(a) $w_j(t, cdot)$ is a holomorphic function $mathbb{C}^* to mathbb{C}^*$ and
(b) $w_j(t, cdot)$ has a holomorphic antiderivative, say $F: mathbb{C}^* to mathbb{C}$.
I want to find a function $u: [0,1] times mathbb{C}^* to mathbb{C}^*$ that such that, for fixed $t in [0,1]$, $u(t, cdot)$ is still holomorphic and still has a holomorphic antiderivative.
My first thought was to use partitions of unity with respect to this cover, say ${p_j}_{j in J}$ and to set $u(t, cdot) = sum_{j} p_j(t)w_j(t, cdot)$. Of course, such a $u$ does still have an antiderivative for each $t$ but now we can't be sure whether the image of $u$ still lies in $mathbb{C}^*$.
I get that this is a bit vague, but I'd appreciate any good advice and/or ideas...can someone think of another way of constructing $u$. Or perhaps I can pursue the partitions of unity idea, provided $w_j$ and/or the open cover have some additional features, and if so, what kinds of features would enable this? I'd appreciate any ideas.
complex-analysis algebraic-geometry algebraic-topology complex-geometry riemann-surfaces
asked Dec 20 '18 at 5:38
ActonActon
29619
$endgroup$
I'll denote by $mathbb{C}^*$ the punctured complex plane $mathbb{C} setminus {0}$. Say that I've got some open cover ${V_j}_{j in J}$ of the closed unit interval $[0,1]$, and continuous functions $w_j: V_j times mathbb{C}^* to mathbb{C}^*$ such that, for any $t in V_j$,
(a) $w_j(t, cdot)$ is a holomorphic function $mathbb{C}^* to mathbb{C}^*$ and
(b) $w_j(t, cdot)$ has a holomorphic antiderivative, say $F: mathbb{C}^* to mathbb{C}$.
I want to find a function $u: [0,1] times mathbb{C}^* to mathbb{C}^*$ that such that, for fixed $t in [0,1]$, $u(t, cdot)$ is still holomorphic and still has a holomorphic antiderivative.
My first thought was to use partitions of unity with respect to this cover, say ${p_j}_{j in J}$ and to set $u(t, cdot) = sum_{j} p_j(t)w_j(t, cdot)$. Of course, such a $u$ does still have an antiderivative for each $t$ but now we can't be sure whether the image of $u$ still lies in $mathbb{C}^*$.
I get that this is a bit vague, but I'd appreciate any good advice and/or ideas...can someone think of another way of constructing $u$. Or perhaps I can pursue the partitions of unity idea, provided $w_j$ and/or the open cover have some additional features, and if so, what kinds of features would enable this? I'd appreciate any ideas.
complex-analysis algebraic-geometry algebraic-topology complex-geometry riemann-surfaces
complex-analysis algebraic-geometry algebraic-topology complex-geometry riemann-surfaces
asked Dec 20 '18 at 5:38
ActonActon
29619
asked Dec 20 '18 at 5:38
ActonActon
29619
asked Dec 20 '18 at 5:38
ActonActon
29619
asked Dec 20 '18 at 5:38
ActonActon
29619
asked Dec 20 '18 at 5:38
ActonActon
29619
29619
$begingroup$
You don't seem to require any compatibility relations between this $u$ and the previous data, so it's unclear to me what kind of $u$ do you want to construct. Could you give some additional precisions ?
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:57
$begingroup$
P.S I think your previous idea should work with some additional modifications, for example you can assume that $J$ is countable and fix an arbitrary ordering on it. Now it should be possible to inductively build the $p_j$ so that the function is still $Bbb C^*$ valued.
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:59
$begingroup$
Thanks for your remarks! Is it actually possible to build the partition of unity inductively so that the function is still nonzero? I'm not quite sure if I understand what you mean!
$endgroup$
– Acton
Dec 21 '18 at 2:00
add a comment |
$begingroup$
You don't seem to require any compatibility relations between this $u$ and the previous data, so it's unclear to me what kind of $u$ do you want to construct. Could you give some additional precisions ?
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:57
$begingroup$
P.S I think your previous idea should work with some additional modifications, for example you can assume that $J$ is countable and fix an arbitrary ordering on it. Now it should be possible to inductively build the $p_j$ so that the function is still $Bbb C^*$ valued.
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:59
$begingroup$
Thanks for your remarks! Is it actually possible to build the partition of unity inductively so that the function is still nonzero? I'm not quite sure if I understand what you mean!
$endgroup$
– Acton
Dec 21 '18 at 2:00
$begingroup$
You don't seem to require any compatibility relations between this $u$ and the previous data, so it's unclear to me what kind of $u$ do you want to construct. Could you give some additional precisions ?
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:57
$begingroup$
You don't seem to require any compatibility relations between this $u$ and the previous data, so it's unclear to me what kind of $u$ do you want to construct. Could you give some additional precisions ?
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:57
$begingroup$
P.S I think your previous idea should work with some additional modifications, for example you can assume that $J$ is countable and fix an arbitrary ordering on it. Now it should be possible to inductively build the $p_j$ so that the function is still $Bbb C^*$ valued.
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:59
$begingroup$
P.S I think your previous idea should work with some additional modifications, for example you can assume that $J$ is countable and fix an arbitrary ordering on it. Now it should be possible to inductively build the $p_j$ so that the function is still $Bbb C^*$ valued.
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:59
$begingroup$
Thanks for your remarks! Is it actually possible to build the partition of unity inductively so that the function is still nonzero? I'm not quite sure if I understand what you mean!
$endgroup$
– Acton
Dec 21 '18 at 2:00
$begingroup$
Thanks for your remarks! Is it actually possible to build the partition of unity inductively so that the function is still nonzero? I'm not quite sure if I understand what you mean!
$endgroup$
– Acton
Dec 21 '18 at 2:00
add a comment |
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$begingroup$
You don't seem to require any compatibility relations between this $u$ and the previous data, so it's unclear to me what kind of $u$ do you want to construct. Could you give some additional precisions ?
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:57
$begingroup$
P.S I think your previous idea should work with some additional modifications, for example you can assume that $J$ is countable and fix an arbitrary ordering on it. Now it should be possible to inductively build the $p_j$ so that the function is still $Bbb C^*$ valued.
$endgroup$
– Nicolas Hemelsoet
Dec 20 '18 at 11:59
$begingroup$
Thanks for your remarks! Is it actually possible to build the partition of unity inductively so that the function is still nonzero? I'm not quite sure if I understand what you mean!
$endgroup$
– Acton
Dec 21 '18 at 2:00