Revisions to Is there a discrete Cerf theory?

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edited Jun 22, 2022 at 7:16

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Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condition that critical points be non-degenerate. The theory has proven quite useful in that it has allowed Morse theoretic arguments to be used in discrete settings, and it forms a part of computational topology.
My attention was recently drawn to the technique as a result of reading a preprint of Conant, Schneiderman, and Teichner a preprint of Conant, Schneiderman, and Teichner. Therefore, this question might well be hopelessly naïve (it's also possible that it's open).

Is there a discrete version of Cerf Theory? Are there at least partial results in this direction? Conversely, is it known that no such theory can exist?

My motivation is that I would imagine that a discrete proof of Kirby's Theorem, among other results which use Cerf Theory, might prove quite valuable in quantum topology (I'd love such a result at my fingertips!). I know that this doesn't (yet?) exist, or I would surely have heard about it. Additionally, a topological "machine" can only be used by a computer if it requires only finite information.

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condition that critical points be non-degenerate. The theory has proven quite useful in that it has allowed Morse theoretic arguments to be used in discrete settings, and it forms a part of computational topology.
My attention was recently drawn to the technique as a result of reading a preprint of Conant, Schneiderman, and Teichner. Therefore, this question might well be hopelessly naïve (it's also possible that it's open).

Is there a discrete version of Cerf Theory? Are there at least partial results in this direction? Conversely, is it known that no such theory can exist?

My motivation is that I would imagine that a discrete proof of Kirby's Theorem, among other results which use Cerf Theory, might prove quite valuable in quantum topology (I'd love such a result at my fingertips!). I know that this doesn't (yet?) exist, or I would surely have heard about it. Additionally, a topological "machine" can only be used by a computer if it requires only finite information.

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condition that critical points be non-degenerate. The theory has proven quite useful in that it has allowed Morse theoretic arguments to be used in discrete settings, and it forms a part of computational topology.
My attention was recently drawn to the technique as a result of reading a preprint of Conant, Schneiderman, and Teichner. Therefore, this question might well be hopelessly naïve (it's also possible that it's open).

Is there a discrete version of Cerf Theory? Are there at least partial results in this direction? Conversely, is it known that no such theory can exist?

My motivation is that I would imagine that a discrete proof of Kirby's Theorem, among other results which use Cerf Theory, might prove quite valuable in quantum topology (I'd love such a result at my fingertips!). I know that this doesn't (yet?) exist, or I would surely have heard about it. Additionally, a topological "machine" can only be used by a computer if it requires only finite information.

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edited Jun 15, 2020 at 7:27

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Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condition that critical points be non-degenerate. The theory has proven quite useful in that it has allowed Morse theoretic arguments to be used in discrete settings, and it forms a part of computational topology.
My attention was recently drawn to the technique as a result of reading a preprint of Conant, Schneiderman, and Teichner. Therefore, this question might well be hopelessly naïve (it's also possible that it's open).

Is there a discrete version of Cerf Theory? Are there at least partial results in this direction? Conversely, is it known that no such theory can exist?

My motivation is that I would imagine that a discrete proof of Kirby's Theorem, among other results which use Cerf Theory, might prove quite valuable in quantum topology (I'd love such a result at my fingertips!). I know that this doesn't (yet?) exist, or I would surely have heard about it. Additionally, a topological "machine" can only be used by a computer if it requires only finite information.

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condition that critical points be non-degenerate. The theory has proven quite useful in that it has allowed Morse theoretic arguments to be used in discrete settings, and it forms a part of computational topology.
My attention was recently drawn to the technique as a result of reading a preprint of Conant, Schneiderman, and Teichner. Therefore, this question might well be hopelessly naïve (it's also possible that it's open).

Is there a discrete version of Cerf Theory? Are there at least partial results in this direction? Conversely, is it known that no such theory can exist?

My motivation is that I would imagine that a discrete proof of Kirby's Theorem, among other results which use Cerf Theory, might prove quite valuable in quantum topology (I'd love such a result at my fingertips!). I know that this doesn't (yet?) exist, or I would surely have heard about it. Additionally, a topological "machine" can only be used by a computer if it requires only finite information.

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condition that critical points be non-degenerate. The theory has proven quite useful in that it has allowed Morse theoretic arguments to be used in discrete settings, and it forms a part of computational topology.
My attention was recently drawn to the technique as a result of reading a preprint of Conant, Schneiderman, and Teichner. Therefore, this question might well be hopelessly naïve (it's also possible that it's open).

Is there a discrete version of Cerf Theory? Are there at least partial results in this direction? Conversely, is it known that no such theory can exist?

My motivation is that I would imagine that a discrete proof of Kirby's Theorem, among other results which use Cerf Theory, might prove quite valuable in quantum topology (I'd love such a result at my fingertips!). I know that this doesn't (yet?) exist, or I would surely have heard about it. Additionally, a topological "machine" can only be used by a computer if it requires only finite information.