Revisions to Exotic analytic triangulations of $S^5$?

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edited May 14, 2021 at 14:46

aglearner

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I would like to understand a bit better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and $M$ a smooth manifold. We say that a map $f \colon K \to M$ is piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to imposemodify the condition insteaddefinition. First, ask that $f$ restricts analytically to each simplex (stronger condition). But onlySecond ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is aNow, the question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

I would like to understand a bit better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and $M$ a smooth manifold. We say that a map $f \colon K \to M$ is piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to impose the condition instead that $f$ restricts analytically to each simplex (stronger condition). But only ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is a question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

I would like to understand better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and $M$ a smooth manifold. We say that a map $f \colon K \to M$ is piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

I would like to modify the definition. First, ask that $f$ restricts analytically to each simplex (stronger condition). Second ask that the restriction has injective differential in the interior of each simplex (weaker condition). Now, the question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

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edited May 11, 2021 at 22:01

aglearner

14.3k
8
40
99

I would like to understand a bit better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and M$M$ a smooth manifold. We say that a map $f \colon K \to M$ is piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to impose the condition instead that $f$ restricts analytically to each simplex (stronger condition). But only ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is a question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

I would like to understand a bit better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and M a smooth manifold. We say that a map $f \colon K \to M$ is piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to impose the condition instead that $f$ restricts analytically to each simplex (stronger condition). But only ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is a question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

I would like to understand a bit better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and $M$ a smooth manifold. We say that a map $f \colon K \to M$ is piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to impose the condition instead that $f$ restricts analytically to each simplex (stronger condition). But only ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is a question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

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occurred May 8, 2021 at 15:22

Poincare 3-sphere —> Poincaré homology sphere

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edited May 6, 2021 at 23:40

aglearner

14.3k
8
40
99

I would like to understand a bit better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and M a smooth manifold. We say that a map $f \colon K \to M$ is called piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to impose the condition instead that $f$ restricts analytically to each simplex (stronger condition). But only ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is a question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

I would like to understand a bit better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and M a smooth manifold. We say that a map $f \colon K \to M$ is called piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to impose the condition instead that $f$ restricts analytically to each simplex (stronger condition). But only ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is a question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

I would like to understand a bit better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and M a smooth manifold. We say that a map $f \colon K \to M$ is piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to impose the condition instead that $f$ restricts analytically to each simplex (stronger condition). But only ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is a question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)