# Questions tagged [pl-manifolds]

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9
questions

**10**

votes

**0**answers

128 views

### Torus trick without surgery theory

It follows from surgery theory that in dimension $\geq 5$ every closed PL manifold homotopy equivalent to a torus has a finite cover which is PL homeomorphic to a torus. This is an important ...

**13**

votes

**1**answer

359 views

### Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent?

Summary
Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the ...

**3**

votes

**0**answers

86 views

### Reference request concerning PL tangent Stiefel-Whitney classes

I am hoping for a reference for the fact that PL manifolds have tangent Stiefel-Whitney classes.
I understand this as follows: they have tangent microbundles, which in turn lead to spherical ...

**2**

votes

**0**answers

191 views

### Are there analogs of smooth partitions of unity and good open covers for PL-manifolds?

Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds.
Are there analogs of these tools for piecewise linear manifolds?
A PL ...

**14**

votes

**1**answer

488 views

### What is the status of the PL-pseudoisotopy stability theorem?

Suppose that $M$ is a compact PL-manifold (possibly with boundary) and let $C^{PL}(M)$ denote the (simplicial) group of PL isomorphisms of $M \times I$ relative to $M \times \{0\} \cup \partial M \...

**8**

votes

**0**answers

175 views

### PL surface projections - is there a theory of folds and cusps?

For smooth surfaces, the generic singularities of a map of one surface to another are folds and cusps (Whitney). It is a standard result in singularity theory that the generic isotopy of such a map is ...

**6**

votes

**1**answer

303 views

### What is known about the distribution of average edge-degrees for 3-manifold triangulations (with the number of 3-simplices less than a fixed constant)?

This is my first question on mathoverflow! It relates to a project I'm undertaking with a student.
Work by Tamura (extending results by Luo and Stong) shows that for any closed 3-manifold $M$ and any ...

**3**

votes

**2**answers

668 views

### uniqueness of regular/tubular neighborhood with equivariant boundary

Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is ...

**3**

votes

**1**answer

327 views

### When is the neighbourhood of a set a ball?

In euclidean n-space, it's easy to show that given a set $S$ of radius $< r$, the $a$-neighbourhood of $S$ is a ball, for any $a \geq 2r$.
Proof: Let $S$ be contained in $B_r(y)$, $y \in \mathbb{...