Questions tagged [pl-manifolds]

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14 votes
1 answer
483 views

Can the product of an exotic torus and a circle be the standard torus?

As discussed in this question from last week, if $M$ is a closed manifold such that $M\times S^1$ is homeomorphic to the torus $T^{n+1}$, then $M$ is homeomorphic to $T^n$. Is the corresponding ...
6 votes
1 answer
349 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 ...
1 vote
0 answers
47 views

When is a PL function linear?

We have a simplicial decomposition of a $n$-dimensional disk $D_n$ an a PL function $\mu$ on the decomposition. For each set of vertices $v_1,\dots,v_n,v_{n+1}$, so that both $v_1,\dots,v_{n-1},v_n$ ...
7 votes
0 answers
352 views

Understanding that a simplicial complex is a PL manifold via links

Suppose $X$ it a simplicial complex homeomorphic to a topological $n$-manifold. Suppose we know that the link of each $k$-simplex $\Delta^k$ is homeomorphic (as a topological space) to the sphere $S^{...
4 votes
0 answers
119 views

Is finding boundary-reducing discs for PL 3-manifolds with boundary pattern computationally efficient?

I am working with manifolds with boundary pattern, as defined by Matveev in his book Algorithmic Topology and Classification of 3-Manifolds. A manifold with boundary pattern is a pair $(M, P)$ where $...
10 votes
0 answers
454 views

Exotic analytic triangulations of $S^5$?

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/~...
7 votes
1 answer
215 views

Singularities of PL embedding of surface in a contractible 4-manifold

I am trying to understand the article "A solution to a conjecture of Zeeman" by Akbulut, but I am not an expert in PL-geometry. As far as I understand, two statements should be true, but I ...
11 votes
0 answers
227 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 ...
16 votes
1 answer
687 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
113 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
233 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
613 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
193 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 ...
3 votes
2 answers
824 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
334 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{...